Supply and Demand Management in Ride-Sourcing Markets 0443189374, 9780443189371

Table of contents :
Front Cover
Supply and Demand Management in Ride-Sourcing Markets
Supply and Demand Management in Ride-Sourcing Markets
Copyright
Contents
Contributors
About the authors
Preface
1 - Introduction of ride-sourcing markets
1.1 Background
1.2 Theoretical developments
1.2.1 Stationary equilibrium state
1.2.2 Monopoly optimum, social optimum, and Pareto-efficient solutions
1.2.3 Regulations
1.2.4 Ride-pooling services
1.2.5 Congestion externalities
1.2.6 Platform competition and platform integration
1.2.7 Ride sourcing and public transit
1.2.8 On-demand matching and its key decision variables
1.3 Outline of this book
References
2 - Fundamentals of ride-sourcing market equilibrium analyses
2.1 Introduction
2.1.1 Passenger demand
2.1.2 Driver supply
2.2 Matching frictions (inductive approaches)
2.2.1 Perfect matching function
2.2.2 Production functions
2.3 Matching frictions (deductive approaches)
2.3.1 Queuing models
2.3.2 First-come-first-served (FCFS)
2.3.3 Batch-matching process
2.4 Market measures
2.4.1 Monopoly optimum
2.4.1.1 Production-function-based model
2.4.1.2 Queuing model
2.4.1.3 FCFS-based model
2.4.2 Social optimum
2.4.2.1 Production-function-based model
2.4.2.2 Queuing model
2.4.2.3 FCFS-based model
2.4.3 Pareto-efficient solutions
2.4.3.1 Production function-based model
2.4.3.2 Queuing model
2.4.3.3 FCFS-based model
2.5 Discussion
Glossary of notation
References
3 - Calibration and validation of matching functions for ride-sourcing markets
3.1 Introduction
3.2 Matching functions and market metrics
3.2.1 Base model
3.2.2 Matching functions
3.2.2.1 Perfect matching
3.2.2.2 Cobb–Douglas production function
3.2.2.3 M/M/1 queuing model
3.2.2.4 M/M/1/k queuing model
3.2.2.5 M/M/N queuing model
3.2.2.6 First-come-first-served (FCFS) model
3.2.2.7 Batch-matching model
3.2.2.8 Summary of matching functions
3.2.3 Key market metrics
3.3 Experimental settings
3.3.1 Simulator
3.3.2 Experiment
3.4 Analysis of experimental results
3.4.1 Market segmentation
3.4.2 Best-fit models for estimation of matching rate
3.4.3 Best-fit models for the estimation of matching time
3.4.4 Best-fit models for the estimation of passenger pick-up time
3.4.5 Best-fit models for the estimation of passengers' total waiting time
3.5 Summary
3.6 Discussion and conclusion
Appendix 3.A
Glossary of notation
References
4 - Government regulations for ride-sourcing services
4.1 Properties of the pareto-efficient solutions
4.2 An alternative method to obtain and analyse pareto-efficient solutions
4.3 Regulations
4.3.1 Price-cap regulation
4.3.2 Fleet size regulation
4.3.3 Wage regulation
4.3.4 Income regulation
4.3.5 Commission regulation
4.3.6 Commission ratio regulation
4.3.7 Minimum utilisation rate regulation
4.3.8 Demand regulation
4.3.9 Summary
4.3.10 Numerical illustrations
4.4 Discussion and conclusion
Glossary of notation
References
5 - Equilibrium analysis for ride-pooling services
5.1 Introduction
5.2 Pool-matching schemes
5.2.1 En-route pool-matching scheme
5.2.1.1 General model
5.2.1.2 Probabilistic model
5.2.2 Pre-assigned pool-matching with meeting points
5.2.3 Comparisons
5.3 Equilibrium analyses
5.3.1 Supply and demand function
5.3.2 Market equilibrium
5.3.3 Comparative static effects of regulatory variables
5.4 Market measures
5.4.1 Monopoly optimum
5.4.2 Social optimum
5.4.3 Pareto-efficient solutions
5.5 Numerical illustrations
5.5.1 Experimental settings
5.5.2 Detour-unconstrained scenario
5.5.3 Detour-constrained scenario
5.6 Conclusion
Glossary of notation
References
6 - Ride-pooling services and traffic congestion
6.1 Introduction
6.2 Equilibrium analyses
6.2.1 Demand function
6.2.2 Speed function
6.2.3 Supply function
6.2.4 Equilibrium solution
6.3 Market measures
6.3.1 Monopoly optimum
6.3.2 Social optimum
6.3.3 Pareto-efficient solutions
6.4 Conclusion
Glossary of notation
References
7 - Equilibrium analysis for ride-pooling services in the presence of traffic congestion
7.1 Introduction
7.2 Equilibrium analyses
7.2.1 Vehicle conservation
7.2.2 Demand function
7.2.3 Supply function
7.2.4 Equilibrium solution
7.3 Market measures
7.3.1 Monopoly optimum (MO)
7.3.2 Social optimum (SO)
7.3.3 Pareto-efficient solutions
7.4 Numerical studies
7.4.1 Equilibrium outcomes
7.4.2 Optimal operating strategies (non-pooling market)
7.4.3 Optimal operating strategies (ride-pooling market)
7.4.4 Effects of matching window
7.5 Conclusion and remarks
Glossary of notation
References
8 - Revisiting government regulations for ride-sourcing services under traffic congestion
8.1 Introduction
8.2 Theoretical analyses
8.2.1 Monopoly optimum
8.2.2 Social optimum
8.2.3 Pareto-efficient solutions
8.3 Numerical studies
8.3.1 Settings
8.3.2 Market with drivers with heterogeneous reservation rates and no traffic congestion
8.3.3 Market with drivers with heterogeneous reservation rates and traffic congestion
8.3.4 Effects of driver rationing
8.3.5 Summary and discussion
8.4 Conclusion
Glossary of notation
References
9 - Third-party platform integration in ride-sourcing markets
9.1 Background
9.2 Market equilibrium and optimal strategies
9.2.1 Market without platform integration
9.2.2 Market with platform integration
9.3 Evaluation of the performance of platform integration
9.3.1 Effect of vehicle fleet size at the Nash equilibrium/social optimum
9.3.2 Effect of platform integration at the Nash equilibrium
9.3.3 Effect of platform integration at the Social optimum
9.4 Numerical studies
9.4.1 Effect of market fragmentation
9.4.2 Effect of vehicle fleet size
9.4.3 Effect of commission fee
9.5 Conclusion
Appendix 9.A. Proof of Lemma 9-1
Appendix 9.B. Proof of theorem 9-1
Appendix 9.C. Proof of Lemma 9-2
Appendix 9.D. Proof of theorem 9-2
Appendix 9.E. Proof of Lemma 9-3
Appendix 9.F. Proof of Lemma 9-4
Appendix 9.G. Proof of theorem 9-3
Appendix 9.H. Proof of Lemma 9-5
Appendix 9.I. Proof of theorem 9-4
Appendix 9.J. General matching function
Glossary of notation
References
10 - Ride-sourcing services and public transit
10.1 Background
10.2 Model description
10.3 Optimal strategy design
10.3.1 Monopoly optimum
10.3.2 Social optimum
10.3.3 Second-best solution
10.4 Numerical case study
10.4.1 Analysis of equilibrium states
10.4.2 Analysis of profit- and/or social welfare-maximising strategies
10.5 Conclusion
Glossary of notation
References
Appendix 10.A
11 - Optimization of matching-time interval and matching radius in ride-sourcing markets
11.1 Research problem
11.2 Modelling and optimising the matching process
11.2.1 Online matching process
11.2.2 Matched passenger–driver pairs
11.2.3 Expected pick-up distance
11.2.4 System performance measure
11.2.5 General model properties
11.2.5.1 Effect of matching radius
11.2.5.2 Effect of matching-time interval
11.3 Model properties in imbalanced scenarios
11.3.1 Effects of matching-time interval
11.3.2 Properties of optimal matching-time interval
11.3.3 Further discussion
11.4 Numerical studies
11.4.1 Balanced scenario
11.4.2 Imbalanced scenarios
11.4.3 Model performance in a dynamic simulation environment
11.5 Conclusion
Glossary of notation
References
12 - Labour supply analysis of ride-sourcing services
12.1 Background
12.1.1 Motivation
12.1.2 Research questions
12.1.3 Methodology
12.1.4 Results
12.1.5 Main contributions
12.2 Related literature
12.3 Labour supply model
12.3.1 Optimal decisions on hours worked based on income targets
12.3.2 Importance of the extensive margin in the labour supply model
12.4 Modelling endogeneity of income rates and self-selected participation in the labour force
12.4.1 Methodological implications of self-selection and endogeneity
12.4.1.1 Modelling self-selected participation in the labour force
12.4.1.2 Modelling the endogeneity of the hourly income rate
12.4.2 Model of labour supply elasticity on a ride-sourcing platform
12.5 Research design
12.5.1 Research context
12.5.2 Large-scale natural experiment
12.5.2.1 Income multiplier
12.5.2.2 Exogenous shocks in natural experiments
12.5.3 Data description
12.5.3.1 Variable description
12.5.4 Driver classification along the extensive and intensive margins
12.5.5 Empirical analysis
12.5.5.1 Basic model
12.5.5.2 Consideration of potential sample selection bias
12.5.5.3 Identification of the outcome equation
12.6 Results and discussion
12.6.1 Model estimation
12.6.1.1 Evidence of sample selection and endogeneity bias
12.6.1.2 Validity of IVs
12.6.2 Estimates of labour supply elasticity in the presence of driver heterogeneity
12.6.3 Labour supply in subgroups
12.7 Conclusion
Glossary of notation
References
13 - Some empirical laws of ride-pooling services
13.1 Introduction
13.2 Literature review
13.3 Optimisation framework and data descriptions
13.3.1 Definitions of the key measures
13.3.2 Optimisation algorithms
13.3.3 Random matching without an optimisation objective
13.3.4 Experimental settings and data description
13.4 Empirical laws
13.4.1 Law of passenger detour distance
13.4.2 Law of average vehicle routing distance
13.4.3 Law of pool-matching probability
13.4.3.1 Discussions on matching probability
13.4.4 Effect of matching radius
13.4.5 Discussion on empirical laws
13.5 Conclusions
Appendices
Appendix 13.A. Probabilistic density distribution of Δl¯ under objective P1
Appendix 13.B. Probabilistic density distribution of l¯d under objective P1
Appendix 13.C. Empirical law of Δl¯ under objective P1
Appendix 13.D. Empirical law of l¯d under objective P1
Appendix 13.E. Empirical fitting of p under objective P1
Appendix 13.F. Probabilistic density distribution of Δl¯ under objective P2
Appendix 13.G. Probabilistic density distribution of l¯d under objective P2
Appendix 13.H. Empirical law of Δl¯ under objective P2
Appendix 13.I. Empirical law of l¯d under objective P2
Appendix 13.J. Empirical fitting of p under objective P2
Appendix 13.K. Probabilistic density distribution of Δl¯ under objective P3
Appendix 13.L. Probabilistic density distribution of l¯d under objective P3
Appendix 13.M. Empirical law of Δl¯ under objective P3
Appendix 13.N. Empirical law of l¯d under objective P3
Appendix 13.O. Empirical fitting of p under objective P3
Appendix 13.P. Proof of ∂M/∂N﹥0 and ∂M/∂N≤1
Appendix 13.Q. Empirical laws for the downtown area of Manhattan
Glossary of notation
References
14 - Summary
Glossary of abbreviations
Index
A
B
C
D
E
F
G
H
I
K
L
M
N
O
P
Q
R
S
T
U
V
W
Back Cover

Citation preview

Supply and Demand Management in Ride-Sourcing Markets

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Supply and Demand Management in Ride-Sourcing Markets

Jintao Ke Assistant Professor, Department of Civil Engineering, University of Hong Kong, Hong Kong

Hai Yang Chair Professor, Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong

Hai Wang Associate Professor, School of Computing and Information Systems, Singapore Management University, Singapore

Yafeng Yin Professor at Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright Ó 2023 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-443-18937-1 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Joseph P. Hayton Acquisitions Editor: Kathryn Eryilmaz Editorial Project Manager: Rupinder K. Heron Production Project Manager: Swapna Srinivasan Cover Designer: Mark Rogers Typeset by TNQ Technologies

Contents Contributors About the authors Preface

1.

xiii xv xvii

Introduction of ride-sourcing markets Jintao Ke, Hai Yang, Hai Wang and Yafeng Yin 1.1 1.2

Background Theoretical developments 1.2.1 Stationary equilibrium state 1.2.2 Monopoly optimum, social optimum, and Pareto-efficient solutions 1.2.3 Regulations 1.2.4 Ride-pooling services 1.2.5 Congestion externalities 1.2.6 Platform competition and platform integration 1.2.7 Ride sourcing and public transit 1.2.8 On-demand matching and its key decision variables 1.3 Outline of this book References

2.

1 3 4 6 7 8 10 12 13 14 16 19

Fundamentals of ride-sourcing market equilibrium analyses Jintao Ke, Yafeng Yin, Hai Yang and Hai Wang 2.1

2.2

2.3

Introduction 2.1.1 Passenger demand 2.1.2 Driver supply Matching frictions (inductive approaches) 2.2.1 Perfect matching function 2.2.2 Production functions Matching frictions (deductive approaches) 2.3.1 Queuing models 2.3.2 First-come-first-served (FCFS) 2.3.3 Batch-matching process

25 25 26 27 27 28 31 31 33 36

v

vi Contents 2.4

Market measures 2.4.1 Monopoly optimum 2.4.2 Social optimum 2.4.3 Pareto-efficient solutions 2.5 Discussion Glossary of notation References

3.

39 40 43 47 50 52 53

Calibration and validation of matching functions for ride-sourcing markets Shuqing Wei, Siyuan Feng, Jintao Ke and Hai Yang 3.1 3.2

Introduction Matching functions and market metrics 3.2.1 Base model 3.2.2 Matching functions 3.2.3 Key market metrics 3.3 Experimental settings 3.3.1 Simulator 3.3.2 Experiment 3.4 Analysis of experimental results 3.4.1 Market segmentation 3.4.2 Best-fit models for estimation of matching rate 3.4.3 Best-fit models for the estimation of matching time 3.4.4 Best-fit models for the estimation of passenger pick-up time 3.4.5 Best-fit models for the estimation of passengers’ total waiting time 3.5 Summary 3.6 Discussion and conclusion Appendix 3.A Glossary of notation References

4.

55 57 58 59 64 65 65 67 68 68 69 72 74 75 77 77 78 82 82

Government regulations for ride-sourcing services Jintao Ke, Xinwei Li, Hai Yang and Yafeng Yin 4.1 4.2 4.3

Properties of the pareto-efficient solutions An alternative method to obtain and analyse pareto-efficient solutions Regulations 4.3.1 Price-cap regulation 4.3.2 Fleet size regulation 4.3.3 Wage regulation 4.3.4 Income regulation 4.3.5 Commission regulation 4.3.6 Commission ratio regulation

87 91 94 94 95 96 97 98 99

Contents vii

4.3.7 Minimum utilisation rate regulation 4.3.8 Demand regulation 4.3.9 Summary 4.3.10 Numerical illustrations 4.4 Discussion and conclusion Glossary of notation References

5.

100 101 101 101 103 104 106

Equilibrium analysis for ride-pooling services Jintao Ke, Hai Yang and Hai Wang 5.1 5.2

Introduction Pool-matching schemes 5.2.1 En-route pool-matching scheme 5.2.2 Pre-assigned pool-matching with meeting points 5.2.3 Comparisons 5.3 Equilibrium analyses 5.3.1 Supply and demand function 5.3.2 Market equilibrium 5.3.3 Comparative static effects of regulatory variables 5.4 Market measures 5.4.1 Monopoly optimum 5.4.2 Social optimum 5.4.3 Pareto-efficient solutions 5.5 Numerical illustrations 5.5.1 Experimental settings 5.5.2 Detour-unconstrained scenario 5.5.3 Detour-constrained scenario 5.6 Conclusion Glossary of notation References

6.

107 108 108 111 112 113 113 115 116 118 119 120 123 124 124 125 126 130 130 131

Ride-pooling services and traffic congestion Jintao Ke and Hai Yang 6.1 6.2

Introduction Equilibrium analyses 6.2.1 Demand function 6.2.2 Speed function 6.2.3 Supply function 6.2.4 Equilibrium solution 6.3 Market measures 6.3.1 Monopoly optimum 6.3.2 Social optimum 6.3.3 Pareto-efficient solutions 6.4 Conclusion Glossary of notation References

133 134 134 135 138 139 140 140 141 143 144 145 145

viii Contents

7.

Equilibrium analysis for ride-pooling services in the presence of traffic congestion Jintao Ke 7.1 7.2

Introduction Equilibrium analyses 7.2.1 Vehicle conservation 7.2.2 Demand function 7.2.3 Supply function 7.2.4 Equilibrium solution 7.3 Market measures 7.3.1 Monopoly optimum (MO) 7.3.2 Social optimum (SO) 7.3.3 Pareto-efficient solutions 7.4 Numerical studies 7.4.1 Equilibrium outcomes 7.4.2 Optimal operating strategies (non-pooling market) 7.4.3 Optimal operating strategies (ride-pooling market) 7.4.4 Effects of matching window 7.5 Conclusion and remarks Glossary of notation References

8.

147 149 149 151 151 152 154 154 155 157 158 159 160 161 162 163 164 164

Revisiting government regulations for ride-sourcing services under traffic congestion Jintao Ke, Xinwei Li, Hai Yang and Yafeng Yin 8.1 8.2

Introduction Theoretical analyses 8.2.1 Monopoly optimum 8.2.2 Social optimum 8.2.3 Pareto-efficient solutions 8.3 Numerical studies 8.3.1 Settings 8.3.2 Market with drivers with heterogeneous reservation rates and no traffic congestion 8.3.3 Market with drivers with heterogeneous reservation rates and traffic congestion 8.3.4 Effects of driver rationing 8.3.5 Summary and discussion 8.4 Conclusion Glossary of notation References

167 168 172 173 175 178 178 179 182 185 186 188 188 189

Contents

9.

ix

Third-party platform integration in ride-sourcing markets Yaqian Zhou, Jintao Ke, Hai Yang and Hai Wang 9.1 9.2

Background Market equilibrium and optimal strategies 9.2.1 Market without platform integration 9.2.2 Market with platform integration 9.3 Evaluation of the performance of platform integration 9.3.1 Effect of vehicle fleet size at the Nash equilibrium/social optimum 9.3.2 Effect of platform integration at the Nash equilibrium 9.3.3 Effect of platform integration at the Social optimum 9.4 Numerical studies 9.4.1 Effect of market fragmentation 9.4.2 Effect of vehicle fleet size 9.4.3 Effect of commission fee 9.5 Conclusion Appendix 9.A. Proof of Lemma 9-1 Appendix 9.B. Proof of theorem 9-1 Appendix 9.C. Proof of Lemma 9-2 Appendix 9.D. Proof of theorem 9-2 Appendix 9.E. Proof of Lemma 9-3 Appendix 9.F. Proof of Lemma 9-4 Appendix 9.G. Proof of theorem 9-3 Appendix 9.H. Proof of Lemma 9-5 Appendix 9.I. Proof of theorem 9-4 Appendix 9.J. General matching function Glossary of notation References

10.

191 194 195 198 207 207 208 209 210 210 214 217 219 220 221 222 223 224 227 229 231 233 233 236 237

Ride-sourcing services and public transit Jintao Ke, Zhu Zheng and Hai Yang 10.1 10.2 10.3

Background Model description Optimal strategy design 10.3.1 Monopoly optimum 10.3.2 Social optimum 10.3.3 Second-best solution 10.4 Numerical case study 10.4.1 Analysis of equilibrium states 10.4.2 Analysis of profit- and/or social welfare-maximising strategies 10.5 Conclusion Glossary of notation References Appendix 10.A

239 241 245 245 247 248 249 250 254 256 257 257 258

x Contents

11.

Optimization of matching-time interval and matching radius in ride-sourcing markets Jintao Ke and Hai Yang 11.1 11.2

Research problem Modelling and optimising the matching process 11.2.1 Online matching process 11.2.2 Matched passengeredriver pairs 11.2.3 Expected pick-up distance 11.2.4 System performance measure 11.2.5 General model properties 11.3 Model properties in imbalanced scenarios 11.3.1 Effects of matching-time interval 11.3.2 Properties of optimal matching-time interval 11.3.3 Further discussion 11.4 Numerical studies 11.4.1 Balanced scenario 11.4.2 Imbalanced scenarios 11.4.3 Model performance in a dynamic simulation environment 11.5 Conclusion Glossary of notation References

12.

261 262 263 264 265 266 267 271 271 273 274 275 275 277 278 280 282 282

Labour supply analysis of ride-sourcing services Sun Hao, Hai Wang and Zhixi Wan 12.1

Background 12.1.1 Motivation 12.1.2 Research questions 12.1.3 Methodology 12.1.4 Results 12.1.5 Main contributions 12.2 Related literature 12.3 Labour supply model 12.3.1 Optimal decisions on hours worked based on income targets 12.3.2 Importance of the extensive margin in the labour supply model 12.4 Modelling endogeneity of income rates and self-selected participation in the labour force 12.4.1 Methodological implications of self-selection and endogeneity 12.4.2 Model of labour supply elasticity on a ride-sourcing platform 12.5 Research design 12.5.1 Research context 12.5.2 Large-scale natural experiment

283 284 285 285 285 286 286 289 289 291 292 292 297 300 300 300

Contents

12.5.3 Data description 12.5.4 Driver classification along the extensive and intensive margins 12.5.5 Empirical analysis 12.6 Results and discussion 12.6.1 Model estimation 12.6.2 Estimates of labour supply elasticity in the presence of driver heterogeneity 12.6.3 Labour supply in subgroups 12.7 Conclusion Glossary of notation References

13.

xi 302 304 304 308 309 316 317 318 320 320

Some empirical laws of ride-pooling services Jintao Ke, Zhengfei Zheng and Hai Yang 13.1 13.2 13.3

Introduction Literature review Optimisation framework and data descriptions 13.3.1 Definitions of the key measures 13.3.2 Optimisation algorithms 13.3.3 Random matching without an optimisation objective 13.3.4 Experimental settings and data description 13.4 Empirical laws 13.4.1 Law of passenger detour distance 13.4.2 Law of average vehicle routing distance 13.4.3 Law of pool-matching probability 13.4.4 Effect of matching radius 13.4.5 Discussion on empirical laws 13.5 Conclusions Appendices Appendix 13.A. Probabilistic density distribution of Dl under objective P 1 Appendix 13.B. Probabilistic density distribution of l d under objective P 1 Appendix 13.C. Empirical law of Dl under objective P 1 Appendix 13.D. Empirical law of l d under objective P 1 Appendix 13.E. Empirical fitting of p under objective P 1 Appendix 13.F. Probabilistic density distribution of Dl under objective P 2 Appendix 13.G. Probabilistic density distribution of l d under objective P 2 Appendix 13.H. Empirical law of Dl under objective P 2 Appendix 13.I. Empirical law of l d under objective P 2 Appendix 13.J. Empirical fitting of p under objective P 2

323 326 328 329 330 333 333 336 336 338 341 345 349 349 350 350 350 350 350 351 351 352 353 353 354

xii Contents Appendix 13.K. Probabilistic density distribution of Dl under objective P 3 Appendix 13.L. Probabilistic density distribution of l d under objective P 3 Appendix 13.M. Empirical law of Dl under objective P 3 Appendix 13.N. Empirical law of l d under objective P 3 Appendix 13.O. Empirical fitting of p under objective P 3 Appendix 13.P. Proof of vM=vN > 0 and vM=vN  1 Appendix 13.Q. Empirical laws for the downtown area of Manhattan Glossary of notation References

14.

Summary

354 355 356 356 357 357 358 369 369 373

Jintao Ke, Hai Yang, Hai Wang and Yafeng Yin

Glossary of abbreviations Index

377 379

Contributors Siyuan Feng, Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China Sun Hao, Faculty of Business, The University of Hong Kong, Pokfulam, Hong Kong, China Jintao Ke, Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China Xinwei Li, School of Economics and Management, Beihang University, Beijing, China Zhixi Wan, Faculty of Business, The University of Hong Kong, Pokfulam, Hong Kong, China Hai Wang, School of Computing and Information Systems, Singapore Management University, Bras Basah, Singapore Shuqing Wei, Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China Hai Yang, Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China Yafeng Yin, Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI, United States Zhengfei Zheng, Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China Zhu Zheng, Department of Civil Engineering, Zhejiang University, Hangzhou, China Yaqian Zhou, School of Automation, Chongqing University, Chongqing, China

xiii

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About the authors Dr Jintao Ke is an Assistant Professor in the Department of Civil Engineering, the University of Hong Kong. He received his bachelor’s degree from Zhejiang University, China, and his PhD from the Hong Kong University of Science and Technology. His research interests include smart transportation, smart city, urban computing, shared mobility, machine learning in transportation, operational management for transportation studies, etc. He has published over 30 SCI/SSCI-indexed papers in top transportation journals, including Transportation Research Part AeE, IEEE Transactions on Intelligence Transportation System. He serves as an Advisory Board Member of Transportation Research Part C and Transportation Research Part E. Professor Hai Yang is a Chair Professor in the Department of Civil and Environmental Engineering, the Hong Kong University of Science and Technology. He received his bachelor’s degree from Wuhan University, China, and his PhD from Kyoto University, Japan. Professor Yang is internationally known as an active scholar in the field of transportation, with more than 300 papers published in SCI/SSCI-indexed journals. He has received a number of national and international awards, including the JSCE Outstanding Paper Award (1991); the Distinguished Overseas Young Investigator Award from the National Natural Science Foundation of China (2004); the National Natural Science Award bestowed by the State Council of China (2011); HKUST School of Engineering Research Excellence Awards (2012); 2020 Frank M. Masters Transportation Engineering Award, American Society of Civil Engineers (ASCE) and 2021 ASCE Francis C. Turner Award. Professor Yang served as the Editor-in-Chief of Transportation Research (TR) Part B: Methodological from 2013 to 2018, a prestigious journal in the field of transportation. Currently, Professor Yang serves on the Distinguished Editorial Board of TR Part B, Scientific Council of TR Part C: Emerging Technologies, and serves as an Advisory Editor of Transportation Science. Dr Hai Wang is an Associate Professor in the School of Computing and Information Systems at Singapore Management University and a visiting faculty at the Heinz College of Information Systems and Public Policy at Carnegie Mellon University. He received his bachelor’s degree from Tsinghua University and his PhD from MIT. His research focuses on the methodologies of analytics and optimisation, data-driven modelling, computational xv

xvi About the authors

algorithms and machine learning and relevant applications in smart cities, including innovative transportation, advanced logistics, modern e-commerce and intelligent healthcare. He serves as the Department Editor for Transportation Research Part E, Associate Editor for Transportation Science and Service Science, Special Issue Editor for Transportation Research Part B, Transportation Research Part C and Service Science and Editorial Board Member for Transportation Research Part C. Dr Wang was selected as Chan Wu and Yunying Rising Star Fellow in Transportation, received Lee Kong Chian Research Excellence Awards, was nominated for MIT’s Top Graduate Teaching Award and won the Excellent Teaching Award at SMU. During his PhD at MIT, he also served as the Co-President of the MIT Chinese Students and Scholars Association and Chair of MIT-China Innovation and Entrepreneurship Forum. Dr Yafeng Yin is a Professor of Civil and Environmental Engineering and Professor of Industrial and Operations Engineering at the University of Michigan, Ann Arbor. His research aims to analyse and enhance multimodal transportation systems towards efficiency, resilience and environmental sustainability. Currently, he focuses on developing innovative mobility solutions and services by leveraging vehicle connectivity and automation. Dr Yin has published nearly 150 refereed papers in leading academic journals. He was the Editor-in-Chief of Transportation Research Part C: Emerging Technologies between 2014 and 2020 and currently serves as Area Editor of Transportation Science and Associate Editor of Transportation Research Part B: Methodological, another two flagship journals in the transportation domain. Professor Yin received his PhD from the University of Tokyo, Japan, in 2002, and his master’s and bachelor’s degrees from Tsinghua University, Beijing, China, in 1996 and 1994, respectively.

Preface Current disruptive trends in transportation, such as driving automation, increased connectivity, vehicle electrification and shared mobility, are altering traditional thinking on transportation and changing our daily travel modes. For example, the prevalence of shared mobility services, which typify emerging on-demand ridesourcing services, has dramatically increased over the past decade. This has led to research in many interesting areas, including the modelling of passengers’ mode choices and drivers’ mode participation; investigations of ride-sourcing platforms’ optimal decisions on pricing, wages and matching; the design of effective government regulations and analyses of the social effects of ridesourcing services. Ride-sourcing markets are challenging to research as they are two-sided markets in which demand and supply interact in a complex manner. Therefore, there is a pressing need for mathematical models that can precisely characterise passengers’ and drivers’ ride-sourcing behaviours and perform efficient on-demand matching of passengers and drivers. Such models will assist ride-sourcing platforms to develop operational strategies to maximise their profits and assist governments to design effective regulatory schemes to enhance social welfare. This book addresses this need by detailing the methodological development of a series of advanced mathematical models that delineate the complex and intriguing relationship between a system’s endogenous variables (such as effective passenger demand and driver supply) and a platform’s decision variables (such as price, wage and matching rules) in the stationary equilibrium state of ride-sourcing markets. These models are intended to enable effective research on several current topics, including Pareto-efficient frontiers and the design of Pareto-efficient government regulation schemes; pricing and matching operations for ride-pooling services; the effects of traffic-congestion externalities on ride-sourcing markets; the design of operational strategies for ride-pooling services in the presence of traffic congestion; the reanalysis of government regulations by considering traffic congestion and the heterogeneity of drivers’ reservation rates and the optimisation of on-demand matching of passengers with drivers. This book also describes the first systematic modelling framework for ridesourcing markets, which aims to address key operational and planning aspects from the viewpoint of ride-sourcing platforms or social planners. This framework is based on state-of-the-art research that has primarily been conducted by

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us and our colleagues. Accordingly, this book will be a useful reference for all students, scholars, scientists, and professionals studying and/or working in the fields of shared mobility, especially those focused on ride-sourcing services. We are sincerely grateful to several researchers for their collaborations with us in this endeavour. In particular, we thank Dr Xinwei Li from Beihang University, Dr Zheng Zhu from Zhejiang University, Dr Hao Sun and Prof. Zhixi Wan from the University of Hong Kong and Dr Xiaoran Qin, Dr Yaqian Zhou, Mr Zhengfei Zheng, Ms Shuqing Wei and Mr Siyuan Feng from the Hong Kong University of Science and Technology for their valuable contributions to the theoretical analyses and numerical studies presented in this book. Finally, we acknowledge funding support from the Research Grants Council of the Hong Kong Special Administrative Region (HKSAR). Jintao Ke Hai Yang Hai Wang Yafeng Yin January, 2023

Chapter 1

Introduction of ride-sourcing markets Jintao Ke1, Hai Yang2, Hai Wang3 and Yafeng Yin4 1

Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China; Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China; 3School of Computing and Information Systems, Singapore Management University, Bras Basah, Singapore; 4Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI, United States 2

1.1 Background Urban mobility has undergone dramatic changes around the world in recent years with the introduction of ride-sourcing services (or on-demand ride-hailing services) provided by transportation network companies (TNCs). These companies, such as DiDi, Uber, Grab, Lyft, Careem and Ola, efficiently connect passengers and dedicated drivers via online platforms. Ride-sourcing services are playing an increasingly important role in meeting on-demand mobility needs and reshaping the conventional taxi industry. Moreover, the accelerating development of mobile Internet-based technologies has led to a rapid expansion in ride-sourcing services. For example, since its official launch in 2011, Uber has expanded its business to more than 700 metropolitan areas in 65 countries and served over five billion on-demand trips (DMR, 2019a). It offers riders a menu of services, including UberX (the basic service provided by four-seater sedans), Uber Black (an executive luxury service), UberPool (a ride-pooling service that enables one driver to serve two or more passengers with different requests in each ride), SUV (a six-seater vehicle luxury service) and Taxi (an e-hailing taxi service requested via the platform). Similarly, DiDi is the largest ride-sourcing company in China, and since its launch in 2012 has served more than 550 million users in 400 cities throughout China (DMR, 2019b). Its services include Express (a basic service), Premier (an upgraded service), Luxe (an executive luxury service), ExpressPool (a ride-pooling service), Hitch (a ride-sharing service offered by non-dedicated drivers with their own trip plans) and Minibus (an on-demand minibus for shared ride services). In addition, these companies are now deploying electrified vehicles to reduce fuel costs and developing autonomous vehicles to reduce human labour costs. Supply and Demand Management in Ride-Sourcing Markets. https://doi.org/10.1016/B978-0-443-18937-1.00013-9 Copyright © 2023 Elsevier Inc. All rights reserved.

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On a typical ride-sourcing platform, a passenger makes a request that contains his/her detailed trip information, i.e., origin and destination location, departure time and service option. At the same time, drivers affiliated with the platform are cruising around the city waiting for requests or parked in a specific waiting area (e.g., an airport or railway station). Thus, the platform matches waiting passengers and idle drivers and aims to maximise the matching rate (the number of driverepassenger pairs matched per unit time) and minimise the average pick-up time (the time taken for drivers to pick up their assigned passenger(s)). The platform also collects trip fares from passengers and pays wages to drivers, and the difference between the fares collected and wages paid is the commission, which represents the platform’s profit. Wang and Yang (2019) developed a general framework to describe the intrinsic and complex interactions between various endogenous and decision variables of ride-sourcing market stakeholders and agents. As shown in Fig. 1.1, a ride-sourcing market is a typical two-sided market that comprises a supply side (drivers) and a demand side (passengers) that interact with each other. On the demand side, potential passengers compare ride-sourcing services with other transportation services, such as conventional street-hailing taxis and public transit, by evaluating the trip fare and quality of these services (e.g., waiting time until pick-up). On the supply side, drivers decide whether to work and when and for how long to work, according to their expected income level or hourly wage. A ride-sourcing platform therefore aims to maximise profit or social welfare by leveraging various decision variables,

FIGURE 1.1 General framework of ride-sourcing markets. Adapted from Wang, H., Yang, H., 2019. Ridesourcing systems: a review and framework. Transportation Research Part B: Methodological 129, 122e155.

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such as the trip fare charged to passengers and the hourly wage paid to drivers, or by directly controlling the vehicle fleet size, while taking into account the effect of its decisions on passengers and drivers. An essential characteristic of a ride-sourcing system is the matching friction between passengers and drivers. Although ride-sourcing platforms perform efficient on-demand matching that often reduces matching friction more than in conventional street-hailing taxi services, matching friction cannot be completely eliminated. In particular, the equilibrium quantity of services supplied by drivers is generally greater than that consumed by passengers, which results in a certain surplus supply. This surplus is measured in terms of idle vehicle hours and is an important measure of service quality, as it governs passengers’ waiting time until pick-up. Thus, the supply surplus indirectly influences passengers’ generalised cost and passenger demand, which means that passenger demand, service quality (measured as passengers’ waiting time until pick-up), and drivers’ idle time (or number of idle vehicle-hours) interact with each other in a complex manner. These are key endogenous variables that influence platforms’ operational efficiency, and therefore serve as crucial references that guide platform operations and decisions on aspects such as pricing (trip fares and drivers’ wages), vehicle fleet-size management, empty vehicle repositioning, ride-pooling assignments and fare splitting, and dispatching and matching. The design of operational strategies for maximising ride-sourcing platform profit or achieving optimal social welfare therefore requires a precise understanding of the intricate relationships between a platform’s decision variables and a system’s endogenous variables. Thus, there is a pressing need for the development of efficient mathematical models to describe ride-sourcing markets, which can be used to determine optimal operating strategies and regulatory policies for ride-sourcing platforms.

1.2 Theoretical developments Various aggregate models (e.g., Zha et al., 2016) and disaggregate models (for example, He et al., 2018; Xu et al., 2021) have been proposed to describe supplyedemand conditions and properties at stationary equilibria (He et al., 2018; Xu et al., 2017; Zha et al., 2016). Due to the similarities between ridesourcing markets and conventional taxi markets, studies have been rooted in research on street-hailing taxi services (Yang and Yang, 2011; Yang et al., 2010) and Internet-based-hailing (e-hailing) taxi services (He and Shen, 2015; He et al., 2018; Wang et al., 2017). Aspects of ride-sourcing markets that have been examined include the coordination of supply and demand using prices and wages (Bai et al., 2019; Taylor, 2018); pricing and surge-pricing strategies (Cachon et al., 2017; Castillo et al., 2017; Yang et al., 2020b; Zha et al., 2016); government regulations and policies (Yu et al., 2020); the effects of ridesourcing markets on conventional taxi markets (Nie, 2017; Wallsten, 2015);

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geometrical matching and order dispatching (Lyu et al., 2019; Xu et al., 2017; Xu et al., 2018; Yang et al., 2020a; Zha et al., 2018; Zhang et al., 2017); driver labour supply (Sun et al., 2019a,b; Zha et al., 2017); supply and demand predictions (Ke et al., 2017, 2019b; Tong et al., 2017); repositioning and subsidies for empty vehicles (Wang and Wang, 2020; Zhu et al., 2021); ridepooling services (Ke et al., 2020b); and electrified ride-sourcing vehicles (Ke et al., 2019a). We do not exhaustively review this large and growing body of literature here; instead, we outline several important topics and some relevant analytical studies. Readers are invited to refer to Wang and Yang (2019) for a general framework and comprehensive review of research problems in ridesourcing markets.

1.2.1 Stationary equilibrium state Most previous studies have focused on a stationary equilibrium state in which the rate of arrival of passengers, the service quality (i.e., passengers’ waiting time) and the combined number of idle/in-trip vehicles are invariant over time. This equilibrium state is affected by decision variables, such as trip fares and vehicle fleet sizes, and exogenous variables, such as potential demand, trip distances, and city sizes and topologies. As a result, the equilibrium states at different times of the day or on different days of the week can be compared by putting different exogenous variables into a model. On the supply side, at any given instant of the equilibrium state (which can be obtained by taking a snapshot of the market), a vehicle is in one of three phases: an idle phase (i.e., waiting for passengers, and thus parked at a specific region or being driven around a city), an in-trip phase (delivering a passenger to his/her destination) or a pick-up phase (en route to pick up a passenger assigned by online matching). On the demand side, service quality is measured in terms of passengers’ waiting time, which consists of two parts: the time passengers spend waiting online, after submitting a request for transport, to be matched with drivers, and the time passengers spend waiting to be picked up by drivers with whom they have been matched. We denote the first part of waiting time the matching time and the second part of waiting time the pick-up time. The distributions of vehicles in each phase on the supply side and the service quality on the demand side are endogenously and interactively dependent. First, the average matching and pick-up times depend on both the number of idle vehicles and the number of waiting passengers. The more idle vehicles and waiting passengers there are the shorter the average pick-up time is, as under these circumstances it is easier for a platform to match vehicles and passengers. Second, the average matching and pick-up times are crucial service-quality measures that influence passenger demand, which in turn affects the distribution of vehicles in each phase, e.g., the higher the passenger demand, the higher the number of in-trip vehicles (i.e., vehicles transporting

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passengers), and thus the lower the number of idle or pick-up vehicles. Moreover, the interdependencies of these parameters, such as the relationship between the average pick-up time and the number of idle vehicles and waiting passengers, are also influenced by the matching technologies and algorithms implemented by a platform. Over the past few decades, various mathematical models have been developed to analyse aspects of the market equilibria of ride-sourcing services (or taxi services). Arnott (1996) studied the marketplace offered by a taxi call centre that operated its service based on a first-come-first-served (FCFS) mechanism, which immediately matches a passenger who submits a request with the closest idle taxi driver. Arnott (1996) further assumed that idle drivers’ entry into the marketplace followed a spatial Poisson process, and thus developed an analytical approximation to the average waiting time that is inversely proportional to the square root of the number of idle vehicles. However, this model implicitly neglects the effect of the time vehicles spend in pick-up on their utilisation; that is, it assumes that vehicles are either in an intrip or idle phase, and never in a pick-up phase. To deal with this problem, Castillo et al. (2017) considered a modified FCFS matching mechanism that assumes a vehicle is idle, en route (to pick up passengers) or in-trip. They found that this created a wild goose chase regime, as matching failure occurred when idle drivers were matched with distant passengers because drivers wasted substantial time in the pick-up phase. Castillo et al. (2017) showed that such a matching failure caused the trip supply curve to bend backwards, and that the failure could be alleviated or prevented by the use of well-designed surge pricing. In actual operations, instead of using an FCFS scheme, ride-sourcing platforms such as DiDi and Uber use a batch-matching mechanism that accumulates a certain number of waiting passengers and idle vehicles in a matching pool before performing online matching. In addition, some platforms divide a marketplace into numerous subregions and then perform matching between idle drivers and waiting passengers within each subregion. These matching strategies can effectively prevent distant matching and reduce pickup time. However, a ride-sourcing market that uses these matching strategies cannot be well characterised by the models that have been proposed by Arnott (1996) and Castillo et al. (2017), as pick-up time depends on the number of waiting passengers in addition to the number of idle vehicles. Thus, the matching efficiency measured in terms of passengers’ waiting time is governed by the size of these two groups of agents. Yang and Yang (2011) and Zha et al. (2016) have both used a Cobbe Douglas-type meeting function to characterise the searching and meeting frictions between drivers and passengers in such a two-sided matching mechanism. In their models, the rate of matching between drivers and passengers is an increasing function of the number of waiting passengers and the number of idle drivers. The matching functions can exhibit increasing,

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constant or decreasing returns to scale. Under certain conditions, such matching function-based models can be reduced to the model of Arnott (1996). However, this two-sided matching model is unable to consider the effects of pick-up time on the distribution of vehicle phases, which means that although it is a good approximation for markets in which matchings are made between drivers and passengers within small blocks (where the pick-up time is relatively short, and thus can be neglected), it is not a good approximation for markets in which matchings are made between drivers and passengers who are distant from each other (where the pick-up time is long, and thus cannot be neglected). Alternatively, a queuing model can be used to approximate the waiting time. For example, Bai et al. (2019) proposed a queueing model that analytically approximates the average waiting time of passengers by assuming that drivers are servers and passengers are arriving customers. In another example, Banerjee et al. (2015) combined a theoretical queueing model with underlying stochastic dynamics to determine the stationary equilibrium solutions that capture the choices of drivers and passengers, and the maximum profit for a platform. Xu et al. (2019) constructed a double-ended queueing model to analyse the supply curve of an e-hailing system with a constrained matching radius, which revealed that a smaller matching radius decreased the backward bending of the supply curve. Such queueing theoretical models are a flexible and trackable framework with which to describe the matching process and market equilibria and can generate interesting analytical results, but they rely on strict assumptions regarding the birth and death processes of a queue, and the spatiotemporal distributions of the arrivals of passengers and drivers. In sum, the complexity of the marketplace means that there is always a trade-off in the use of mathematical models for describing ride-sourcing market equilibria, as these models must balance interpretative ability with mathematical tractability.

1.2.2 Monopoly optimum, social optimum, and Pareto-efficient solutions Analyses of ride-sourcing markets must determine optimal operating strategies by tuning platform decision variables, such as trip fares, wages, vehicle fleet sizes, and matching strategies. However, ride-sourcing platforms (private firms) and governments (the public sector) may have different interests and objectives: the former are typically only interested in maximising their profits, whereas the latter also wish to maximise the total social welfare of a ridesourcing system, which requires maximising the benefits of various stakeholders and agents (such as passengers’ surplus and drivers’ welfare). The set of platform decision variables that generate the maximum profit for a platform in a monopoly market is regarded as the monopoly optimum solution, whereas the set of platform decision variables that generate the maximum social

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welfare is regarded as the social optimum solution. It is also important to analyse the Pareto-efficient frontier, along which neither stakeholder (platform or government) can increase its own benefit (profit or social welfare) without decreasing the other stakeholder’s benefit. As a result, Pareto-efficient solutions are a set of operating strategies that achieve the best results when considering both the platform and the government, as deviating from these strategies cannot simultaneously improve the results for both stakeholders. The monopoly optimum and social optimum are thus the two polar points of the Pareto-efficient frontier. Yang and Yang (2011) sought a set of Pareto-efficient solutions by simultaneously considering two objectivesethe maximisation of platform profit and the maximisation of social welfareewhich naturally gave rise to a bi-criteria or bi-objective maximisation problem. They showed that the utilisation rate of vehicles and the service quality (measured in terms of the waiting/searching time of passengers) were constant along the Pareto-efficient frontier and equal to that at the social optimum. Analyses of the properties of the Pareto-efficient frontier would assist governments to design suitable regulations, such as a price cap, a minimum wage level, a maximum-allowed fleet size or a minimum vehicle utilisation rate, to achieve a desirable level of social welfare without deflecting the optimal strategies from the Pareto-efficient frontier, thereby preserving market efficiency.

1.2.3 Regulations The emergence of ride-sourcing services brings convenience to travellers but also creates many questions and challenges. A major question is whether and how a government should regulate a ride-sourcing market. Regulations have already been established in some locations, particularly in metropolitan cities. For example, New York City requires ride-sourcing platforms to guarantee that the hourly wage of drivers is higher than the minimum wage (US$15/h). This was extended to a ‘minimum per-trip formula’ stipulating that the wage per trip should not be less than US$23 for a 30 min/7.5 mile ride. In June 2019, New York City imposed a more stringent regulation on Uber, Lyft and their competitors, which requires drivers to carry a passenger at least 69% of the time they are operating in Manhattan below 96th Street, or the companies will be subject to penalties. Similarly, in January 2020, California Assembly Bill (AB5) was passed, which classifies hundreds of thousands of independent contractors, including ride-sourcing drivers, as full-time employees. Uber and Lyft were denied exemption from this legislation, but nevertheless refused to reclassify their drivers as employees and declared that they planned to continue ‘business as usual’, which exposes them to litigation from state agencies. In addition, since 2016, the authorities of Beijing and Shanghai have required DiDi to only employ drivers who are registered residents in their cities. This regulation is similar to the fleet-size control rules imposed in taxi

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markets, which allow only drivers with local taxi licences to provide ride services. The effects of existing and proposed regulations have been extensively studied and debated. Li et al. (2019) argued that although imposing a minimum wage can motivate ride-sourcing companies to hire more drivers and serve more passengers, it will cause companies’ profits to decrease. They also found that fleet size control (i.e., maximum fleet size) regulation reduces driver income, as it motivates a platform to hire cheaper labour by reducing drivers’ average pay. Parrott and Reich (2018) examined the likely effects of the regulations imposed in New York City by carrying out simulation studies based on TNC administrative data. This revealed that a regulation guaranteeing minimum wage will increase drivers’ income by 22.5%, but will also increase passengers’ trip fares and waiting times by 5% and 12e15 s, respectively. Yu et al. (2020) argued that traditional taxi services will die out if no government regulation is applied, and therefore agreed that the Chinese government’s new regulations effectively balance multiple objectives, namely, business and job creation, the viability of taxi services and consumer welfare. These relatively mixed empirical findings on the effects of regulation on ride-sourcing markets are partially due to the fact that passenger demand, driver supply and other characteristics of markets vary from city to city. Therefore, it is critical that mathematical models are established to investigate the effects of regulatory policies currently applied by various cities to their respective ride-sourcing markets, as this will facilitate the development of new regulatory policies that better coordinate the interests of all stakeholders in these markets.

1.2.4 Ride-pooling services Recently, several TNCs have launched on-demand ride-pooling services (Chen et al., 2017), which enable a driver to serve two or more passengers per ride. Typical examples include UberPool, DiDi Express Pool, Lyft Line and GrabShare, with Lyft aiming for 50% of its rides to be shared by 2022 (Schaller, 2018). Ride-pooling services are expected to improve vehicle utilisation and alleviate traffic congestion, and help solve the first-/last-mile problem in public transit (Wang et al., 2019; Wang and Odoni, 2016). However, this new mobility service brings new challenges, such as determining a discounted trip fare that will attract ride-pooling passengers. When passengers launch a TNC application (app), they can choose to submit an order for a ride-pooling service or a normal (non-pooling) service. The fare for a ride-pooling service will typically be discounted to a predetermined cost that is less than the trip fare of a non-pooling service for the same distance. A key concern for ride-sourcing platforms is the pool-matching probability, i.e., the proportion of passengers who are pool-matched and thus share their rides with other ride-pooling passengers, as ride-pooling can have

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adverse effects on passengers’ service experience or platforms’ profits. For example, pool-matched passengers may experience a longer trip time than they would by using a non-pooling service, due to detours being made to service the transportation needs of their fellow pool-matched passengers. Alternatively, platforms may suffer a loss of revenue because up-front discounts decrease predetermined fares. The relationships between the system endogenous variables and decision variables of ride-pooling services are more complicated than those between the variables of regular ride-sourcing services. First, the pool-matching probability depends on the passenger demand for ride-pooling and the poolmatching strategies, which generally impose matching radii to pool-match passengers with similar origins and destinations. A platform will also typically perform pool-matching after a certain period of waiting (by passengers) for a certain length of matching window, as this accumulates passengers to enable better pool-matching. Thus, it is crucial to determine the length of the matching window and the matching radius, as these influence the poolmatching probability, passengers’ detour time and vehicle utilisation. Second, the predetermined fare discount rate directly affects a platform’s profits and passenger demand, which in turn affects pool-matching probability and thereby indirectly affects a platform’s profits. A precise understanding of the intricate relationships between a platform’s decision variables (trip fare, vehicle fleet size, matching intervals and matching radii) and a system’s endogenous variables (passenger demand, pool-matching probability and detour time/distance experienced by passengers) is essential for the maximisation of a platform’s profit, trip throughput and social welfare. Compared with normal (non-pooling) ride-sourcing services, the emerging ride-pooling services have received less research attention, and warrant further examination. Jacob and Roet-Green (2021) proposed a model for a market with a ride-pooling service and a non-pooling service, and identified the optimal priceeservice menus, i.e., the ride services to be provided and their corresponding prices. They showed that it was optimal to offer both types of service to passengers if the congestion level was not severe and the distribution of passengers’ preferences was not skewed. However, their model focuses on the design of priceeservice menu mechanisms, and highly simplifies the poolmatching process and the associated intricate relationships in ride-pooling services. Ke et al. (2020b) compared the equilibrium states and optimal operating strategies in ride-sourcing markets with and without ride-pooling, which revealed that the monopoly optimum and social optimum in a ridepooling market are generally lower than those in a non-pooling market. Their model gives a probabilistic approximation of the pool-matching probability and average detour time experienced by passengers who opt for ride-pooling. Yan et al. (2020) simplified the pool-matching process by introducing a dynamic waiting mechanism, whereby two pool-matched passengers are picked up and dropped off at certain meeting points. This means

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that drivers only need to drive to one meeting point to pick up pool-matched passengers, rather than having to go to a separate pick-up point for each passenger. The Uber ExpressPool service uses such a dynamic waiting mechanism, which is an alternative service to UberPool. Furthermore, ride-pooling services are distinct from traditional ridesharing or car-pooling programmes, which have been extensively studied (Ferguson, 1997; Huang et al., 2000; Konishi and Mun, 2010; Yang and Huang, 1999). A key difference is that ride-pooling services are offered by dedicated drivers who cruise urban streets, ready to serve passenger requests to earn money, whereas traditional ride-sharing programmes are offered by nondedicated drivers who have their own trip plans and serve on-demand passengers while fulfilling these trips to share fuel costs. In actual operation, a platform operating ride-sharing services focuses only on matching drivers and riders who have similar routes, whereas a platform operating ride-pooling services must also improve vehicle utilisation by reducing drivers’ idle time, and balance the probability of successful pool-matching and the detours experienced by passengers. Considering the unique characteristics of ridepooling services, it would be of great interest to develop analytical models to investigate how a platform can maximise its profit or social welfare by optimising the values of its operational decision variables, such as the trip fare discount for ride-pooling services, the wage paid to drivers and the poolmatching strategies.

1.2.5 Congestion externalities Most studies have assumed that the travel speed for ride-sourcing vehicles is exogenous, which ignores the effects of interactions between a market equilibrium and traffic congestion caused by ride-sourcing vehicles and regular private cars (e.g., other background traffic). However, the presence of traffic congestion externalities may influence an equilibrium and the decisions of a platform. Specifically, in the absence of traffic congestion, the trip times of passengers are regarded as an exogenous parameter. In contrast, in the presence of traffic congestion, an increase in vehicle fleet size will add additional vehicle miles to a road network and thus increase traffic congestion. This will also affect passengers’ trip times and thus their mode choices, such that trip times are no longer exogenously given but determined endogenously. As mentioned, a ride-sourcing platform maximises its profit or achieves optimal social welfare by leveraging trip fares and vehicle fleet size (or the wage paid to drivers) to directly influence passenger demand, or changes service quality to indirectly influence passenger demand. By also taking into account the effect of traffic congestion, platform operators can adjust their operating strategies (e.g., trip fare and vehicle fleet size) for profit or welfare maximisation. In addition, the monopoly optimum and social optimum solutions of ride-sourcing markets may exhibit different properties in the absence

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and presence of traffic congestion. As traffic congestion is greatest in metropolitan areas, incorporating traffic congestion into models of ride-sourcing markets will enable ride-sourcing platforms and governments to better manage supply and demand by applying appropriate operating strategies. Moreover, the accelerating growth of ride-sourcing services has triggered debate on various issues, in particular the effects of ride-sourcing services on traffic congestion. Advocates claim that well-designed ride-sharing services (traditional ride-sharing services offered by non-dedicated drivers and ridepooling services offered by dedicated drivers) can significantly reduce traffic congestion by improving vehicle utilisation rates, such as by using one vehicle to serve two or more passengers. For example, Alexander and Gonza´lez (2015) examined the influence of ride-sharing on network-wide traffic congestion by analysing passengers’ smartphone mobility records. They compared the originedestination trips of travellers who used private cars and other travel modes, which revealed that ride-sharing was adopted more by travellers with cars than those without cars. This indicates that ride-sharing can reduce the total vehicle miles travelled on a road network, and thus reduce traffic congestion. However, critics argue that ride-pooling has little or no ability to alleviate traffic congestion. For example, Schaller (2018) found that ridesourcing services add 2.8 vehicle miles to a road network for each mile of driving removed and that the inclusion of ride-pooling services results in only a marginal reduction in the increased miles (i.e., 2.6 vehicle miles added to a road network for each mile of driving removed). This indicates that ridepooling services do not offset the traffic congestion caused by ride-sourcing services. The possible reasons for these negative effects are as follows: (1) the pool-matching probability is too low, meaning that many pooling passengers are not pool-matched with other pooling passengers; (2) pool-matched rides may inevitably add additional vehicle miles to a road network because of the extra detours that are required to pick up and drop off two or more passengers with different routes; and (3) the typically low trip fares of ridepooling services may attract passengers to ride-pool rather than use other more space-efficient transportation modes, such as bicycles and public transit. Thus, it remains unclear to what extent ride-pooling can alleviate traffic congestion and thereby influence the trip times of ride-sourcing passengers and other road users. These problems have been examined in some studies. Ke et al. (2020b) proposed a static model that utilises a macroscopic fundamental diagram (MFD) to capture the speededensity relationship and its effects on the trip time of ride-sourcing passengers and private car users. They compared markets with and without ride-pooling to identify the critical level of passenger demand at which the implementation of ride-pooling generates an optimal situation for all stakeholders. Ke et al. (2020a) incorporated a traffic congestion model into an equilibrium model for ride-sourcing markets and studied how a platform maximises its profit or social welfare by leveraging platform decision

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variables, such as the trip fare, vehicle fleet size and pool-matching window length.

1.2.6 Platform competition and platform integration Only a few studies have been focused on analysing a ride-sourcing market with platform competition. For example, Zha et al. (2016) studied a duopoly market in which a Nash equilibrium is established, such that the two platforms cannot further increase their profits by unilaterally changing the values of two key variables (the trip fare and the vehicle fleet size). Zha et al. (2016) argued that in this scenario, competition does not necessarily decrease the trip fare or increase the social welfare because competition segments the groups of passengers and drivers, and thus increases matching frictions. Cohen and Zhang (2022) investigated a duopoly market in which two ride-sourcing platforms choose the trip fares they charge to passengers and the wages they pay to drivers, and thus compete for both demand and supply. Se´journe´ et al. (2018) showed that the splitting of demand between a range of platforms makes a market thinner, and thus leads to market fragmentation and inefficiency. Many studies that have examined the economics of a general two-sided market are relevant to platform integration (Rochet and Tirole, 2003; Armstrong, 2006; Armstrong and Wright, 2007), and some seminal models have been extended to enable an analysis of ride-sharing and ride-sourcing markets. For example, Jeitschko and Tremblay (2020) studied a two-sided market in which passengers endogenously determine whether they use only one platform or simultaneously use two competing platforms and showed that passengers prefer the platform that offers the lowest price. Bernstein et al. (2021) examined multi-homing on the supply side by analysing and comparing two settings: one in which drivers serve passengers on one platform, and one in which drivers serve passengers on both platforms. They found that although individual drivers may increase their income by choosing multi-homing, no drivers increase their income if all of them choose multi-homing. In general, these studies have developed bespoke models to describe two groups of agents (i.e., demand and supply in markets) and offer interesting managerial insights. However, these studies have not characterised certain specific processes in the ride-sourcing market, such as the matching of drivers with passengers, and thus may generate some biased conclusions when applied in a ride-sourcing context. Mobility as a service (MaaS) integrates multiple modes of transport into seamless trip chains, thereby enabling users to plan, book, and pay for multiple types of mobility services. Thus, transportation services from public and private transportation providers can be combined into a single portal that creates and manages users’ trips. For example, Rasulkhani and Chow (2019) generalised a prescriptive many-to-one assignment game to consider routes containing multiple segments with line capacities for offline operating design analysis. Pantelidis et al. (2020) devised a model that allows travellers to make

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multimodal multi-operator trips, which results in stable cost allocations between competing network operators and thus provides users with MaaS. In contrast to MaaS, platform integration is the integration of private ridesourcing platforms. Unlike MaaS operators, which collectively manage all legs of a user’s trip (i.e., multimodal trips), platform integrators only provide ridesourcing services for one leg of a user’s trip (i.e., the leg comprising unimodal trips). Nevertheless, MaaS operators and platform integrators both attempt to integrate multiple transport service providers into a single application, as this obviates the need for users to interact with multiple portals to obtain the optimal trip. Previous studies have examined ride-sourcing markets with MaaS. In contrast, in Chapter 9 of this book, we mathematically investigate the equilibrium state of a ride-sourcing market with platform integration. That is, whereas previous studies have examined how to combine various travel modes into sequential segments for a single trip that traverses only a few nodes in a network, we examine trips in which one segment follows a Nash equilibrium game, i.e., where different platforms compete by setting suitable fares. In addition, as a third-party integrator aims to mitigate market fragmentation and thus reduce market friction in bilateral matching of passengers with vehicles, we investigate the effects of the emergence of a third-party integrator on such a Nash equilibrium.

1.2.7 Ride sourcing and public transit To achieve an unbiased understanding of how ride-sourcing reshapes urban mobility, the interaction of ride-sourcing with public transit must be determined. In this context, a key consideration is that as the popularity of ride-sourcing increases among passengers, it may have both positive and negative effects on public transit. That is, ride-sourcing services may act as feeders and thus solve first- or last-mile problems but may also draw passengers away from public transit. Some studies have treated traditional transportation modes (e.g., taxis and shuttles) and advanced transportation modes (e.g., automated vehicles) as complements of public transit (Yap et al., 2016; Wang et al., 2017; Chen and Wang, 2018). Compared with traditional and advanced transportation modes, ride-sourcing services provided by TNCs are more convenient and practical for passengers. Thus, some public transit operators have considered integrating fixed route/point/zone ride-sourcing services into public transit systems to provide sufficient geographic coverage (Li and Quadrifoglio, 2010; Chen and Nie, 2017; Maheo et al., 2017). In addition, governments and TNCs have recently collaborated to improve service quality (McCoy et al., 2018). However, these studies have focused on the microscopic-level development of efficient dispatching, scheduling, and routing algorithms. Therefore, these studies have devised equilibrium-based models for system-level ride-sourcing and public transit ridership analysis in which public transit is regarded as a minor mode,

14 Supply and Demand Management in Ride-Sourcing Markets

and thus its characteristics are not formulated (Zha et al., 2016; Wang et al., 2018). In addition, Zhu et al. (2020) examined the effects of dynamic ridesharing on public transit via a network model. They found that long-distance ride-sharing services provided by TNCs are absolute competitors for public transit and that TNCs can help to maintain a high level of public transit usage by implementing certain operating strategies. Some empirical studies have demonstrated how ride-sourcing can complement and substitute for public transit, in addition to revealing the influence of ride-sourcing services on traffic congestion. Li et al. (2016) used urban mobility reports and Uber ride-sourcing data to demonstrate that Uber ridesharing services have significantly reduced traffic congestion in urban areas in the United States. Hall et al. (2018) employed a difference-in-differences approach to show that Uber is, on average, a complement for public transit, as it increased public transit ridership by 5% over the 2 years subsequent to its emergence. Moreover, the positive effects of Uber’s emergence on transit ridership have been more significant in larger cities and for smaller transit agencies than in smaller cities and for larger transit agencies. However, Schaller (2018) claimed that ride-sourcing services increase vehicle usage by adding 2.8 new vehicle miles on a road for each mile of automobile travel subtracted from the road. Thus, these empirical studies have reached mixed or inconsistent conclusions, which are partially attributable to variations in the demand patterns, supply patterns and urban transportation network topologies of the cases they have examined. In summary, there is an urgent need for comprehensive analytical modelling frameworks that can help researchers determine the mechanisms whereby ride-sourcing markets affect the usage of public transit and how they affect multi-modal transportation markets. Chapter 10 fills this research gap by developing an equilibrium-based mathematical model that can delineate passengers’ choices in a transportation market with ride-sourcing and public transit. This model provides insights into the conditions and properties of equilibrium between ride-sourcing services, public transit services and combinations of these services. Moreover, the associated research investigates operational designs by using a TNC’s decision variables in a multi-modal transportation market to maximise profit or social welfare.

1.2.8 On-demand matching and its key decision variables The recent rapid growth of technology-enabled mobility services has enabled TNCs to offer convenient and efficient on-demand ride services via online platforms that reduce information asymmetry and uncertainty. For example, TNCs dynamically match drivers and passengers by tracking their real-time locations, thereby reducing search friction. Accordingly, a key concern in ride-sourcing markets is the design of matching strategies that minimise vehicle usage, fleet size and passengers’ waiting time.

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Bilateral searching and the matching of idle drivers and waiting passengers have been studied in traditional taxi markets (Yang et al., 2010a,b) and in ehailing and ride-sourcing markets (He and Shen, 2015; Wang et al., 2017, 2018a; Zha et al., 2016). In a traditional taxi market, idle drivers cruise streets searching for waiting passengers, which generates search friction. In contrast, in e-hailing and ride-sourcing markets, drivers are matched with passengers via online platforms, which efficiently reduces search friction. Ride-sourcing platforms generally use one of two matching modes: broadcast mode or dispatch mode. In broadcast mode, ride-sourcing platforms serve as transaction intermediaries, by collecting ride requests from passengers and then broadcasting these requests to idle drivers. Idle drivers aim to select a broadcast request that optimises their individual utility (in terms of pick-up distance, order value and destination, among other factors). In dispatch mode, ride-sourcing platforms match idle drivers with waiting passengers to optimise overall system efficiency. This mode is currently adopted by many ride-sourcing TNCs, such as DiDi and Uber. Advanced mobile technologies assist online platforms to trace the status of each passenger request and each idle driver, such as their real-time location and cumulative waiting/idle time, and to detect the current supplyedemand situation, such as the numbers of idle drivers and waiting passengers. Then, future short-term supplyedemand conditions, such as the expected arrival rates of new requests and idle drivers within the next 10 min, can be predicted on the basis of historical supplye demand trends and real-time external features (such as weather). The accuracy of such predictions can reach 85% (Tong et al., 2017). Therefore, matching strategies can be dynamically adjusted to improve system efficiency. Several combinatorial optimisation-based online matching strategies have been recently proposed for taxi or ride-sourcing markets. Due to the complexity of the matching process, these strategies are based on diverse assumptions and have different objectives. Agatz et al. (2011) performed a simulation study of 2008 travel demand data in Atlanta that revealed sophisticated optimisation methods substantially outperform simple greedy matching rules in ride-sharing systems. Stiglic et al. (2015) determined that the use of meeting points in ride-sharing systems significantly increases the number of successfully matched participants and reduces system-wide driving distances. Wang et al. (2017) introduced the concept of matching stability in a dynamic ride-sharing system and proposed several mathematical programming methods that generated stable or nearly stable ride-sharing. Vazifeh et al. (2018) developed a vehicle-sharing network to calculate the minimum number of vehicles needed to serve a collection of on-demand requests. They also developed and verified efficient algorithms for computing optimal or nearoptimal solutions using a dataset from New York City that contained more than 150 million taxi trips. Sequential decision-making algorithms have also been used to adapt to the dynamics and stochasticity of large-scale matching. For example, Wong and

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Bell (2006) developed a rolling horizon model to optimise total pick-up time in taxi dispatching, which is based on heuristic approaches that simultaneously consider the expectation of future requests and real-time traffic dynamics. These methods exhibited excellent performance in numerical experiments. Miao et al. (2016) proposed a receding-horizon control framework to maximise the matching of supply and demand with minimum idle taxi-driving distance. This framework incorporates spatiotemporal passenger demand, real-time Global Positioning System locations and taxi occupancy. Model-free approaches have also been recently implemented in taxi and ride-sourcing vehicle dispatch systems, such as the Markov decision process and reinforcement learning (Jindal et al., 2018; Ke et al., 2022; Tang et al., 2019; Wang et al., 2018b; Xu et al., 2018). The abovementioned studies have focused on the mechanism and design of algorithms for instant and batch matching, but have not considered the joint optimisation of the two control variables, i.e., the matching time interval (the time interval within which waiting passengers and idle drivers are accumulated and then subjected to peer-to-peer matching) and the matching radius (the maximum allowable pick-up distance within which waiting passengers and idle drivers are matched), which govern the performance of an on-demand matching strategy. By appropriately lengthening the matching time interval, a platform can accumulate a large pool of waiting (i.e., unserved) passengers and a large pool of idle drivers, which can then be matched to afford shorter expected pick-up distances. However, if the matching time interval is excessively long, some passengers may become impatient and abandon their requests. Similarly, by shortening the matching radius, a platform can shorten the expected pick-up distance, but this may also decrease the matching rate. Therefore, the matching time interval and matching radius must be jointly optimised to enhance system efficiency in terms of passenger waiting time, vehicle utilisation and matching rate (Yang et al., 2020a).

1.3 Outline of this book This book proposes an analytical modelling framework for characterising the nature of ride-sourcing market equilibria and solves several important practical problems related to the design, operation and management of ridesourcing services. Insights obtained from these theoretical developments and numerical studies will assist the managers of ride-sourcing platform to design effective operating strategies (such as pricing and matching strategies) to maximise profits and assist governments to impose appropriate regulations on platforms to balance the trade-offs between social welfare and platform profit. This book comprises 14 chapters. Current approaches to ride-sourcing studies are first detailed, and then these approaches are extended by addressing new problems or developing new methodologies.

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In Chapter 2, we present a fundamental framework for analysing the market equilibria of ride-sourcing markets. We summarise current models to characterise their matching frictions and market equilibrium states. Then, we determine the comparative static effects of regulatory variables on key system endogenous variables and find the sets of regulatory variables at the monopoly optimum, the social optimum, and along the Pareto-efficient frontier. We explain the economic meanings of the pricing formula at the monopoly optimum and the social optimum and obtain several managerial insights from the Pareto-efficient solutions. In Chapter 3, we develop a simulation platform and conduct sensitivity studies to calibrate, validate, and compare various matching functions that have been widely used in previous studies, including those presented in the previous chapter. This chapter will help researchers to select the most appropriate and accurate matching functions for modelling a market scenario according to the intensity of demand and supply. In Chapter 4, we investigate the regulations that a government can impose to induce a platform to choose a predetermined or targeted Pareto-efficient strategy. We examine and compare various regulations, such as price-cap, wage, fleet size, utilisation rate and commission regulations, only some of which are shown to be Pareto-efficient. This enables us to offer practical suggestions to assist governments in better regulating ride-sourcing markets and enhanciing social welfare. In Chapter 5, we examine ride-pooling services and their effects on market equilibria, and design operating strategies that can maximise a platform’s profit or social welfare. We also construct a model for approximating the poolmatching process in ride pooling and for analytically describing the intricate interrelationships between the key endogenous variables that influence market equilibria: pool-matching probability, passenger demand, detours and matching strategies. In Chapter 6, we focus on the traffic congestion externality and its effects on the design of a platform operating strategy. We propose a MFD model to characterise the relationship between speed and density, and seamlessly incorporate this model into the equilibrium analyses discussed above. We examine the aggregate effects of traffic congestion on the trip times of ridesourcing passengers and other travellers in a road network. We also discuss how a platform may change the value of decision variables, such as trip fare, vehicle fleet size and pool-matching window, in response to changes in the level of traffic congestion. In Chapter 7, we examine the effects of traffic congestion on ridesourcing markets that offer ride-pooling services. We analyse to what extent the implementation of a ride-pooling service alleviates traffic congestion and whether this affects a market equilibrium. We then discuss the monopoly optimum, social optimum and Pareto-efficient solutions of a ride-sourcing market that offers a ride-pooling service and compare these

18 Supply and Demand Management in Ride-Sourcing Markets

solutions with those of a ride-sourcing market that does not offer a ridepooling service. In Chapter 8, we re-examine the effects of government regulations on ridesourcing markets with traffic congestion externality and driver heterogeneity. We employ theoretical and numerical analyses to investigate the effects of important regulatory approacheseprice-cap regulation, minimum wage guarantees and fleet size control e on platform profit, consumer surplus, driver welfare and social welfare. We highlight that the outcome of a given regulatory approach may vary according to the level of traffic congestion. In Chapter 9, we introduce and examine a novel business modeleplatform integrationein which a third-party integrator allows passengers to simultaneously hail rides from multiple ride-sourcing platforms. Platform integration can maintain competition between ride-sourcing platforms and thus prevent them from setting prices that are too high and can also mitigate the market fragmentation caused by platform competition. Our theoretical and numerical results demonstrate that in most cases, platform integration benefits passengers, drivers and ride-sourcing platforms. In Chapter 10, we extend the modelling framework developed in Chapter 3 to analyse the complementary and substitutional relationships between ridesourcing services and public transit services. We also explore which operating strategies are optimal for ride-sourcing platforms, given that these platforms provide direct-ride services that compete with public transit and first-/last-mile ride services that act as feeders to public transit. In Chapter 11, we study the joint optimisation of two key decision variables ematching time interval and matching radiusdin an on-demand matching process. We first establish a mathematical model that approximates the batch-matching process and then analyse how these two key decision variables affect system performance measures, i.e., matching rate, passengers’ matching time and pick-up time. Then, we design an optimisation strategy to dynamically adjust the two key decision variables in response to real-time supply-demand conditions. In Chapter 12, we discuss the key factors that affect both the participation decision and working-hour decision and evaluate the effects of hourly income rate on labour supply. Instead of using the abovementioned theoretical model, in this chapter, we use a tailored regression model and a natural experiment with exogenous shocks on a ride-hailing platform to analyse the effects of hourly income rates on labour supply. In Chapter 13, we discover empirical rules that govern trends in poolmatching probability, the average detour distance of passengers and drivers’ average routing distance with respect to passenger demand for on-demand ride-pooling services. The findings are well validated by the results of simulation studies based on actual mobility in Chengdu and Haikou (both China), and Manhattan, New York City (USA). These empirical laws will help other

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researchers to establish theoretical models for the on-demand ride-pooling services that are extensively discussed in Chapter 5. Chapter 14 summarises the major contents of this book.

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Zhu, Z., Ke, J., Wang, H., 2021. A mean-field Markov decision process model for spatial-temporal subsidies in ride-sourcing markets. Transportation Research Part B: Methodological 150, 540e565. Zhu, Z., Qin, X., Ke, J., Zheng, Z., Yang, H., 2020. Analysis of multi-modal commute behavior with feeding and competing ridesplitting services. Transportation Research Part A: Policy and Practice 132, 713e727.

News and reports DMR, 2019a. 110 Amazing Uber Stats and Facts (2019) by the Numbers, Access in 2019. https:// expandedramblings.com/index.php/uber-statistics/. DMR, 2019b. 25 Amazing DiDi Facts and Statistics (2019) by the numbers, access in 2019. https:// expandedramblings.com/index.php/didi-chuxing-facts-statistics/.

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Chapter 2

Fundamentals of ride-sourcing market equilibrium analyses Jintao Ke1, Yafeng Yin2, Hai Yang3 and Hai Wang4 1

Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China; Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI, United States; 3Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China; 4School of Computing and Information Systems, Singapore Management University, Bras Basah, Singapore 2

2.1 Introduction In this section, we briefly overview the major components of a ride-sourcing market: passenger demand, which depends on the pricing and service quality; and driver supply, which depends on the level of driver income. In the following Sections (2.2 and 2.3), we discuss matching frictions between supply and demand under different model settings and the resulting market equilibrium outcomes.

2.1.1 Passenger demand We consider a stationary equilibrium in which each vehicle operated by a ridesourcing platform serves only one passenger per ride. Passenger demand is assumed to be elastic and decreases with the generalised cost, which is primarily governed by trip fare and service quality (measured in terms of waiting time). As mentioned, waiting time is the sum of matching time (the interval of time between a passenger requesting an order and the passenger being matched online with a driver) and pick-up time (the interval of time between a passenger being matched online with a driver and the passenger being picked up by the assigned driver). We use w to denote the average waiting time of passengers, F to denote the average trip fare, and t to denote the average trip time. We further assume that passengers have a homogeneous value of time, denoted b, which means that the average generalised cost of passengers is given by F þ bðw þ tÞ. We also use Q to denote passenger demand, which represents the arrival rate of passengers who opt for ride-sourcing services and

Supply and Demand Management in Ride-Sourcing Markets. https://doi.org/10.1016/B978-0-443-18937-1.00016-4 Copyright © 2023 Elsevier Inc. All rights reserved.

25

26 Supply and Demand Management in Ride-Sourcing Markets

can be written as a monotonically decreasing function of the average generalised cost, as follows: Q ¼ f ðF þ b $ ðw þ tÞÞ

(2.1) f 0 ð $Þ

where f ð $Þ is a strictly decreasing function, i.e., < 0; F is a decision variable; b is a constant; t is assumed to be a constant in the absence of traffic congestion, and w is an endogenous variable that depends on the service quality of a platform. The service quality and thus w largely depend on the vehicle fleet size (N), which is the number of vehicles a platform has available for ride services, and on Q. This implies that w is a function of N and Q, i.e., w ¼ wðQ; NÞ, which depends on the matching frictions between demand and supply (discussed in the next section). It can easily be found that there is a feedback loop between Q and w, as the value of Q influences w via the level of service, and w in turn affects Q via the generalised cost. This endogenous feedback loop results from the matching frictions, which can be characterised by several different models and are discussed in the following sections.

2.1.2 Driver supply Drivers determine whether to enter the market to supply ride services according to their average income level. We use E to denote the average wage (expense) paid to a driver per order, N to denote the number of drivers participating in the platform (i.e., the vehicle fleet size), and mcv to denote the meeting rate (the number of passengeredriver pairs matched per hour). Accordingly, the average income per driver per hour, denoted by U, is given by: U¼

Emcv N

(2.2)

where Emcv is the total wage paid to all drivers per hour. We further assume that drivers’ reservation rate (r) follow a certain distribution over ½0; þNÞ, where r is the lowest E for which a potential ride-sourcing service supplier (driver) is willing to supply ride-sourcing services. Specifically, we assume that a potential driver enters the market if U  r. Thus, the proportion of potential ride-sourcing drivers who choose to provide ride-sourcing services is given by ProbfU  rg, which can be treated as a non-decreasing function of U. We let GðUÞ ¼ ProbfU  rg, where G0 ðUÞ  0. If the total number of potential drivers is N, then the supply function is given by:
cv  Em N ¼ NG (2.3) N which is equivalent to: Em

cv


 N ¼ NG N 1

(2.4)

where Emcv is the total E paid to all drivers, which also represents the operational cost of the ride-sourcing platform. Clearly, N is an elastic supply

Fundamentals of ride-sourcing market equilibrium analyses Chapter | 2

27

that is governed by the function GðRÞ, which takes various forms depending on assumptions about the labour supply. For example, as highlighted by Zha et al. (2016), if the supply is sufficient, r is homogeneous and thus drivers’ entry into the market is free, which means that drivers will continue entering the market until their net earnings become zero. In this case, r is homogeneous and equal to a constant vehicle operation cost (c), and the market equilibrium will be reached at the point where Emcv ¼ cN. This indicates that Emcv is linearly proportional to the realised value of N, and thus the marginal cost of recruiting an additional driver is constant. In contrast, Bai et al. (2019) assumed that the value of r is uniformly distributed over ½0; 1 and scaled from a maximum  cv cv ¼ N 2 U N. This reservation rate U, and thus N ¼ N Em , which yields Em NR indicates that Emcv is proportional to the square of N, and thus the marginal cost of recruiting an additional driver increases with N.

2.2 Matching frictions (inductive approaches) Matching frictions are one of the key characteristics that distinguish ridesourcing markets from many other markets. These frictions arise due to imperfect information and spatial heterogeneity of customers and drivers. The total trip hours supplied by drivers is greater than the total trip hours demanded by passengers, leading to a supply surplus that governs service quality. The service quality is generally represented by the waiting time of customers w and affects the passenger demand Q. Service quality also governs the utilisation rate of vehicles, which influences the driver supply via drivers’ income level. Two types of approaches are used to determine the matching frictions between supply and demand: inductive approaches and deductive approaches. Inductive approaches directly apply a matching function/model (such as a CobbeDouglas matching function) to characterise the matching frictions without specifying a matching mechanism and considering its microfoundation, whereas deductive approaches first assume a physical matching process as the micro-foundation and then deduce the matching frictions. In this section, we discuss the most widely used inductive approaches for the modelling of matching frictions and their equilibrium outcomes; in the following Section (2.3), we discuss deductive approaches for the modelling of matching frictions.

2.2.1 Perfect matching function The perfect matching function assumes that there is no matching friction between supply and demand, such that any demand (supply) is matched if

28 Supply and Demand Management in Ride-Sourcing Markets

there is ample supply (demand). Let T v denote the arrival rate of drivers, and thus the matching rate for perfect matching (mcv ) is simply given by: mcv ¼ minðQ; T v Þ

(2.5)

The underlying assumption of perfect matching is that drivers and passengers arrive at the marketplace with the rates Q and T v , respectively, and are then either matched or immediately exit the market. This implies that there are no waiting passengers or idle/vacant vehicles, which means that Q only depends on F, not on the service quality measured by w. Moreover, the pick-up distance between matched passengers and drivers is ignored. As there are no idle vehicles, T v is given by the ratio of N to t, i.e., T v ¼ N=t. This shows that the perfect matching function makes assumptions that are too strong, as it fails to consider the unique characteristic of ride-sourcing markets, i.e., matching frictions. Consequently, the perfect matching function can only be applied to a special point-meeting market (such as an airport or taxi station) with drivers and impatient passengers, in which drivers and arriving passengers are either immediately matched or exit the market.

2.2.2 Production functions Production functions are widely used in labour economics as they can approximate the matching process and reveal matching frictions. Intuitively, mcv depends on the size of two pools in a stationary equilibrium: the number of vacant vehicles (N v ) on the supply side and the number of waiting passengers (N c ) on the demand side. We let wv denote the average idle/vacant time of drivers. Thus, as the examined market is stationary, N c can be approximated as the product of Q and w, i.e., N c ¼ Qw, and N v can be approximated as the product of T v and wv , i.e., N v ¼ T v wv . Then, mcv can be written as a function of N v and N c , as follows: mcv ¼ MðN v ; N c Þ ¼ MðT v wv ; QwÞ

(2.6)

> 0 and > 0. The elasticities of the matching where function for N v and N c are denoted by a1 and a2 , respectively, and are given by: vmct =vN v

vmct =vN c

a1 ¼

vM N v vN v M

(2.7)

a2 ¼

vM N c vN c M

(2.8)

Fundamentals of ride-sourcing market equilibrium analyses Chapter | 2

29

By assuming a1 and a2 to be constant, mcv can be characterised by the following CobbeDouglas production function: mct ¼ A$ðN v Þa1 $ðN c Þa2 ¼ A$ðT v wv Þa1 $ðQwÞa2

(2.9)

where A is a scaling parameter that captures external features that affect the matching outcomes, such as vehicular speed, and the size and topology of an urban area. It can be generally assumed (Hu and Zhou, 2020) that a1 ˛ ð0; 1Þ, a2 ˛ ð0; 1Þ. Furthermore, the market clearing between supply and demand at equilibrium implies that: mct h QhT v

(2.10)

The inductive model further assumes that each vehicle is either idle or intrip (i.e., occupied by passengers), and as a result, the phase transitions of N active vehicles should satisfy the following conservation equation: N ¼ Qt þ N v

(2.11)

Note that Eq. (2.11) does not explicitly incorporate the pick-up phase, as this may render the subsequent derivations analytically intractable. Nevertheless, as indicated by Zha et al. (2016), this inductive matching function can be used to describe the frictions for both matching and pick-up. In other words, the matching time and pick-up time are not differentiated. For passengers, w capture the time between when a passenger makes a request and when a driver shows up; for drivers, wv captures the time between the last drop off and this pick up. This process is regarded as the whole meeting/matching process and mcv can represent the meeting rate at which passengers physically meet with drivers, rather than the matching rate at the online matching stage. Waiting passengers and idle/searching vehicles randomly arrive at such a market and spread across the market, such that at any instant only those passengers and drivers with a pick-up distance within the matching radius will be matched, while others will remain in the market and continue to wait for matching. The values of a1 and a2 are proportional to the matching radius, which means that the length of w before online matching is the major focus and is governed by the matching frictions described in Eq. (2.9), whereas the pick-up time is relatively short and can be viewed as a constant that constitutes a part of the trip time (t). By substituting Eq. (2.10) and Eq. (2.11) into Eq. (2.9), we can express w as a function of Q and N, as follows: wðQ; NÞ ¼ A

a1

2

Q

1a2 a2

ðN  QtÞ

a

a1 2

(2.12)

The properties of the function wðQ; NÞ, without considering endogeneity, are:
 vw w 1  a2  a1 a1 N ¼ þ (2.13) vQ Q a2 a2 N  Qt

30 Supply and Demand Management in Ride-Sourcing Markets

vw a1 w ¼ vN a2 N  Qt

(2.14)

Given a pair of ðF; NÞ, the market equilibrium is given by a combination of Eqs. (2.1) and (2.14), which yields: 0 0 11 a B B  1 1a2 CC  1 Q ¼ f @F þ b $ @A a2 Q a2 ðN  QtÞ a2 þ tAA

(2.15)

which can be regarded as an implicit equation of Q. For a given pair of (F; N), the left-hand-side (LHS) of the above equation increases with Q, while the right-hand-side (RHS) is a decreasing function of Q. The LHS becomes smaller than the RHS (which equals a positive number) as Q ¼ 0, whereas the LHS becomes larger than the RHS (which is approximately equal to zero) as Q ¼ N. This indicates that there is one and only one equilibrium solution for any given (F; N) in a market equilibrium described by the inductive matching model. We next examine the comparative static effects of F and N. Taking the partial derivatives of both sides of Eq. (2.15) with respect to F and N gives rise to: vQ ¼ vF

f0






(2.16)

1 bf 0 W2c vQ Q
 ¼ N c vN c 0 1 þ bf  W1 þ 2 W2 Q

(2.17)

1þ

bf 0



W1c

N þ 2 W2c Q

where W1c and W2c are abbreviations that simplify the expressions and are given by: W1c ¼

w 1  a1  a2 Q a2

W2c ¼ 

a1 wQ a1 w ¼ a2 N  Qt a2 wv

(2.18) (2.19)

Clearly, if a1 þ a2  1 (i.e., the matching function exhibits constant or decreasing returns to scale), vQ=vF < 0 and vQ=vN > 0; otherwise, a1 þ a2 > 1 (i.e., the matching function exhibits increasing returns to scale), and the signs of both vQ=vF and vQ=vN are undetermined. In addition, by taking the partial derivatives of Eq. (2.15) with respect to F and N (i.e., considering

Fundamentals of ride-sourcing market equilibrium analyses Chapter | 2

31

the endogeneity of Q), and then combining these partial derivatives with Eqs. (2.13) and (2.14), we obtain:
 N f 0  W1c þ 2 W2c vw vw vQ Q
 ¼ ¼ (2.20) N c vF vQ vF c 0 1 þ bf  W1 þ 2 W2 Q
 vw N vQ 1 c ¼ W1c  2 W2c þ W ¼ vN Q vN Q 2


1þ

bf 0

W2c 

1 Q

W1c

N þ 2 W2c Q



(2.21)

which shows that vw=vF < 0 and vw=vN < 0 if the matching function exhibits constant or decreasing returns to scale, i.e., a1 þ a2  1; whereas the signs of vw=vF and vw=vN are undetermined if the matching function exhibits increasing returns to scale, i.e., a1 þ a2 > 1:

2.3 Matching frictions (deductive approaches) As mentioned, deductive matching models attempt to construct matching models based on the micro-foundations of the matching process. However, as the matching process is highly complex and varies between matching mechanisms, some assumptions must be made to specify the matching mechanism to ensure that a model is analytically tractable. We briefly introduce these aspects in the following section.

2.3.1 Queuing models Queuing models assume that passengers randomly arrive at a platform to request services, with each service request comprising a random number of service units offered by a service provider (measured by, for example, the travel distance (in km) of a unit in the ride-sourcing market). Queuing models are based on the assumption that the ride order requested by passengers (denoted here as customers) can be met by any of the available drivers (denoted here as service providers). In reality, the queueing process in a ridesourcing market can be characterised as a G/G/n queuing system, in which the first G represents a general random-arrival process, the second G represents a general random-service process, and n represents multiple service providers. However, to obtain neat or closed-form equilibrium solutions, a simple M/M/1 queuing system has been widely used in the literature, where each M represents a Markovian process (of arrival or service), and one indicates that there is only a single aggregate service provider. We use the M/M/1 queuing model as an example to illustrate how it approximates and characterises matching frictions.

32 Supply and Demand Management in Ride-Sourcing Markets

Consider an M/M/1 queuing system in which the customer arrival rate of the queue equals Q and the service rate of each service provider (average number of customers served per hour) is given by 1=t. If N active drivers are willing to provide ride services, the service rate of one aggregate service provider is given by N=t, and thus the utilisation of these N drivers is equal to Q=ðN =tÞ. To guarantee system stability, we assume that Qt < N, i.e., the service rate is always greater than Q. This assumption is analogous to the assumption made by matching models that a supply surplus (the difference between the total supply, represented by N, and the consumed supply, represented by Qt) always exists and governs the service quality. Based on the properties of an M/M/1 queue, w can be approximated by the following concise closed-form expression: w¼

Q 
N N Q t t

(2.22)

Clearly, w is an important measure of service quality and depends on the difference between Q and N=t, and on Q=ðN =tÞ. By combining Eqs. (2.1) and (2.22), we obtain the following market equilibrium condition for F and N: 0 0 11 Q B B C  þ tC Q ¼ f @F þ b $ @
AA N N Q t t

(2.23)

which can be regarded as an implicit equation for Q. For a given pair of (F; N), it can be found that the LHS increases with Q and the RHS is a decreasing function of Q. As Q ¼ 0, the LHS (0) is less than the RHS (a positive number); as Q ¼ N, the LHS is greater than the RHS (0). Thus, we can conclude that the equilibrium condition stated by Eq. (2.23) yields one and only one equilibrium solution. As mentioned, although the two directly controlled decision variables are F and E, the platform maximisation problem, social welfare maximisation problem, and bi-objective maximisation problem can be reorganised as problems that aim to maximise the objectives as a function of F and N. We next examine the comparative static effects of the latter two regulatory variables on Q and w. By taking the partial derivatives of both sides of Eq. (2.23) with respect to F and N, respectively, we obtain: vQ ¼ vF

f0 1  bf 0
N t

1

2 Q

(2.24)

Fundamentals of ride-sourcing market equilibrium analyses Chapter | 2

vQ ¼  bf 0 vN

Uð2  UÞt 2

3

6 0 ð1  UÞ2 N 2 6 41  bf


33

(2.25)

7 2 7 5 N  Q t 1

where f 0 is the derivative of the Q function and U is the utilisation rate, which equals Qt=N. From the queuing stability condition Qt  N, we can easily prove that vQ=vF < 0 and vQ=vN > 0, which indicates that Q strictly decreases with F and increases with N. These observations are intuitive, as a higher F directly discourages passengers from using ride-sourcing services, while a larger N improves the service rate, reduces w, and thus attracts more passengers. Moreover, by taking the partial derivatives of Eq. (2.22) with respect to F and N, and combining the result with Eqs. (2.24) and (2.25), we readily obtain: vw vQ 1 f0 ¼ 2 ¼
2
vF vF N N Q  Q  bf 0 t t

(2.26)

2 3
N Q 7 vw vQ 1 Uð2  UÞt Uð2  UÞt 6 t 6
7 ¼ ¼ 2  
4 5 2 2 2 vN vN N ð1  UÞ N 2 ð1  UÞ N 2 N 0 Q  Q  bf t t (2.27) 2



From U ¼ Qt=N < 1 and f 0 < 0, we can easily prove that w decreases with F, i.e., vw=vF < 0, and decreases with N, i.e., vw=vN < 0. This is because a higher F will reduce Q and thus decrease w (given the properties of Eq. 2.22), whereas an increase in N indicates a higher service rate and thus results in higher service quality, i.e., a decrease in w.

2.3.2 First-come-first-served (FCFS) Another simplified but widely adopted assumption is the FCFS immediate dispatch mechanism (Castillo et al., 2017; Ke et al., 2020), in which a passenger is immediately matched to the nearest idle driver after the passenger submits a ride request. This model implicitly assumes that the matching radius is infinite and thus one idle vehicle always exists in the marketplace. Under this assumption, the matching time is negligible, and w is equal to the pick-up time, which depends on the mass or density of idle drivers. We denote the number of idle vehicles at any instant of the stationary equilibrium state as N v , and thus w can be written as a decreasing function of N v , i.e., w ¼ wðN v Þ, and w0 < 0.

34 Supply and Demand Management in Ride-Sourcing Markets

On the supply side, each vehicle can be in one of three phases (Castillo et al., 2017): an idle phase (driving on the streets or waiting in a specific location), a pick-up phase (on the way to pick up a passenger), or an in-trip phase (transporting a passenger). Note that in market equilibrium, Q is equal to the trip throughput supplied by N working drivers, and thus the numbers of drivers in the pick-up and in-trip phases in a stationary equilibrium state are Qw and Qt, respectively. Then, from the following time-conservation condition of N: N ¼ N v þ Q$wðN v Þ þ Q$t

(2.28)

we have Q¼

N  Nv wðN v Þ þ t

(2.29)

which shows that Q can be written as a univariate function of the endogenous variable N v . It can be easily found that this univariate function is not necessarily monotonic and that there are two driving forces: an increase in N v decreases N  N v and thus decreases Q, whereas an increase in N v decreases wðN v Þ and thus increases Q. By taking the partial derivative of Q with respect to N v , we obtain: vQ  ðQw0 þ 1Þ ¼ vN v wþt

(2.30)

Conversely, N v can also be viewed as an explicit function of Q, and the partial derivative of N v with respect to Q is given by: vN v ðw þ tÞ ¼ vQ Qw0 þ 1

(2.31)

Clearly, w0 < 0, Q > 0, w > 0, and t > 0, but the signs of vQ=vN v and vQ=vN v are undetermined as the term Qw0 þ 1 can be either positive or negative, depending on the relative absolute magnitude of Qw0 and 1. If Qw0 þ 1 < 0, then Q strictly increases with N v , or equivalently, N v strictly increases with Q, which indicates a wild goose chase (WGC) regime (Castillo et al., 2017). As mentioned, the WGC represents a market failure of the ridesourcing system that is manifest as an extremely low density of vacant vehicles and substantial vehicle hours wasted on the pick-up phase. In contrast, if Qw0 þ 1 > 0, then Q strictly decreases with N v , or equivalently, N v strictly decreases with Q, which indicates a normal (non-WGC) regime. In a traditional street-hailing taxi market or ride-sourcing market with a small matching radius (which eliminates distant matching), the pick-up phase can be ignored in the vehicle conservation function, which is thus approximated by N ¼ N v þ Q$t. In this case, vN v =vQ ¼ t < 0, which implies that a traditional

Fundamentals of ride-sourcing market equilibrium analyses Chapter | 2

35

market always falls into the normal regime. By combining Eqs. (2.1) and (2.28), we can obtain the following market equilibrium condition: N  Nv ¼ f ðF þ b $ ðwðN v Þ þ tÞÞ wðN v Þ þ t

(2.32)

Clearly, the market equilibrium can be regarded as an implicit function of the endogenous variable N v . The LHS, as mentioned above, is increasing with N v in the WGC regime but decreasing with N v in the normal regime, whereas the RHS is a monotonically increasing function of N v . The market equilibrium point is at the intersection of the non-monotonic curve defined by the LHS and the increasing curve defined by the RHS. As mentioned, a ride-sourcing platform can leverage two proxy major decision variables, F and N (which is a proxy decision variable governed by E), to affect a market equilibrium state. We next examine the comparative static effects of these two regulatory variables. Taking the partial derivative of both sides of Eq. (2.32) with respect to F and N affords: vN v f 0 ðw þ tÞ ¼ 0 vF ðQw þ 1Þ þ f 0 bw0 ðw þ tÞ

(2.33)

vN v 1 ¼ vN ðQw0 þ 1Þ þ f 0 bw0 ðw þ tÞ

(2.34)

As Q can be regarded as an explicit function of N v (Eq. 2.29), the partial derivatives of Q with respect to F and N can be written as: vQ vQ vN v f 0 ðQw0 þ 1Þ ¼ v ¼ vF vN vF ðQw0 þ 1Þ þ f 0 bw0 ðw þ tÞ

(2.35)

vQ 1 vQ vN v f 0 bw0 ¼ þ v ¼ 0 vN w þ t vN vN ðQw þ 1Þ þ f 0 bw0 ðw þ tÞ

(2.36)

Moreover, w is by definition a function of N v , and thus the partial derivatives of w with respect to F and N are as follows: vw vN v f 0 w0 ðw þ tÞ ¼ w0 ¼ vF ðQw0 þ 1Þ þ f 0 bw0 ðw þ tÞ vF

(2.37)

vw vN v w0 ¼ w0 ¼ vN vN ðQw0 þ 1Þ þ f 0 bw0 ðw þ tÞ

(2.38)

In the normal regime with Qw0 þ 1 > 0, we can easily prove that Q increases with N and decreases with F (i.e., vQ=vN > 0, vQ=vF < 0), N v increases with both N and F (i.e., vN v =vN > 0, vN v =vF > 0), and w decreases with N and F (i.e., vw=vN < 0, vw=vF < 0). However, these monotonic properties do not necessarily hold in the WGC regime with Qw0 þ 1 < 0.

36 Supply and Demand Management in Ride-Sourcing Markets

2.3.3 Batch-matching process In actual operations, some ride-sourcing platforms may implement a batchmatching process to accumulate waiting passengers and idle drivers in the matching pool until the end of a batch window, at which point matching is performed. Xu et al. (2017) established a deductive model to delineate the matching process of a batch-wise matching scheme. This assumes that a platform matches all mutually closest pairs of drivers and passengers at each matching time and that pairs in which at least one member is not ‘optimal’ for the other member are carried over to the next matching-time interval. Yang et al. (2020) further extended the model to take into account the effects of two key decision variables d matching (batch)-time interval and matching radius d on matching efficiency. They assumed that a platform accumulates several waiting passengers and idle drivers over a matching time interval, and only matches drivers and passengers with a pick-up distance less than the matching radius at the end of the matching time interval. Those passengers and drivers who are not matched in the current matching-time interval either leave the market or are carried over to the next interval. The models of Xu et al. (2017) and Yang et al. (2020) assume that idle drivers are uniformly distributed and that waiting passengers follow a spatial Poisson point process (Chiu et al., 2013). However, Xu et al. (2017) also attempted to describe a stationary market equilibrium state in which the arrival rates of passengers and drivers, and N c and Nv, remain unchanged over time intervals. In contrast, Yang et al. (2020) considered a dynamic system in which the arrival rates of passengers and drivers and N c and Nv may vary across time. Moreover, the model of Xu et al. (2017) was designed to assist a government to impose appropriate regulations at the planning level, whereas the model of Yang et al. (2020) was designed to help a platform better operate its ridesourcing system by tuning the matching time interval and radius at the operational level. Next, we briefly introduce the core settings of these two models and demonstrate how they approximate w, which comprises matching time and pick-up time. We focus on the model of Xu et al. (2017) to present the fundamentals of equilibrium analyses in ride-sourcing markets. To facilitate our analysis, we assume that the entire analysis period is discretised into various time intervals of equal length s, and consider a simplified matching process. Given N c and N v , the platform loops through each waiting passenger and finds the closest idle driver. The model delineating the matching process should satisfy the following condition: sequence dependence does not affect the matching outcomes. As illustrated in Fig. 2.1A, we consider a matching time interval with two passengers (A and B) and one vehicle (C) in the matching pool. If the platform first matches A and then B, A is matched with C, and B is unmatched. However, the pick-up distance between B and C is shorter than that between A and C. Thus, the A-to-B

Fundamentals of ride-sourcing market equilibrium analyses Chapter | 2

37

FIGURE 2.1 Illustrations of the matching process, where R ¼ matching radius. (A) Sequence dependence (B) Dominant zones of passengers. Adapted from Yang, H., Qin, X., Ke, J., Ye, J., 2020. Optimizing matching time interval and matching radius in on-demand ride-sourcing markets. Transportation Research Part B: Methodological 131, 84e105.

matching sequence has less efficient system performance than the opposite matching sequence (B-to-A). To avoid these conflicts of sequence dependence, we introduce the notion of a dominant zone and a few criteria for matching every mutually closest driverepassenger pair at each time step. The dominant zone of each passenger refers to the passenger’s neighbouring area, within which the distance from any point to the passenger is shorter than that from the point to any other waiting passenger (Fig. 2.1B). During each time interval, a waiting passenger is matched with an idle driver if and only if the following conditions are satisfied. First, the idle driver must lie within the dominant zone of the waiting passenger. Second, the idle driver must be the closest idle driver to the waiting passenger. We assume that waiting passengers are uniformly distributed over the examined space, whereas the spatial distribution of idle drivers follows a spatial Poisson point process; this assumption has been widely used in the literature (Arnott, 1996; Xu et al., 2017; Yang et al., 2020). Suppose that the area of the studied space is A; then, the densities of passengers and drivers (estimated values), which are respectively denoted by rc and rv , are given by rc ¼ N c =A and rv ¼ N v =A. As the area of each passenger’s dominant zone equals ðrc Þ1 ¼ A=N c and the density of drivers is rv , the mean N v within each dominant zone is equal to ðrc Þ1 $rv ¼ N v =N c . Therefore, the probability that n idle drivers are within the dominant zone of each waiting passenger can be written as follows:
v 
v n 1 N N Pfng ¼ exp  c  c (2.39) n! N N

38 Supply and Demand Management in Ride-Sourcing Markets

The probability of each passenger being matched equals the probability of having at least one idle driver within each passenger’s matching area, which is given by:
v N PM ¼ 1 Pf0g ¼ 1 exp  c (2.40) N In this model, passengers’ matching time and pick-up time are separately considered. Let wc denote the average passenger matching time, wm denote the average pick-up time, and wv denote drivers’ average idle/waiting time. Given s, wc is given by: s
v wc ¼ (2.41) N 1 exp  c N In a stationary process, the arrival rates of waiting passengers and idle drivers are equal, and thus the relationship N c =N v ¼ wc =wv always holds. By substituting this relationship into Eq. (2.41), we obtain: s
v  wv ¼ c  (2.42) N N 1 exp  Nv Nc Hence, both wc and wv depend on the ratio N v =N c . If N v [N c , then wc y s and wv yN v s=N c . This indicates that if N v is greater than N c , passengers can be matched almost immediately after they submit an order, whereas idle drivers must wait for several intervals before they are matched, and vice versa. We next derive wm as a function of N v and N c , based on the same assumption discussed above. The distance from each waiting passenger to the closest driver, which is denoted as x, has a cumulative distribution function (CDF) HðxÞ and a probability density function (PDF) hðxÞ. If the distribution of idle drivers follows a spatial Poisson distribution, the CDF HðxÞ and PDF hðxÞ are as follows:   HðxÞ ¼ 1 Pf0g ¼ 1 exp px2 rv ; x  0 (2.43)   hðxÞ ¼ 2pxrv exp px2 rv ; x  0 (2.44) As mentioned, the waiting passengers are assumed to be distributed uniformly, which enables us to approximate the dominant zone of each passenger pffiffiffiffiffiffiffiffi as a disc with a radius ðrÞ ¼ 1= prc . Thus, the expected pick-up distance is m given by w , as follows:  Z r xhðxÞdx wm y9 HðxÞ1 0

Fundamentals of ride-sourcing market equilibrium analyses Chapter | 2

39


rffiffiffiffivffi
v  9 1 r 1 r
v  pffiffiffiffivffi erf ffiffiffiffiffiffiffi ffi p  exp  c r 2 r prc rc r 1 exp  c r 3 2
rffiffiffiffiffivffi
v 9 1 N 1 N 7
v 6 ¼ rffiffiffiffiffiffiffifficffi exp  c 5 (2.45) 4 rffiffiffiffiffivffi erf c N N N N pN 1 exp  c 2 N A A R x t2 2 where erf ðxÞ ¼ pffiffipffi 0 e dt is a Gaussian error function and 9 is a detour ¼

ratio, which is the ratio of the actual road distance to the Euclidean distance. Yang et al. (2018) found that the distribution of the detour ratios can be characterised by a universal horn-shaped distribution law. The mean of these detour ratios must be inversely proportional to the Euclidean distance with an intercept, such that the mean approaches a constant of approximately 1.27 when the Euclidean distance is sufficiently long. This constant has been empirically and theoretically identified in the literature (e.g., Arnott, 1996). Accordingly, w can be expressed as a sum of matching time and pick-up time, as follows: 3 2
rffiffiffiffiffivffi
v 9 1 N 1 N 7
v 6 wy rffiffiffiffiffiffiffifficffi exp  c 5 4 rffiffiffiffiffivffi erf c N N N N pN 1 exp  c 2 N (2.46) A A s
v þ N 1 exp  c N where the RHS is a complex non-linear function of N v and N c . The equilibrium can therefore be solved using a system of the following nonlinear simultaneous equations: the Q function (Eq. 2.1), the supply function (Eq. 2.2), the market-clearing condition (Eq. 2.10), the vehicle conservation equation (Eq. 2.28), and the w approximation (Eq. 2.46). However, although this deductive model effectively approximates w and describes the market equilibrium in a realistic batch-matching setting, it is too complicated for analytical derivations.

2.4 Market measures The interests of a government and a monopoly platform may partially conflict, as a government is most interested in achieving the social optimum (SO) that maximises the total social welfare, while a monopoly platform is most interested in achieving the monopoly optimum (MO) that maximises its profit. The SO is generally associated with a negative platform profit and is thus

40 Supply and Demand Management in Ride-Sourcing Markets

unsustainable; as a result, a government may attempt to strike a balance between the MO and SO. To this end, a government can impose regulations to induce a monopoly platform to choose a set of decision variables that lead to a desirable market outcome. Such a set of decision variables is said to be Paretoefficient if the resulting platform profit cannot be improved without reducing the social welfare, and vice versa. In this section, we discuss the properties of MO, SO, and Pareto-efficient solutions for different models. We only give the analytical results of the models that are analytically tractable, as some models are analytically intractable.

2.4.1 Monopoly optimum We first consider a monopoly market in which a ride-sourcing platform aims to maximise its profit (P) by optimising the twomajor decision variables: F and E. The optimal operating strategies of a monopoly platform can be determined by solving the following optimisation problem: max PðF; EÞ ¼ ðF  EÞQ

(2.47)

subject to the Q function (Eq. 2.1), supply function (Eq. 2.3) and an endogenous waiting-time function w ¼ wðQ; NÞ depending on the matching frictions. Based on Eq. (2.4), Eq. (2.47) can be rewritten as:
 1 N max PðF; NÞ ¼ FQ NG (2.48) N subject to Eq. (2.1) and w ¼ wðQ; NÞ. Then, we can determine the MO trip  ) and vehicle fleet size (N  ). The resulting monopoly optimal defare (Fmo mo mand (Qmo ) can be obtained from the equilibrium equations (Eq. 2.1 and w ¼  ) can be obtained from Eq. (2.4). For wðQ; NÞ), and then the optimal wage (Emo the sake of simplicity, we let
 N CðNÞ ¼ NG1 (2.49) N where CðNÞ is a strictly increasing function of N. It can be shown that CðNÞ exhibits constant returns to scale, namely CðNÞ ¼ cN, under the assumptions of sufficient supply and homogeneous r made by Zha et al. (2016). In contrast, CðNÞ exhibits increasing returns to scale for any increasing function Gð $Þ, which can be derived from a heterogeneous distribution of r (such as the assumption made by Bai et al., 2019). To summarise, the MO solutions can be determined by solving max PðF; NÞ ¼ FQ CðNÞ

(2.50)

where Q is endogenously given by Eq. (2.1) and w ¼ wðQ; NÞ, while CðNÞ represents the cost of recruiting N drivers from the labour market and exhibits

Fundamentals of ride-sourcing market equilibrium analyses Chapter | 2

41

constant or increasing returns to scale depending on the distinct assumptions regarding supply. As discussed above, Eq. (2.47) can be solved by equivalently solving Eq. (2.50), which treats F and N as proxy decision variables. Therefore, for the sake of neat and concise mathematical expressions, we hereafter solve Eq. (2.50) rather than Eq. (2.47). The first-order conditions of Eq. (2.50) are given by: vP vQ ¼Q þ F ¼0 vF vF

(2.51)

vP vQ ¼F C 0 ðNÞ ¼ 0 vN vN

(2.52)

where vQ=vF and vQ=vN depend on the market equilibrium and have different formulations in models that use different ways to characterise  , N  , and matching frictions. The first-order conditions determine Qmo , Fmo mo  the average waiting time at the MO (wmo ). Next, we analytically examine the first-order conditions in various models with distinct assumptions and settings.

2.4.1.1 Production-function-based model In the inductive model with a CobbeDouglas-type matching function, the comparative static effects of regulatory variables on Q are given by Eqs. (2.16)e(2.19). Substituting these equations into the first-order conditions Eqs. (2.51) and (2.52) gives rise to c    a1 Nmo C 0 Nmo ¼b v a2 Nmo

   v  1  a1  a2 0    v Qmo  wmo þ t þ ¼ C0 Nmo C Nmo wmo  0 Fmo a1 fmo

(2.53) (2.54)

v c where wv idle mo , Nmo , and Nmo are drivers’ average idle time, a number of   0 and C 0 N  vehicles, and a number of waiting passengers at the MO, and fmo mo are the derivatives of the Q function and driver-recruiting cost function at the MO. Eq. (2.54) follows the form of the Lerner formula (Lerner, 1934), with the RHS consisting of three terms:the marginal  cost of recruiting a driver to serve  a new passenger (C 0 Nmo wv þ t ), a ‘matching externality’ term mo    v  0 1a1 a2 0 ), which ( a1 C Nmo wmo ), and the monopoly markup (Qmo fmo represents the market power of the monopoly platform to determine the price. The sign of the matching externality with respect to the matching rate is related to the return to scale given by a1 þ a2 ; that is, if the matching function exhibits increasing, constant, or decreasing returns to scale (a1 þ a2 > 1, a1 þ a2 ¼ 1, or a1 þ a2 < 1), the matching externality is negative, zero, or positive, respectively.

42 Supply and Demand Management in Ride-Sourcing Markets

2.4.1.2 Queuing model In the M/M/1 queuing model, the comparative static effects of regulatory variables on Q are given by Eqs. (2.24)e(2.25). Substituting these equations into the first-order conditions Eq. (2.51) and Eq. (2.52) gives rise to    2     2  Umo Umo 0 C Nmo ¼ b  (2.55) 2   Nmo 1  Umo    t C 0 Nmo Qmo  Fmo ¼ (2.56)  2  0  1 fmo 1  Umo  is the utilisation rate at the MO. Eq. (2.56) represents the optimal where Umo  ) as the sum of two terms: a term associated with t, trip fare at the MO (Fmo     ; and a moUmo , and the marginal cost of recruiting a new driver C 0 Nmo  0 ). The former term may indicate the marginal nopoly markup term (Qmo fmo cost of recruiting a driver to serve a new passenger, taking into account the U of drivers.

2.4.1.3 FCFS-based model The comparative static effects of regulatory variables on Q in the deductive model based on the FCFS scheme are given by Eqs. (2.35) and (2.36). By substituting these two equations into the first-order conditions (Eqs. 2.51 and 2.52), we readily obtain:      0  C 0 Nmo (2.57) Qmo wmo þ 1 ¼  bQmo w0mo  Q      Fmo ¼ C0 Nmo wmo þ t  0mo  fmo

(2.58)



where w0mo is the derivative of the waiting function at the MO. Eq. (2.58) follows the form of the Lerner formula (Lerner, 1934), in which the RHS consists of two terms: the marginal cost of recruiting a driverto serve a  wmo þ t ) passenger in both the in-trip phase and the pick-up phase (C 0 Nmo   v   0 ). In addition, as w0  and the monopoly mark-up (Qmo fmo mo ¼ dwmo dNmo , Eq. (2.57) can be re-written as: 
    dwmo dw (2.59) Qmo v þ 1 ¼  bQmo mo C0 Nmo v dNmo dNmo which is equivalent to:     v  $ dNmo þ Qmo dwmo ¼ bQmo dwmo C0 Nmo

(2.60)

The LHS of Eq. (2.60) represents the sum of the marginal cost of  $dN v ) and pickrecruiting and operating vehicles in the idle phase (C0 Nmo mo    Q dw ), while the RHS represents the marginal w cost up phase (C 0 Nmo mo mo

Fundamentals of ride-sourcing market equilibrium analyses Chapter | 2

43

(bQmo dwmo ). This implies that the total marginal cost of operating vehicles in the idle and pick-up phases (which governs the service quality in terms of passengers’ pick-up time) is equal to the marginal pick-up time cost of pas sengers at the MO. In addition, from Eq. (2.57) and the fact that w0mo < 0, we  have Qmo w0mo þ 1 > 0, which leads to the following lemma. Lemma 2e1. Under the FCFS scheme, the MO in the non-pooling ridesourcing market is always located in the normal regime rather than in the WGC regime. This lemma indicates that in the ideal scenario in which a platform can freely choose an F and N to maximise its P, it will make optimal decisions such that the market does not fall into the inefficient (i.e., WGC) regime.

2.4.2 Social optimum We now consider the first-best SO, which is an ideal scenario in which the regulatory variables (F and E or N) are chosen such that the total social welfare (S) is maximised. In ride-sourcing markets, S is generally defined as the sum of the consumer (passenger) surplus, provider (driver) surplus, and P. The consumer surplus (CS) is the integral of the Q function from the market equilibrium point to the maximum generalised cost, which is written as: Z Q CS ¼ f 1 ðzÞdz ½F þ b$ðw þ tÞQ (2.61) 0

In addition, the surplus of a provider who participates in the platform with r  R ¼ EQ=N is given by EQ=N  r, and thus the total provider surplus (PS) is given by:  ZN=N EQ  r dGðrÞ ¼ EQ  N rdGðrÞ N

N=N


Z PS ¼ N 0

(2.62)

0

For simplicity, we let: Z

N=N

Cp ðNÞ ¼ N

rdGðrÞ

(2.63)

0

which is denoted the PS cost function, and only depends on (and is clearly an increasing function of) N. Specifically, if the supply is sufficient and r is homogeneous, as assumed by Zha et al. (2016), r ¼ R ¼ EQ=N at equilibrium. Thus, PS ¼ 0 and Cp ðNÞ ¼ EQ ¼ CðNÞ. As mentioned, in this scenario the driver-recruiting cost function (CðNÞ) exhibits constant returns to scale, i.e., it is linearly proportional to N and can be written as CðNÞ ¼ cN. S, and thus equals CS þ PS þ P. The first-best SO can be obtained by solving the following equation to maximise S as a function of F and E:

44 Supply and Demand Management in Ride-Sourcing Markets

Z

Q

max SðF; EÞ ¼ ðF  EÞQ þ "

0

f 1 ðzÞdz ½F þ b$ðw þ tÞQ # Z N=N

þ EQ 

 (2.64)

rdGðrÞ 0

subject to the Q function (2.1), the supply function (2.3), and an endogenous w function (w ¼ wðQ; NÞ) that depends on the matching frictions. Clearly, we can also write Eq. (2.64) as a maximisation problem of F and N, as follows: Z Q f 1 ðzÞdz b$ðw þ tÞQ Cp ðNÞ (2.65) max SðF; NÞ ¼ 0

where the endogenous variables Q and w are given by the market equilibrium and governed by both F and N, and Cp ðNÞ only depends on N. This equation  ) and vehicle fleet size (N  ), which we can use obtains the SO trip fare (Fso so with equilibrium conditions to calculate the resulting SO passenger demand  ). Similar to the MO, the SO can be solved via Eq. (2.64) (Qso ) and wage (Eso or Eq. (2.65); hereafter, we solve Eq. (2.65), which treats F and N as proxy decision variables, to obtain concise mathematical expressions. The first-order conditions of the S maximisation problem (Eq. 2.65) are given by: vS vQ vw ¼ 00F ¼ bQ$ vF vF vF

(2.66)

vS vQ vw ¼ 00Cp0 ðNÞ ¼ F  bQ$ vN vN vF

(2.67)

Similarly, the first-order conditions of the SO depend on the comparative static effects of regulatory variables (F and N) on endogenous variables such as Q and w, which vary depending on the assumptions and approximations  , N  , and w used in models. These first-order conditions will obtain Qso , Fso so so at the SO. Next, we examine the analytical first-order conditions of the SO in various models with distinct assumptions and settings.

2.4.2.1 Production-function-based model The general formulations of the first-order conditions at the SO are given by Eqs. (2.66) and (2.67), which do not have specific formulations of vQ= vF, vQ=vN, vw=vF, or vw=vN. By substituting the specific partial derivatives in the production function-based model given by Eqs. (2.16)e(2.21) into the first-order conditions of Eqs. (2.66) and (2.67), we obtain: c   a1 Nso Cp0 Nso ¼b v a2 Nso

(2.68)

Fundamentals of ride-sourcing market equilibrium analyses Chapter | 2

   v  1  a1  a2 0    v  Fso ¼ Cp0 Nso Cp Nso wso wso þ t þ a1

45

(2.69)

 , N  , wv , Q , N v , and N c are the trip fare, the vehicle fleet size, where Fso so so so so so the drivers’ average idle time, the passenger demand, the number of idle vehicles,   and the number of waiting passengers at the SO, respectively, and Cp0 Nso are the derivatives of the PS cost function at the SO. Clearly, Eq. (2.68) at the SO has the same form as Eq. (2.53) at the MO, but unlike the pricing formula (Eq. 2.54) at the MO, the SO pricing formula (Eq. 2.69) does not contain the monopoly mark-up. Next, we examine P at the first-best SO (Pso ), which is given by:     Qso C Nso Pso ¼ Fso (2.70)

Substituting Eq. (2.69) into Eq. (2.70) yields:    v  1  a1  a2 0    v   wso þ t þ Cp Nso wso C Nso Pso ¼ Cp0 Nso a1

(2.71)

if the supply is sufficient and r is homogeneous, then Specifically,  so  ¼ c and C N   Cp0 Nso so ¼ cNso , and thus P becomes: Pso ¼

1  a1  a2 v cwso a1

(2.72)

In this case, Pso < 0 if the matching function exhibits increasing returns to scale (i.e., a1 þ a2 > 1), Pso ¼ 0 if the matching function exhibits constant returns to scale (i.e., a1 þ a2 ¼ 1), and Pso > 0 if the matching function exhibits decreasing returns to scale (i.e., a1 þ a2 < 1). This implies that if the matching function exhibits increasing returns to scale, the SO is unachievable without a government subsidy.

2.4.2.2 Queuing model By substituting the specific partial derivatives in the M/M/1 queuing model given by Eqs. (2.24)e(2.27) into the first-order conditions in Eq. (2.66) and Eq. (2.67), we find that:    2    2  Uso Uso 0 Cp Nso ¼ b  (2.73) 2   Nso 1  Uso   t Cp0 Nso  Fso ¼ (2.74)  2  1  Uso  1 

 is the utilisation rate at the SO and f 0 is the derivative of the Q where Uso so function at the SO. Clearly, Eq. (2.73) at the SO has the same form as Eq. (2.55) at the MO, but unlike the pricing formula at the MO (Eq. 2.56), the

46 Supply and Demand Management in Ride-Sourcing Markets

pricing formula at the SO (Eq. 2.74) does not contain the monopoly mark-up. We next examine Pso , which is given by:     Qso C Nso Pso ¼ Fso (2.75) Substituting Eq. (2.74) into Eq. (2.75) yields:     t Cp0 Nso so  P ¼  2 Qso C Nso  1 1  Uso

(2.76)

Clearly, the sign of Pso depends on the forms of Cp ðNÞ and CðNÞ. In   ¼c particular, if the supply is sufficient and r is homogeneous, then Cp0 Nso    , and thus Pso becomes: and C Nso ¼ cNso  Uso 1  0 and C 0 ðNÞ > 0, we must have Q w0 þ 1 > 1, which leads to the following lemma. Lemma 2e3. Under the FCFS scheme, all solutions along the Pareto-efficient frontier in the non-pooling ride-sourcing market lie in the normal regime rather than in the WGC regime. This lemma states that the use of a Pareto-efficient set of regulatory variables ensures that the ride-sourcing market does not fail, i.e., the WGC regime can be avoided over the long term by the use of operating strategies that meet the objectives of both a government and a ride-sourcing platform. However, in short-term operations, the time-varying fluctuations in Q mean that a WGC regime can occur in peak-usage periods due to rapid Q surges. In these situations, surge pricing can be utilised to suppress Q and thereby rescue the system from market failure.

2.5 Discussion The above-described models each have advantages and disadvantages, and thus specific contexts in which they are applicable. The perfect matching model is simple but fails to capture the major characteristic of ride-sourcing markets: the matching frictions between demand and supply. The production function-based model is another type of inductive approach and uses a hypothesised matching function to delineate matching frictions. Its major advantages are that it (1) captures the effects of the pool size of two groups of agents (passengers and drivers) on matching efficiency, and (2) effectively describes system performance under different matching conditions that exhibit increasing, constant, or decreasing returns to scale. Its major disadvantages are that it (1) lacks micro-foundations, and (2) does not differentiate between matching time and pick-up time. Thus, it cannot reflect the endogenous interactions between the pick-up time and vehicle availability and U. In contrast to the above inductive approaches, which directly define the matching functions, deductive approaches derive the matching functions/ models based on certain assumed micro-foundations. For example, queuing models assume that the matching process is characterised by a queuing system, in which drivers are treated as service providers and passengers are treated as randomly arriving customers. However, only a few queuing models, such as

Fundamentals of ride-sourcing market equilibrium analyses Chapter | 2

51

the M/M/1 queuing model, can derive analytical solutions for optimal operating strategies. In addition, queuing models focus only on the matching time, which means that they implicitly assume that w after online matching (i.e., the pick-up time) is negligible in comparison with the matching time, or treat the pick-up time as a constant and as part of t. In contrast, the FCFS model assumes that upon arrival a passenger is immediately matched to the nearest idle driver, and thus the matching time is negligible. Thus, the FCFS model has the advantage of effectively defining the vehicle conservation equation, in which the vehicle fleet consists of vehicles in one of three phases: an idle phase, a pick-up phase, or an in-trip phase. This means that the FCFS model can describe a market failure called a WGC, in which drivers spend an excessive length of time in a pick-up phase. However, the FCFS model has the disadvantage of strictly assuming that the matching time is zero. The batchmatching model overcomes the disadvantages of the queuing model and FCFS model by simultaneously incorporating the matching time and pick-up time. However, the batch-matching model is too complicated to be analytically tractable. Overall, the FCFS model is a better approximation for a market in which a platform matches passengers rapidly, even if distant matching (and thus a long pick-up time) is required. In contrast, queuing models and the production function-based model are better approximations for a market in which a platform only matches passengers and drivers with short pick-up distances or delays matching to accumulate a sufficient N c and N v in the matching pool to facilitate better matching (i.e., shorter pick-up times). However, although the batch-matching model can precisely characterise the batch-matching mechanism for different matching radii, it is too complicated for theoretical analysis. Accordingly, the WGC phenomenon can only be analytically identified by a deductive model based on the FCFS scheme. Li et al. (2021) developed a general matching model for ride-sourcing markets that can be regarded as a unified version of different inductive and deductive matching models (or functions). Specifically, under certain conditions, the general matching model can be reduced to the CobbeDouglas-based production function, the FCFS model, and a deterministic queuing model. The major assumptions and treatments that are used in the abovementioned models are summarised in Table 2.1. In the next chapter, we analytically examine the properties of Paretoefficient solutions and discuss the effects of various regulations.

52 Supply and Demand Management in Ride-Sourcing Markets

TABLE 2.1 Summary of the inductive and deductive models.

Model

Type

Major assumption

Waiting time governed by

Perfect matching

Inductive

No matching friction.

Zero

Yes

Production function

Inductive

Matching frictions are approximated by a Cobb eDouglas-based production function

Matching and pickup time

Yes

Queuing model

Deductive

A queuing system with drivers as service providers and randomly arriving passengers as customers

Matching time

Yes

FCFS

Deductive

A passenger is immediately matched online based on an FCFS scheme

Pick-up time

Yes

Batch matching

Deductive

A batch-matching mechanism is utilised to match accumulated idle drivers and waiting passengers

Matching time and pick-up time

No

Analytically tractable?

Note. FCFS, first-come-first-served.

Glossary of notation E PS F t b w Q N mcLv U r N U Tv Nv

the average wage (expense) paid to a driver per order total provider surplus average trip fare average trip time passengers have a homogeneous value of time average waiting time of passengers passenger demand vehicle fleet size meeting rate (the number of passengeredriver pairs matched per hour) the average income per driver per hour drivers’ reservation rate total number of potential drivers a maximum reservation rate the arrival rate of drivers number of vacant vehicles

Fundamentals of ride-sourcing market equilibrium analyses Chapter | 2 Nc wv U A rc rv wc wm wv s x Rx 2 erf ðxÞ [ p2ffiffipffi 0 eLt dt 9 P Fmo  Nmo Qmo Emo wv mo v Nmo c Nmo  0 fmo   C0 Nmo Fso  Nso Qso Eso wv so v Nso c Nso Pso  Uso 0 fso x Q F N w wv N v N c f0

53

number of waiting passengers average idle/vacant time of drivers utilisation rate area of the studied space densities of passengers (estimated values) densities of drivers (estimated values) average passenger matching time average pick-up time drivers’ average idle/waiting time matching time interval distance from each waiting passenger to the closest driver a Gaussian error function a detour ratio (ratio of the actual road distance to the Euclidean distance) profit the trip fare at the MO vehicle fleet size at the MO resulting in monopoly optimal demand at the MO the optimal wage at the MO drivers’ average idle time at the MO number of idle vehicles at the MO number of waiting passengers at the MO derivative of the Q function at the MO derivatives of the driver-recruiting cost function at the MO the trip fare at the SO vehicle fleet size at the SO resulting passenger demand at the SO resulting wage at the SO drivers’ average idle time at the SO the number of idle vehicles at the SO the number of waiting passengers at the SO P at the first-best SO utilisation rate at the SO derivative of the Q function at the SO Lagrange multiplier passenger demand at Pareto-efficient solutions the trip fare at Pareto-efficient solutions vehicle fleet size at Pareto-efficient solutions average waiting time of passengers at Pareto-efficient solutions drivers’ average idle time at Pareto-efficient solutions number of idle vehicles at Pareto-efficient solutions number of waiting passengers at Pareto-efficient solutions derivative of the Q function

References Arnott, R., 1996. Taxi travel should be subsidized. Journal of Urban Economics 40 (3), 316e333. Bai, J., So, K.C., Tang, C.S., Chen, X., Wang, H., 2019. Coordinating supply and demand on an ondemand service platform with impatient customers. Manufacturing & Service Operations Management 21 (3), 556e570.

54 Supply and Demand Management in Ride-Sourcing Markets Castillo, J.C., Knoepfle, D., Weyl, G., 2017. Surge pricing solves the wild goose chase. In: Proceedings of the 2017 ACM Conference on Economics and Computation. ACM, pp. 241e242. Chiu, S.N., Stoyan, D., Kendall, W.S., Mecke, J., 2013. Stochastic Geometry and its Applications. John Wiley & Sons. Geoffrion, A.M., 1967. Solving bicriterion mathematical programs. Operations Research 15 (1), 39e54. Hu, M., Zhou, Y., August 31, 2020. Price, wage, and fixed commission in on-demand matching. Available at: SSRN: https://doi.org/10.2139/ssrn.2949513. Ke, J., Yang, H., Li, X., Wang, H., Ye, J., 2020. Pricing and equilibrium in an on-demand ridepooling market. Transportation Research B: Methodological 139, 411e431. Lerner, A.P., 1934. The concept of monopoly and the measurement of monopoly power. The Review of Economic Studies 1, 157e175. Li, X., Ke, J., Yang, H., Wang, H., Zhou, Y., September 1, 2021. A general matching model for ondemand mobility services. Available at: SSRN: https://doi.org/10.2139/ssrn.3915450. Xu, Z., Yin, Y., Zha, L., 2017. Optimal parking provision for ride-sourcing services. Transportation Research Part B: Methodological 105, 559e578. Yang, H., Ke, J., Ye, J., 2018. A universal distribution law of network detour ratios. Transportation Research C: Emerging Technologies 96, 22e37. Yang, H., Qin, X., Ke, J., Ye, J., 2020. Optimizing matching time interval and matching radius in on-demand ride-sourcing markets. Transportation Research Part B: Methodological 131, 84e105. Yang, H., Yang, T., 2011. Equilibrium properties of taxi markets with search frictions. Transportation Research Part B: Methodological 45 (4), 696e713. Zha, L., Yin, Y., Yang, H., 2016. Economic analysis of ride-sourcing markets. Transportation Research Part C: Emerging Technologies 71, 249e266.

Chapter 3

Calibration and validation of matching functions for ridesourcing markets Shuqing Wei1, Siyuan Feng1, Jintao Ke2 and Hai Yang1 1 Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China; 2Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China

3.1 Introduction Ride-sourcing services use information sharing to perform efficient and rapid online matching of waiting passengers with idle drivers, and thus have lower matching frictions than conventional street-hailing taxi services. The development of ride-sourcing services has accelerated in the past decade, and they have become an indispensable component of urban mobility systems and have attracted the attention of academia, industry, the media, and the public (Wang et al., 2021). Ride-sourcing markets are a representative two-sided market and have five groups of stakeholders who need to be considered: drivers, passengers, platforms, policymakers, and the general public (Wang and Yang, 2019). Various aspects of the supply side of ride-sourcing markets have been examined, such as independent drivers’ willingness to participate in markets (Chen et al., 2019; Hall and Krueger, 2018) and their contribution and schedule of working hours (Farber, 2005; Hu and Zhou, 2020; Zha et al., 2017). Similarly, various aspects of the demand side of ride-sourcing markets have been examined, such as the factors affecting passenger demand (Alemi et al., 2018; Gilibert et al., 2017; Peled et al., 2021; Zheng et al., 2019), the prediction of spatiotemporal demand (Feng et al., 2021; Ke et al., 2017, 2021; Saadi et al., 2017; Yao et al., 2018), and passenger travel mode choice (Agarwal et al., 2019; Hwang et al., 2018; Salnikov et al., 2015). Other important areas that have been studied are the design of platform operating strategies (Lee and Savelsbergh, 2015; Vazifeh et al., 2018) and governments’ regulatory policies (Aarhaug and Olsen, 2018; Rogers, 2015; Yu et al., 2020), and how ride-sourcing services affect society (Anderson, 2014; Clewlow and Mishra, 2017). The above studies have demonstrated that ride-sourcing Supply and Demand Management in Ride-Sourcing Markets. https://doi.org/10.1016/B978-0-443-18937-1.00006-1 Copyright © 2023 Elsevier Inc. All rights reserved.

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56 Supply and Demand Management in Ride-Sourcing Markets

markets must be accurately characterised to achieve an understanding of drivers’ and passengers’ behaviours, and to solve problems associated with platform operations and government regulations. A variety of mathematical models have been devised for accurately delineating the on-demand matching process used in ride-sourcing markets. A crucial component of these models is a matching function, which is used to characterise the matching frictions between passengers and drivers. A unique characteristic of ride-sourcing markets is that they involve the online matching of a passenger with a driver, followed by the driver picking up the passenger after a certain period. As such, a matching function is used to describe how the density or mass of supply and demand influence key market metrics, namely the matching rate (the number of orders matched per unit time), passengers’ average matching time (the time from an order being requested to online matching occurring), passengers’ average pick-up time (the time from online matching to a driver picking up a passenger), and passengers’ average total waiting time (the sum of average matching time and pick-up time). A matching function can thus reveal the intriguing relationship between platform decision variables (such as matching rules, price, and wage) and system endogenous variables (such as demand, supply, and key market metrics), and can be used to design optimal operating strategies to maximise platform profit or social welfare. Two types of approaches are used to establish matching functions: inductive and deductive approaches. Inductive approaches make no assumptions about the matching mechanism used by a platform and employ a matching function to characterise the relationship between the matching rate, the number of passengers and drivers, and passengers’ and drivers’ waiting time. Typical examples of matching functions are the perfect matching function (Mo et al., 2020; Yu et al., 2020) and CobbeDouglas-type matching functions (Yang and Yang, 2011; Zha et al., 2016). Conversely, deductive approaches assume that a platform adopts a particular matching mechanism and thus uses the micro-foundations of this mechanism as a basis for the derivation of a matching function. For example, Castillo et al. (2017) assumed that a platform immediately matches a nearby idle driver with a passenger who makes a request and that a market has at least one idle driver. Thus, at equilibrium, drivers must be in a vacant state (idle), a pick-up state (on the way to pick up a passenger), or an occupied state (on the way to drop off a passenger). Due to its analytical tractability, this approach has been widely adopted to address other operational issues, such as those of ride-pooling services (Ke et al., 2020). Xu et al. (2017) assumed that the locations of idle drivers and waiting passengers follow a uniform distribution and a spatial Poisson distribution, respectively, under a batch-matching mechanism; thus, they derived the analytical formulas of idle drivers’ average searching/waiting time, and passengers’ expected matching time and pick-up time. Queuing models have also been employed to approximate passengers’ queuing time or matching time (Bai et al., 2019; Feng et al., 2020).

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Matching functions may differ in terms of their applied ranges, depending on the assumptions and derivations of the models. For example, Castillo et al. (2017) assumed that the driver supply is sufficient to cover all demand and thus their model is based on an instant matching mechanism. This means that the matching or queuing time of passengers is always zero and therefore passengers’ waiting time is governed by their pick-up time. Consequently, this model may not be a good fit for a market in which passengers experience a long matching or queuing time. In contrast, many queuing models have been devised that focus on passengers’ matching time but ignore their pick-up time (Banerjee et al., 2015, 2016; Hu et al., 2020). While most previous studies have assumed that the driver supply is sufficient to satisfy all realised demand, a few modelsdsuch as that of Xu et al. (2020)dconsider market scenarios in which the driver supply is insufficient to serve all passenger requests, such that some passengers have to leave the queue. This scenario may occur in peak hours when there is a long queue and the number of idle drivers is approximately zero, which means that any newly idle driver is immediately dispatched to a passenger. Despite the abovementioned studies’ development of theoretical models and use of these to solve operational problems, there is a lack of empirical evidence on these models’ performance in and applicability to real markets. That is, platforms and policymakers do not know which model best describes a ride-sourcing market with a given level of supply and demand. Traffic simulators have therefore been employed for the evaluation of various models (Gressai et al., 2021; Shi et al., 2021). Accordingly, this chapter conducts a simulation-based sensitivity analysis to evaluate and compare a few widely used matching functions in 420 ride-sourcing market scenarios. After the simulation for each scenario reaches a stationary state, the key market metrics are recorded are analysed. This reveals which model most accurately estimates the key market metrics, namely the matching rate, passengers’ average matching time, passengers’ average pick-up time, and passengers’ average total waiting time. The remainder of this chapter is organised as follows. Section 3.2 reviews the matching functions that are widely used for characterising ride-sourcing markets. Section 3.3 introduces our experiment and the mechanism of our simulator. Section 3.4 presents the experimental results and the analysis of matching functions in various market scenarios. Section 3.5 concludes.

3.2 Matching functions and market metrics In this section, we first briefly introduce the base model of the ride-sourcing market, including the concepts underpinning the model’s major components and the relationships between them. We then introduce the matching functions that have been widely adopted in the literature and which form the core of ride-sourcing models and test the performance of these matching functions in a simulation-based sensitivity analysis. Finally, we present the key market metrics for evaluating how well the models fit reality. We examine only static

58 Supply and Demand Management in Ride-Sourcing Markets

and stationary equilibrium models for ride-sourcing markets; dynamic and non-equilibrium market scenarios are beyond the scope of this work, although they are useful for designing dynamic pricing and matching strategies in shortterm and real-time operational horizons. The key notation of the matching functions is presented below.

3.2.1 Base model The ride-sourcing market is a typical two-sided market, with one side comprising passengers and the other comprising drivers. To describe the interactions between the two sides, a ride-sourcing model usually contains three components: a passenger demand function, a driver supply function, and a matching function. Potential passengers compare the ride-sourcing services offered by the market with other traffic modes and then decide whether to request a ride, while potential drivers determine whether to participate in the ride-sourcing market based on their expected income per hour. A matching function, as the core of the ride-sourcing model, describes the matching frictions between supply and demand. A typical aggregate model for characterising the stationary equilibrium state of ride-sourcing services has three major components: a demand function, a supply function, and a matching model. Let Q denote the passenger demand for ride-sourcing services (or arrival rate) and let N denote the number of active vehicles/drivers in service. Passenger demand Q can be written as a demand function of the trip fare p set by the platform and passengers’ waiting time, which consists of two components: the matching time (wm ) and passengers’ pick-up time (wp ), as follows:
 Q ¼ f p; wm ; wp (3.1) On the supply side, the number of drivers who decide to join the market (N) depends on the wage paid by the platform (u) and the vehicle utilisation rate (Qt=N). That is,   Qt (3.2) N ¼ G u; N where t is the average trip time. In addition to the above demand and supply functions, a matching function is required to characterise the on-demand process of matching supply with demand and its effects on the endogenous variables wm and wp . In general, a matching function comprises the following group of nonlinear simultaneous equations, which take Q and N as inputs and generate wm and wp as outputs: wm ; wp ¼ WðQ; N; QÞ

(3.3)

where Q represents the platform’s matching strategy, which also influences passengers’ waiting time. A matching model typically also includes a vehicle conservation equation, which accounts for the fact that the active vehicles are (1) idle or cruising streets to seek their next passengers, (2) travelling to pick

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up passengers after order confirmation, or (3) transporting passengers from their origin to their destination. Therefore, the stationary equilibrium of an ondemand mobility market can be determined by solving a system of equations consisting of a demand function, a supply function, and a matching function. A platform can thus determine the optimal price and wage by establishing a bilevel mathematical problem that maximises the platform profit or social welfare at the upper level and treats the market equilibrium conditions as constraints at the lower level. Most models assume that passengers wait until they are matched, meaning that mcv is always equal to 1; only some models consider passengers’ balking decisions due to a long waiting time. Previous studies have adopted similar demand functions and supply functions but various matching functions, with the latter based on different assumptions (Bai et al., 2019; Ke et al., 2020; Zha et al., 2016). In this chapter, we focus on how well these matching functions delineate the on-demand matching process of a ride-sourcing market. Therefore, we treat the demand rate Q and vehicle fleet size N as exogenous variables that are fed into the matching functions to generate estimates for the market metrics, i.e., mcv , wm , and wp . In addition, we use a simulation to generate the real market metrics for any inputs of Q and N, and then determine the ability of various matching functions to fit reality by comparing the estimated and real market metrics under various market scenarios.

3.2.2 Matching functions Two main types of matching functions are used to characterise the matching friction of an online ride-sourcing market: inductive and deductive matching functions. In this subsection, we introduce a few widely used inductive and deductive matching functions, which we evaluate later in this chapter.

3.2.2.1 Perfect matching The perfect matching function, which is the basis of an intuitive inductive model, characterises a market without matching friction, i.e., it assumes that at each moment, passengers and drivers arrive according to their respective arrival rates, and that neither customers nor drivers wait. Consequently, when Q is greater than the driver arrival rate, all drivers are matched whereas some passengers are not matched, and the unmatched passengers immediately leave; conversely, when Q is less than the driver arrival rate, all passengers are matched whereas some drivers are not matched, and the unmatched drivers immediately leave. In this scenario, wm and wp are both zero, and as shown in Eq. (3.4), mcv is given by the minimum Q and vacant vehicle arrival rate T v (estimated by service capacity N=t). Although the perfect matching function is usually used for the strategic operational management of ride-sourcing market

60 Supply and Demand Management in Ride-Sourcing Markets

regulations (Yu et al., 2020), it fails to capture passengers’ and drivers’ waiting time and cruising time, respectively.   N mcv ¼ minðQ; T v Þ ¼ min Q; (3.4) t

3.2.2.2 CobbeDouglas production function The CobbeDouglas production function is a well-known production function in labour economics, where it forms the basis of models that are used to measure the influence of input elements on an output. The CobbeDouglastype matching function was first used by Yang and Yang (2011) to represent the bilateral searching and meeting of passengers and taxi drivers, and was then employed by Zha et al. (2016) for characterising the on-demand matching process in ride-sourcing markets. It is worth noting that although Yang and Yang (2011) and Zha et al. (2016) use the same production function, they have different explanations for the variables. In Yang and Yang (2011), the pickup time is ignored because drivers can meet with passengers in a visible range in a street-hailing taxi market. In contrast, the passenger waiting time in Zha et al. (2016)’s model involves matching and pickup time, which however are not differentiated. In this study, we adopt the model proposed by Yang and Yang (2011) to characterize the ride-sourcing markets with a small matching radius, resembling street-hailing taxi markets. In this function, mcv is governed by the size of two pools that consist of idle vehicles and waiting passengers, respectively. A large number of waiting passengers (N c Þ and idle drivers results in a high mcv . The CobbeDouglastype matching function is also the basis of an inductive model, where mcv is calculated as follows: mcv ¼ A$ðN v Þa1 $ðN c Þa2 ¼ A$ðT v wv Þa1 $ðQwm Þa2

(3.5)

where, according to Little’s law, N c is equal to the product of Q and the average waiting time of these passengers (wm ), while the number of idle vehicles (N v ) is equal to T v ; the average waiting/idle time of vehicles is wv ; and A, a1 , and a2 are hyperparameters. At a stationary equilibrium state, the model further assumes that mcv ¼ T v ¼ Q

(3.6)

In addition, the equilibrium should satisfy the following vehicle conservation equation: N ¼ N v þ Qt

(3.7)

Given the values of A, a1 , and a2 , we can obtain the equilibrium solutions of wv and wm by solving a system of simultaneous equations Eqs. (3.5)e(3.7). In this model, mcv is assumed to be equal to the demand rate Q, implying that

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the model assumes that passengers who opt for the ride-sourcing services wait to be matched. It is worth noting that the mcv represents the rate of an eventual physical meeting between customers and drivers, namely, the number of passengers being picked up and served by drivers per unit of time. Before the model is evaluated and analysed, A, a1 , and a1 must be calibrated. To this end, we simulate 420 market scenarios consisting of various combinations of supply and demand, each of which we regard as a sample point. We collect the values of three key metrics (N v , N c , and mcv ) for each sample point, and then use all of the sample points to calibrate the hyperparameters via least-square methods. The values of A, a1 , and a1 that lead to the best calibration performance are then fed into the system of simultaneous equations to obtain the key market metrics for further evaluation.

3.2.2.3 M/M/1 queuing model Queuing models describe the queuing that occurs in service systems and have been adopted to delineate matching frictions in ride-sourcing markets (Bai et al., 2019; Tang et al., 2021). In a ride-sourcing scenario, the queuing time of a queuing model is used to estimate drivers’ idle time (Banerjee et al., 2015, 2016) or passengers’ waiting time (Hu et al., 2020). In this chapter, we only present and evaluate the fundamental queuing models, which we use to estimate wm , and we regard wp as being equal to zero. Queuing models are a kind of deductive model, as they assume that the arrival of passengers and drivers follows a certain distribution and derive the key metrics of this distribution. In the M/M/1 queuing model, we assume that the arrival of passengers follows the Poisson distribution, and regard the entire supply side as one service desk whose service rate is given by N=t. To guarantee the stability of a queuing system, the service rate must be greater than or equal to Q. Thus, the M/M/1 queuing model is only applicable to markets in which N=t > Q; in markets in which N=t  Q, the queuing system is unstable. Moreover, the model assumes that mcv is equal to Q Eq. (3.8), wp is equal to zero, and wm is given by Eq. (3.9). mcv ¼ Q wm ¼

Q   N N Q t t

(3.8) (3.9)

3.2.2.4 M/M/1/k queuing model The assumption of the M/M/1/k queuing model is basically the same as the assumption of the M/M/1 model, except regarding passengers’ balking behaviours. The parameter k represents the maximum queue length, and when the queue length equals k, passengers immediately leave the market rather than

62 Supply and Demand Management in Ride-Sourcing Markets

join the queue. mcv and wm are calculated as follows, and wp is regarded as equal to zero: r¼

(3.10)

1r 1  rkþ1

(3.11)

1r k r 1  rkþ1

(3.12)

P0 ¼ Pk ¼

Q N=t

e ¼ Q  ð1  Pk Þ mcv ¼ Q Ls ¼

k X

r ðk þ 1Þrkþ1  1r 1  rkþ1

iPi ¼

i¼0

Lq ¼

(3.13)

k X ði  1ÞPi ¼ Ls ð1  P0 Þ

(3.14)

(3.15)

i¼0

wm ¼

Lq Qð1  Pk Þ

(3.16)

where Pi is the probability that there are i passengers in the queue, Ls is the average number of passengers in the system (including the passengers being served), and Lq is the average number of passengers in the queue (not including the passengers being served).

3.2.2.5 M/M/N queuing model The M/M/1 model regards the supply side as comprising one service desk with a service rate N=t, which means that all N drivers operate via this service desk. By contrast, the M/M/N model regards each driver as a service desk with a service rate of 1=t, and the platform provides a total of N service desks. Thus, in the M/M/N model, mcv equals Q (implying that all passengers are served), wp equals zero and wm and other variables are given by the following formulas: r¼ " P0 ¼

N 1 X 1

Q N=t

1 1 ðQtÞ þ ðQtÞN k! N! 1  r k¼0 k

(3.17) #1 (3.18)

Calibration and validation for ride-sourcing markets Chapter | 3

Lq ¼

ðNrÞN r N!ð1  rÞ2 wm ¼

P0

Lq Q

63

(3.19)

(3.20)

The M/M/1, M/M/1/k, and M/M/N queueing models directly express mcv and passengers’ average pickup time, and matching rate as explicit functions of Q and N. In contrast, the CobbeDouglas production function and the two models introduced below estimate the market metrics by solving a system of nonlinear simultaneous equations, and thus the metrics are implicit functions of Q and N.

3.2.2.6 First-come-first-served (FCFS) model The FCFS model is another simple and widely adopted deductive model (Castillo et al., 2017; Ke et al., 2020). It assumes that there is always at least one idle driver in the market, such that passengers who raise a ride request are immediately assigned the closest driver, based on an FCFS rule. This is denoted ‘instant matching’ and wm equals zero. Accordingly, mcv is mcv ¼ Q

(3.21)

The relationship between drivers and passengers is therefore used to derive wp . Drivers are in one of three states: idle, on the way to pick up passengers, or on the way to deliver passengers. In contrast, as it is assumed that all passengers undergo instant matching, passengers do not wait for matching and are in one of two states: waiting for pick-up or being served. When the market is at equilibrium, the number of drivers in the pick-up state can be estimated as the product of Q and wp , i.e., Q$wp . In addition, the number of occupied drivers equals the product of Q and the average travel time, i.e., Q$t. Thus, the following vehicle conservation condition must hold: N ¼ N v þ Q$wp þ Q$t The model further assumes i.e.,

wp

(3.22)

is a function of the number of idle vehicles,

wp ¼

9 pffiffiffiffiffiffiffiffiffiffiffi 2v N v =A

(3.23)

where 9 is a detour ratio that could choose 1.27 (Arnott, 1996; Yang et al., 2018) and A is the research area. By combining Eq. (3.22) and Eq. (3.23), we can obtain the equilibrium solutions for N v and wp .

3.2.2.7 Batch-matching model The batch-matching model assumes that a platform adopts a batch-matching mechanism, in which waiting passengers and idle drivers are accumulated in

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the matching pool during a matching interval and are matched at the end of the interval (Xu et al., 2017). In each matching interval, the platform matches all mutually closest pairs consisting of a driver and a passenger. Unmatched passengers or vehicles are carried over to the next time interval. Considering that the market is stationary, the following three relations hold: mcv ¼ Q, N v ¼ Qwv , and N c ¼ Qwm . In addition, the following vehicle-conservation equation must hold: N ¼ Qðwv þ wp þ tÞ Moreover,

wm

and

wp

(3.24)

are given by:

wm ¼

s   c  Nv N 1 exp  v Nc N

(3.25)

3 rffiffiffiffifficffi  c 6 1 9 N 1 N 7  c  6 rffiffiffiffifficffi erf (3.26) wp ¼  rffiffiffiffiffiffiffiffivffi exp  v 7 4 v N N N 5 N pN 1 exp  v v 2 N A0 A0 R 2 x where erfðxÞ ¼ p2ffiffipffi 0 et dt is a Gaussian error function and 9 is a detour ratio 2

that could choose 1.27 (Arnott, 1996; Yang et al., 2018). By solving a system of simultaneous equations consisting of the abovementioned three relations and three equations, and assuming that mcv is equal to Q, we can solve for wp, wm , and wv at equilibrium.

3.2.2.8 Summary of matching functions The calculation of passenger waiting time is governed by wm in some models (such as queuing models) and by wp in other models (such as the FCFS model). A summary of the formulas derived for the key market metrics of these models is given in Table 3.1. 3.2.3 Key market metrics In this chapter, we evaluate various matching functions in terms of four market metrics: mcv , wm , wp , and w (the sum of wm and wp ). mcv is the number of successfully matched orders per unit time, so a high mcv means that many passengers are served per unit of time and thus the system operates at high efficiency. w is regarded as a metric of service quality, as it affects the mode choices of passengers and the final passenger demand for ride-sourcing services. We evaluate how well the above matching functions estimate these key market metrics in a range of market scenarios with various supply and demand conditions.

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TABLE 3.1 Summary of the formulas for key market metrics in various models. Matching rate

Matching time

Pick-up time

Application conditions

Perfect matching

minðQ; N =tÞ

0

0

None

CobbeDouglas production function

Q

Solved by Eqs. (3.5) e(3.7)

0

The system of equations has a solution

M/M/1

Q

Q  N N Q t t

0

N t

M/M/1/k

Q  ð1 Pk Þ

0

None

0

N t

Model

>Q

M/M/N

Q

Lq Qð1Pk Þ Lq Q

FCFS

Q

0

Solved by Eqs. (3.22) e(3.23)

The system of equations has a solution

Batch matching

Q

Solved by Eqs. (3.24) e(3.26)

Solved by Eqs. (3.24) e(3.26)

The system of equations has a solution

>Q

3.3 Experimental settings In this section, we present the basic experimental settings for the empirical studies. First, we establish a simulator, into which we input various combinations of supply and demand to generate the key market metrics, which we regard as the true values for various market scenarios. We also add each combination of supply and demand into the aforementioned matching functions to generate the key market metrics, which we regard as the estimated values for various market scenarios. Subsequently, we determine the differences between the true values and the estimated values to evaluate how well each matching function fits the simulation outcomes, which are treated as proxies for real markets.

3.3.1 Simulator The waiting passengers and drivers accumulate during each time intervald which is set to be 2 seconds in this experimentdand are matched at the end of the interval. Fig. 3.1 illustrates how the simulator generates a simulation of the ride-sourcing market. The main processes in the simulator are described below:

66 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 3.1 Flowchart of the simulator.

Order generation. At the beginning of each time interval, new orders are generated, which simulates the arrival of newly requested orders into a market. The number of new orders follows the Poisson distribution, while the locations of the origin and destination of new orders are uniformly distributed in the research area. New orders are accumulated in the matching pool until the end of the matching time interval (s), at which point the new orders and the unmatched orders carried over from the previous time interval are collated for the matching process. Matching process. The platform matches waiting passengers with idle drivers based on a Lagrange dual decomposition-based algorithm that is similar to that used by Boostanimehr and Bhargava (2014) to solve a resource allocation problem in a heterogeneous network. Matched drivers and passengers disappear from the queues, whereas unmatched passengers are carried over to the next time interval and unmatched drivers remain idle. Cruising and leaving. Idle drivers continue cruising in their current direction, and the cruising distance for each time interval equals the idle drivers’ speed multiplied by the time interval. Idle drivers whose idle time has exceeded the maximum idle time re-select their random direction of cruising while waiting passengers with accumulated waiting time that exceeds wc0 immediately leave the market.

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Updating the status of drivers and passengers. The status of drivers and passengers is updated after the procedures of order generation, matching, cruising and leaving. For example, the status of the idle drivers who are matched to passengers in this batch is updated from ‘idle’ to ‘pickup’.

3.3.2 Experiment Without loss of generality, the experimental setup is as follows. Experimental setup Research area Simulated time Matching time interval Vehicle speed Passengers’ maximum waiting time Drivers’ maximum idle time Maximum pick-up distance The arrival rate of orders per second Number of drivers

a 10 km  10 km square 6h 2s 11.11 m/s (40 km/h) 300 s 60 s 1000 m 0.01 to 2 100 to 2000

We simulate various markets with different supply and demand conditions by setting various order arrival rates and fleet sizes. The arrival rates of new orders range from 0.1 to 2 in increments of 0.1, and an additional rate of 0.01 is set to simulate a situation in which there are almost no orders (a zero arrival rate). The fleet sizes range from 100 to 2000 in increments of 100. We combine 21 arrival rates of new orders and 20 fleet sizes to simulate a total of 420 market scenarios. We run the simulator for each market scenario until a stationary equilibrium is reached, and then identify the key market metrics at this equilibrium. In this simulation, wm is used to judge whether a stationary market equilibrium is reached, and is recalculated at the end of each time interval. When wm converges, the market is regarded as being at equilibrium. For example, consider the market whose arrival rate of orders equals 1.0 and the number of drivers equals 500. The change in the passengers’ average waiting time and the process of reaching the equilibrium in this market are shown in Fig. 3.2; this indicates that the equilibrium is reached at approximately 17,000 s (4.7 h). The blue curve represents the change wm for passengers who are served, and the orange curve represents all of the orders made by passengers, including those who are not served and which result in passengers leaving. This shows that

68 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 3.2 Process of reaching equilibrium in a market.

passengers wait for an average of 150 s; by this time they are matched and thus remain in the market or are unmatched and thus immediately leave the market. When the market reaches equilibrium, the average waiting time for matching is approximately 100 s.

3.4 Analysis of experimental results In this section, we determine which matching function gives the best-fitting performance for each market metric under various supply and demand conditions. Once each market scenario reaches equilibrium, the true values of the key market metrics are recorded; we also calculate these key market metrics using the matching functions presented in Section 2. The matching functions are then evaluated and compared in terms of the mean absolute percentage error (MAPE) for their each of their respective market metrics. MAPE ¼ y^y y  100%; where yb is the estimated value of the market metric (generated by the matching functions), and y is the true value of the market metric (outputted by the simulator). In a given market scenario, it is possible that some matching functions are best for fitting some market metrics, while other matching functions are best for fitting the remaining market metrics.

3.4.1 Market segmentation To facilitate the analysis, we segment the markets into over-supplied, balanced, and under-supplied markets. In balanced markets, the supply is comparable to the demand, i.e., Q is 70%e100% of the service capacity N=t;

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in over-supplied markets, the demand is much less than the supply, i.e., Q is less than 70% of the service capacity; and in under-supplied markets, the demand is much greater than the supply, i.e., Q is greater than 100% of the service capacity. In under-supplied markets, some passengers are unmatched and the system fails to reach a stable equilibrium; therefore, we focus on balanced and over-supplied markets. This market segmentation is depicted in Fig. 3.3 in terms of the true value of Q and service capacity, which are outputted by the simulator. The horizontal axis represents Q (demand), which varies from 0.01 to 2, while the vertical axis represents the number of drivers (supply), which varies from 100 to 2000. The markets closest to the upper left corner have a sufficient supply, whereas those closest to the lower right corner have an insufficient supply.

3.4.2 Best-fit models for estimation of matching rate We calculate the MAPE for each model under various market conditions, which reveals the best model for calculating mcv in various market scenarios. The results of these calculations are presented in Fig. 3.4, in which the horizontal axis, the vertical axis, and the background are the same as in Fig. 3.3. Each point represents an equilibrium market for a certain combination of supply (on the y-axis) and demand (on the x-axis). The colour of each point denotes the best-fit model for estimating mcv in this market, and the size of

FIGURE 3.3 Market segmentation.

70 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 3.4 Best-fit models for estimation of the matching rate in various markets.

each point denotes the MAPE of the model. The larger the point, the greater the deviation of the estimated value from the true value. Fig. 3.4 shows that the perfect matching model and the M/M/1/k queuing model fits the real data well in most markets. In fact, in over-supplied markets, all of the examined models, except for the M/M/1/k queuing model, assume that every arriving passenger is ultimately served and therefore estimate mcv as Q (Table 3.1). These models have the same results as the perfect matching model and thus are omitted from Fig. 3.4 for conciseness. Additionally, all of the models introduced in Section 3.2 generate similar estimates for mcv. Most models estimate mcv as Q, while the M/M/1/k queuing model estimates mcv as Q  ð1 Pk Þ, where Pk represents the proportion of passengers who make an order but are not ultimately matched with a driver, and the ratio 1  Pk is almost equal to 1 when the market has a sufficient supply. This accounts for the fact that the MAPEs for all models represented in Fig. 3.4 are very similar in magnitude. We next fix either N or Q, and then plot the relationship between the former or the latter and mcv . The results, which are also sections of Fig. 3.4, are shown in Fig. 3.5, where the background colours are consistent with those in Fig. 3.4 (i.e., blue for over-supplied markets and grey for balanced markets), and the true values generated by the simulator are indicated by a star and the estimated values are represented by a dot. Fig. 3.5 clearly depicts the variations in the true value of mcv and the differences between various models. With N fixed at 1000 veh (Fig. 3.5A), the true mcv strictly increases with the arrival rate of orders Q, but the initially stable rate of increase eventually

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FIGURE 3.5 Estimated matching rates for various supplyedemand relationships. (A) Relationships between matching rate and demand (the arrival rate of orders) with a fixed supply (fleet size ¼ 1000 veh). (B) Relationships between matching rate and supply (fleet size) with a fixed demand (the arrival rate of orders ¼ 1.0 pax/s).

slows. That is, all requests are served when the supply is sufficient (the blue area) and mcv equals Q; however, as the demand continues to increase, the supply becomes insufficient (the grey area) and thus constrains mcv .

72 Supply and Demand Management in Ride-Sourcing Markets

As mentioned, all models except the M/M/1/k queuing model assume that every arriving passenger is ultimately served and estimated mcv as Q, which is an input in each market scenario. Consequently, in over-supplied markets, the supply is sufficient and all of the models can estimate mcv with a small error because almost all passengers are served. In contrast, in a balanced market, the supply becomes insufficient and thus not all passengers are served. Therefore, if models do not account for passengers leaving the market and thus use Q to estimate mcv , these models generate large errors and may even generate infeasible equilibrium solutions. Models have different applied ranges, so if a model has no feasible solution in a particular market, the point representing this model is not shown and the model comprises only part of a curve in Fig. 3.5. The CobbeDouglas production function, the M/M/1 and M/M/N queuing models, and the batch-matching model can only be applied in over-supplied markets and in some balanced markets, whereas the perfect matching, FCFS, and M/M/1/k queuing models can be applied in all markets. In addition, as the M/M/1/k model is the only model to take passengers’ departure from a market into consideration, it has a lower error than other models when the supply becomes insufficient.

3.4.3 Best-fit models for the estimation of matching time As wm is the interval from a passenger making an order to the passenger being matched online, a long wm increases the likelihood that passengers cancel their order (Fig. 3.6). Overall, the CobbeDouglas production function is most suitable for balanced markets and the batch-matching model is most suitable for over-

FIGURE 3.6 Best-fit models for the estimation of matching time in various markets.

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supplied markets. When a market is close to the red line, i.e., Q is close to the service capacity, the queuing modelsdespecially the M/M/1/k queuing modeldoutperform the other models. This can be explained by fixing the demand and drawing the time curve changing with the supply or by fixing the supply and drawing the time curve changing with the demand (Fig. 3.7). In over-supplied markets (the blue area), wm is close to zero, and all modelsdeven those that regard wm as zerodhave small absolute errors. However, in balanced markets (the grey area), wm is relatively large and difficult to estimate. The CobbeDouglas production function is very close to the true value in terms of its trend and absolute value because it describes the macroscopic relationship between four key variables (Q; N; N v ; wm ) very well. However, this model is unsolvable when the supply in the market is inadequate. That is, due to the vehicle conservation equation and the assumption that all of the passengers are ultimately served, when Q is approximately equal to service capacity N=t, it is difficult to find a solution that simultaneously satisfies both equations of the CobbeDouglas production function. The batch-matching model is the most accurate model in over-supplied markets because the matching mechanism of this model is more similar than those of other models to real-world situations and is the only mechanism that considers s. In batch matching, when the supply is sufficient, mcv is approximately equal to the minimum s, which leads to small errors. However,

FIGURE 3.7 Estimation of matching time for various supplyedemand relationships. (A) Relationships between matching time and demand (the arrival rate of orders) with a fixed supply (fleet size ¼ 1000 veh). (B) Relationships between matching time and supply (fleet size) with a fixed demand (arrival rate of orders ¼ 1.0 pax/s).

74 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 3.7 cont’d

akin to the CobbeDouglas production function, the batch-matching model considers the vehicle conservation equation and thus struggles to find the equilibrium solution when Q approaches the service capacity N=t. In contrast, queuing models do not consider s, and thus their estimates are very close to zero instead of the minimum s. Therefore, queueing models generate larger errors than other models in over-supplied markets. However, queuing models, especially the M/M/1/k model, have a larger applicable range and a smaller MAPE than other models when the supply is insufficient. As a result, when a market is close to the red line, i.e., Q approaches N=t, most models become infeasible but the M/M/1/k queuing modeldbecause it considers passenger departuresdis more consistent with the market.

3.4.4 Best-fit models for the estimation of passenger pick-up time Overall, in the context of wp estimation, the FCFS model is most suitable for over-supplied markets and some balanced markets with a larger supply, whereas the batch-matching model is most suitable for balanced markets. In addition, when the markets are close to the red line, none of the models are able to effectively estimate wp . In particular, the best-fit modelsdthe CobbeDouglas production function and the M/M/1/k queuing model, which regard wm as zerodhave a 100% MAPE (Fig. 3.8). The FCFS model and the batch-matching model are most suitable for calculating wp because they effectively characterise drivers’ idle, pick-up, and in-trip phases, as in Eq. (3.22). As the batch-matching model considers wp and

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FIGURE 3.8 Best-fit models for the estimation of passenger pick-up time in various markets.

wm , it achieves better performance in wp estimation than the FCFS model, which considers only wp and regards wm as zero. When Q is approximately equal to the service capacity, the batch-matching and FCFS models are unsolvable, while the other modelsdwhich regard wp as equal to zerodhave a 100% MAPE (Fig. 3.9).

3.4.5 Best-fit models for the estimation of passengers’ total waiting time As mentioned, w is the sum of wm and wp . The FCFS model performs better than the other models in the estimation of w in over-supplied markets and in some balanced markets with relatively sufficient supply. However, the batchmatching model, the queuing models, and the CobbeDouglas production function perform better than the other models when the supply becomes scarce. In over-supplied markets, orders are matched quickly and thus wp governs w. Given that the FCFS model exhibits good performance in estimating wp , it also outperforms the other models in estimating w in oversupplied markets. In insufficiently supplied markets, orders are not matched quickly and thus w is governed by both wm and wp . However, when the market is close to the red line, none of the models are able to effectively estimate wp but the queuing models effectively estimate wm , so the queuing models perform better than the other models (Fig. 3.10).

76 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 3.9

FIGURE 3.9 Estimates of the pick-up time for various supplyedemand relationships. (A) Relationships between pick-up time and demand (the arrival rate of orders) with a fixed supply (fleet size ¼ 1000 veh). (B) Relationships between pick-up time and supply (fleet size) with a fixed demand (arrival rate of orders ¼ 1.0 pax/s).

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FIGURE 3.10 Best-fit models for the estimation of passengers’ total waiting time in various markets.

3.5 Summary All of the models perform well in estimating mcv in their applied range. In the estimation of wm , wp , and w, the batch-matching model, the Cobbe Douglas production function, and the FCFS model perform better than the other models in the over-supplied and balanced markets. However, the CobbeDouglas production function and the queuing models generate results that are closer to reality than those generated by the other models in the markets at the tipping point, i.e., where Q is approximately equal to the service capacity. The ability of the matching functions to generate results that are close to reality is also evaluated by a simulation study based on an actual mobility dataset, as shown in Appendix 3.A. The results of this simulation study are similar to the abovementioned results obtained using randomly generated trip data. The code of the simulation studies is available at https:// github.com/hku-kejintao/simulator-matching-function-validation.

3.6 Discussion and conclusion A variety of matching functions have been developed to describe and explore the relationships between supply, demand, and key market metrics in ridesourcing markets. A good matching function is essential for the precise characterisation of ride-sourcing markets and can assist platforms and policymakers to design operating and regulating strategies that maximise platform profit and social welfare.

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TABLE 3.2 Summary of best-fit models for various markets. Matching rate

Matching time

Pick-up time

Total waiting time

Over-supplied market

All models

Batch matching

FCFS

FCFS

Balanced market

All applicable models

Cobb eDouglas

FCFS Batch matching

FCFS Batch matching Cobb eDouglas

Market close to Q ¼ N=t

M/M/1/k queuing model

Queuing models

None

Queuing models Cobb eDouglas

Note. FCFS ¼ first-come-first-served.

In this chapter, we conduct a series of simulation-based sensitivity analyses to identify which of several widely used matching functions generates results that best fit reality under a range of market scenarios with various combinations of supply and demand. We evaluate and compare a few key market metrics (mcv , wm , wp , and w) of the studied matching functions in 420 market scenarios. We find that each matching function has different applicability to various market scenarios. The best-fit models for estimating the market metrics in various market scenarios are summarised in Table 3.2. This chapter describes one of the first attempts to obtain empirical evidence to support the application of some matching functions that are widely used in ride-sourcing services. Thus, the results suggest a few new avenues for future exploration. First, the simulation platform could be used to evaluate the performance of other matching functions. Second, it would be intriguing to examine the performance of various theoretical models in market scenarios where a platform sets different parameters for matching (such as a maximum pick-up distance). Third, the simulation framework could be extended to allow the investigation of matching functions in ride-sourcing markets with a variety of ride services (such as a basic ride service and a ride-pooling service), with competition between platforms, or with third-party integrators. This chapter is based on one of our recent publications (Wei et al., 2022).

Appendix 3.A We use order data from Manhattan, New York (USA) from 1 to 30 July 2015 to test the applicability of the models to a real-world dataset. In contrast to the

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previous simulation (Section 3.3.2), in which the research area is a 10 km  10 km square, the research area in this simulation is the 5  22 km rectangle shown in the black box (Fig. 3.A.1). Moreover, based on the local conditions in Manhattan, the vehicle speed is reduced from 11.11 to 6.33 m/s.

FIGURE 3.A.1 Research area in Manhattan, New York (USA).

80 Supply and Demand Management in Ride-Sourcing Markets

Manhattan is a single market. However, to generate a two-dimensional evaluation result for comparison with the original simulation results, we need to sample orders to adjust the demand and to set various values of N to adjust the supply. Thus, the real order records of Manhattan in July 2015 are regarded as an order pool, and in each time interval we sample orders from the order pool according to Q. The simulation result is given below Fig. 3.A.2. As in the previous simulation, all of the models perform well in terms of estimating mcv in their applied range. When the market is close to the red line, the M/M/1/k model performs better than the other models Fig. 3.A.3. Similarly, as in the previous simulation and in terms of estimating wp , when the supply is sufficient, the batch-matching model performs best; when the supply gradually decreases, the CobbeDouglas production function performs best; and when the supply is further reduced and the market is close to the red line, the queuing models perform best (Fig. 3.A.4). Thus, although the results for each specific market in this simulation may not be exactly the same as those in the previous simulation, the trend is similar. This may be attributable to the slower speed in this simulation slowing the drivers’ cruising phase, as under the same supply and demand and compared with a market with a faster speed, a market with a slower speed may have a longer wm because it may be more difficult to drive to find passengers with whom to be matched. In terms of estimating w, the FCFS model performs best in the oversupplied market; the FCFS and batch-matching models perform best when the supply becomes scarce; and no model characterises the market well when it is close to the red line Fig. 3.A.5.

FIGURE 3.A.2 Best-fit models for estimating the matching rates in various markets.

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FIGURE 3.A.3 Best-fit models for estimating matching time in various markets.

FIGURE 3.A.4 Best-fit models for estimating passengers’ pick-up time in various markets.

Overall, the results of this simulation are similar to those for the previous simulation. In an over-supplied market, the best-fit models for w are consistent with the best-fit models for wm ; in the middle area, the FCFS and batchmatching models and the CobbeDouglas function perform best; and when the market is close to the red line, the queuing models and the batch-matching model perform best. Thus, compared with the previous simulation, in this real-

82 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 3.A.5 Best-fit models for estimating passengers’ total waiting time in various markets.

world dataset-based simulation, the shape of the research area, the speed, and the spatial distribution of orders are different but the trend of best-performing models is similar.

Glossary of notation mcLv Q Tv t N Nv Nc wm wp w wv Qe A0 v wc0 s

matching rate of the market (pax/s) arrival rate of passengers/new raised orders (pax/s) arrival rate of vacant vehicles (veh/s) average trip time (s) vehicle fleet size (veh) average number of vacant vehicles at an instant of time (veh) average number of waiting passengers/orders at an instant of time (pax) passengers’ average matching time (s) passengers’ average pick-up time (s) passengers’ average total waiting time (s) drivers’ average idle/searching time (s) arrival rate of passengers who are ultimately matched (pax/s) area of the market considered (m2) average speed of vehicles (m/s) maximum waiting time of passengers (s) matching time interval (s)

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84 Supply and Demand Management in Ride-Sourcing Markets Information Systems and Technologies. Springer, pp. 978e988. https://doi.org/10.1007/978-3319-77703-0_95. Ke, J., Zheng, H., Yang, H., Chen, X.M., 2017. Short-term forecasting of passenger demand under on-demand ride services: a spatio-temporal deep learning approach. Transportation Research C: Emerging Technologies 85, 591e608. https://doi.org/10.1016/j.trc.2017.10.016. Ke, J., Yang, H., Li, X., Wang, H., Ye, J., 2020. Pricing and equilibrium in on-demand ride-pooling markets. Transportation Research B: Methodological 139, 411e431. https://doi.org/10.1016/ j.trb.2020.07.001. Ke, J., Feng, S., Zhu, Z., Yang, H., Ye, J., 2021. Joint predictions of multi-modal ride-hailing demands: a deep multi-task multi-graph learning-based approach. Transportation Research Part C: Emerging Technologies 127, 103063. https://doi.org/10.1016/j.trc.2021.103063. Lee, A., Savelsbergh, M., 2015. Dynamic ridesharing: is there a role for dedicated drivers? Transportation Research Part B: Methodological 81, 483e497. https://doi.org/10.1016/ j.trb.2015.02.013. Mo, D., Yu, J., Chen, X.M., 2020. Modeling and managing heterogeneous ride-sourcing platforms with government subsidies on electric vehicles. Transportation Research Part B: Methodological 139, 447e472. https://doi.org/10.1016/j.trb.2020.07.006. Peled, I., Lee, K., Jiang, Y., Dauwels, J., Pereira, F.C., 2021. On the quality requirements of demand prediction for dynamic public transport. Communications in Transportation Research 1, 100008. https://doi.org/10.1016/j.commtr.2021.100008. Rogers, B., 2015. The social costs of Uber. University of Chicago Law Review Online 82 (1), 6. Saadi, I., Wong, M., Farooq, B., Teller, J., Cools, M., 2017. An Investigation into Machine Learning Approaches for Forecasting Spatio-Temporal Demand in Ride-Hailing Service. arXiv preprint arXiv:1703.02433. https://arxiv.org/abs/1703.02433. Salnikov, V., Lambiotte, R., Noulas, A., Mascolo, C., 2015. Openstreetcab: Exploiting Taxi Mobility Patterns in New York City to Reduce Commuter Costs. arXiv preprint arXiv:1503.03021. https://arxiv.org/abs/1503.03021. Shi, X., Zhao, D., Yao, H., Li, X., Hale, D.K., Ghiasi, A., 2021. Video-based trajectory extraction with deep learning for High-Granularity Highway Simulation (HIGH-SIM). Communications in Transportation Research 1, 100014. https://doi.org/10.1016/j.commtr.2021.100014. Tang, Y., Guo, P., Tang, C.S., Wang, Y., 2021. Gender-related operational issues arising from ondemand ride-hailing platforms: safety concerns and system configuration. Production and Operations Management 30 (10), 3481e3496. Vazifeh, M.M., Santi, P., Resta, G., Strogatz, S.H., Ratti, C., 2018. Addressing the minimum fleet problem in on-demand urban mobility. Nature 557 (7706), 534e538. https://doi.org/10.1038/ s41586-018-0095-1. Wang, H., Yang, H., 2019. Ridesourcing systems: a framework and review. Transportation Research Part B: Methodological 129, 122e155. https://doi.org/10.1016/j.trb.2019.07.009. Wang, Y., Wu, J., Chen, K., Liu, P., 2021. Are shared electric scooters energy efficient? Communications in Transportation Research 1, 100022. https://doi.org/10.1016/j.commtr.2021.100022. Wei, S., Feng, S., Ke, J., Yang, H., 2022. Calibration and validation of matching functions for ridesourcing markets. Communications in Transportation Research 2, 100058. https://doi.org/ 10.1016/j.commtr.2022.100058. Xu, Z., Yin, Y., Zha, L., 2017. Optimal parking provision for ride-sourcing services. Transportation Research Part B: Methodological 105, 559e578. https://doi.org/10.1016/j.trb.2017.10.003. Xu, Z., Yin, Y., Ye, J., 2020. On the supply curve of ride-hailing systems. Transportation Research Part B: Methodological 132, 29e43. https://doi.org/10.1016/j.trb.2019.02.011.

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Chapter 4

Government regulations for ride-sourcing services Jintao Ke1, Xinwei Li2, Hai Yang3 and Yafeng Yin4 1

Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China; School of Economics and Management, Beihang University, Beijing, China; 3Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China; 4Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI, United States 2

4.1 Properties of the pareto-efficient solutions As discussed in Chapter 2, the Pareto-efficient solutions are given by Eqs. (2.95) and (2.96) for the model based on the FCFS scheme. Here we discuss how important decisions and endogenous variables (such as vehicle fleet size N, utilisation rate U, arrival rate of passengers Q and average waiting time w) and market measures (such as profit P and social welfare S) change along the Pareto-efficient frontier from the social optimum (SO) to the monopoly optimum (MO). To ensure analytical tractability, we adopt the assumption in Zha et al. (2016) that the supply is sufficient and the drivers’ reservation rate r is homogeneous, which means that the relationship EQ ¼ CðNÞ ¼ Cp ðNÞ ¼ cN always holds in the following analyses, where E stands for wage per order paid to drivers, EQ refers to the total income of drivers per hour and c is the constant operational cost of one vehicle. Thus the Pareto-efficient solutions in Eqs. (2.95) and (2.96) can be written as: cðQ w0 þ 1Þ ¼  bQ w0

(4.1)

x Q x þ 1 f 0

(4.2)

F  ¼ cðw þ tÞ 

where the constant c represents the average cost of operating one vehicle (or recruiting one driver). We first examine U and the vacancy rate (V) of ridesourcing vehicles, which measure the efficiency of the vehicle usage. U ¼ Qt=N, which is the proportion of all vehicles that are in-trip vehicles, and V ¼ N v =N, which is the proportion of all vehicles that are idle vehicles. Due to the

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existence of the pick-up phase, U þ V < 1; thus, based on the vehicle conservation Eq. (2.28), we have
 1 w 1 N v ¼ N Qt Qw ¼ Qt (4.3) U t 
1 v 1 (4.4) N ¼ N Qt Qw ¼ Qðt þ wÞ 1V We further consider the following specific form of w as a function of N v : wðN v Þ ¼ AðN v Þa

(4.5)

where A and a are two positive parameters. Clearly, the first-order derivative w0 ðN v Þ ¼ dw=dN v ¼ aAðN v Þa1 ¼ aw=N v . Substituting Eq. (4.5) into Eq. (4.1) gives: cN v ¼ aðb þ cÞQ w

(4.6)

From Eqs. (4.3), (4.4) and (4.6), we can then obtain: 
ct 1  v a 1 w ¼ AðN Þ ¼ aðb þ cÞ þ c U  w ¼ AðN v Þa ¼

ct

V 1  V

aðb þ cÞ  c

V 1  V

(4.7)

(4.8)

which indicates that w only depends on U  or V  at the Pareto-efficient frontier. For simplicity, we let f1 ¼

ct aðb þ cÞ þ c

(4.9)

By combining Eq. (2.28) and the definition U ¼ Qt=N, we have: Nv  1  1 t wðN v Þ U

Q¼


(4.10)

which holds for all equilibrium solutions, including the Pareto-efficient solutions. By combining Eq. (4.7) and Eq. (3.10), we obtain:  a1 a11
f1 1 1  Q ¼ 1 (4.11) ðt  f1 Þ U  A

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which implies that Q monotonically increases with U  , and vice versa, along the Pareto-efficient frontier. In addition, by combining Eq. (2.28) and the definition V ¼ N v =N, we have 
1  1 Nv V (4.12) Q¼ wðN v Þ þ t which holds for all equilibrium solutions, including the Pareto-efficient solutions. Combining Eq. (3.8) and Eq. (4.12) thus yields 2 3a1 
1 ct 61 7 
1 4 5  V A 1 aðb þ cÞ V   1  c Q ¼ (4.13) ct 
þt aðb þ cÞ V1  1  c which indicates that Q can be written as a univariate function V  . It can be easily found that Q decreases with V  along the Pareto-efficient frontier. From the analyses above, we derive the following lemma. Lemma 4e1. If wðN v Þ ¼ AðN v Þa , supply is sufficient and r is homogeneous, then the following relationships always hold along the Pareto-efficient frontier: 1. 2. 3. 4.

w decreases with U  and increases with V  ; U  increases with N v , whereas V  decreases with N v ; U  increases with Q , whereas V  decreases with Q ; and w is linearly proportional to the inverse of U  . Proof. By re-organising Eqs. (4.7) and (4.8), we have: U ¼ V ¼

ct ½aðb þ cÞ þ cw þ ct

1 ct c þ1þ aðb þ cÞw aðb þ cÞ

(4.14) (4.15)

This indicates that U  and V  only depend on w . It can be seen from Eqs. (2.88) and (2.89) that U  decreases with w and V  increases with w . Thus, as w is a decreasing function of N v , we can readily prove that U  increases with N v and V  decreases with N v . Moreover, Eq. (4.11) shows that Q increases with U  , and Eq. (4.13) shows that Q decreases with V  . This completes the proof. n Lemma 4-1 indicates several important monotonic relationships between U, V, w and Q along the Pareto-efficient frontier with a Pareto-efficient set of

90 Supply and Demand Management in Ride-Sourcing Markets  , U  V  , V  , E  , E  , F  , F  , Q , regulatory variables. Recall that Umo so mo so mo so mo so mo  v v   Qso , Nmo , Nso , wmo and wso refer to the utilisation rate, the vacancy rate, the average wage per ride, the optimum trip fare, the passenger demand, the number of idle vehicles and the average waiting time at the MO and the SO, respectively, and let Smo , Sso , Pmo , Pso , Pso and Pmo denote the social welfare, the commission per ride and the platform profit at the MO and the SO, respectively. We thereby develop the following proposition. Proposition 4e1. If wðN v Þ ¼ AðN v Þa , supply is sufficient, and r is homogeneous, then as the Pareto-efficient solution moves along the Paretoefficient frontier from the SO to the MO:

1. Q strictly decreases, and Qso > Qmo ; 2. P strictly increases, and Pso < Pmo ; 3. S strictly decreases, and Sso > Smo ;  > U ; 4. U  strictly decreases, and Uso mo    ; 5. V strictly increases, and Vso < Vmo 6. w strictly increases, and wso < wmo ; v > N v ; 7. N v strictly decreases, and Nso mo  > N ; 8. N  strictly decreases, and Nso mo  is higher than F  , i.e., F  < F  ; 9. Fmo so so mo   < E  ; and 10. E strictly increases, and Eso mo 11. P strictly increases, and Pso < Pmo .

Proof. Obviously, when the Pareto-efficient solution moves from the SO to the MO along the Pareto-efficient frontier, P increases and S decreases. As S equals the sum of P, the consumer surplus (CS) and the provider surplus (PS), the CS and PS must decrease as the Pareto-efficient solution moves from the SO to the MO along the Pareto-efficient frontier. In addition, Eq. (2.61) show that the CS and the PS are increasing functions of Q. We thus conclude that Q decreases along the Pareto-efficient frontier from the SO to the MO, which implies that Qso > Qmo . In addition, if the supply is sufficient and r is homogeneous, Lemma 4-1 indicates that U  increases with Q , whereas V  decreases with Q . Thus, we can prove that U  decreases whereas V  increases along the Pareto-efficient  > U  and frontier from the SO to the MO. This also implies that Uso mo     Vso < Vmo . Lemma 4-1 also reveals that w decreases with U and increases with V  . Thus, we show that w increases along the Pareto-efficient frontier from the SO to the MO, and wso < wmo . As w is a decreasing function of N v , we further show that N v decreases along the Pareto-efficient frontier from the v > N v . SO to the MO, and Nso mo  Moreover, as V ¼ N v =N  , V  increases along the Pareto-efficient frontier and N v decreases along the Pareto-efficient frontier, N  must

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strictly decrease along the Pareto-efficient frontier moving from the SO to the  > N  . The optimal pricing formula along the Pareto-efficient MO, and Nso mo frontier given by Eq. (2.96) indicates that as the solution moves from the SO to the MO, the term cðw þ tÞ monotonically increases along the Paretoefficient frontier. Furthermore, the monopoly markup given by Q =f 0 is  is higher than F  . In the always positive, and thus we can prove that Fmo so scenario with sufficient supply and a homogeneous r, EQ ¼ cN, and thus E ¼ cN=Q ¼ ct=U, which indicates that the average driver E per hour is inversely proportional to U. Therefore, we can conclude that E strictly in < E . Moreover, P creases along the Pareto-efficient frontier, and thus Eso mo can be written as PQ, where P is the commission given by F  E. As the Pareto-efficient solution moves from the SO to the MO, P increases but Q decreases, and thus P must increase. This completes the proof. n These findings offer valuable managerial insights for a government that may need to strike a balance between S and P by imposing various regulations, such as a price cap, an entry limitation, a minimum U or a maximum V. Some of these regulations will lead to Pareto-efficient solutions, whereas others will not and are thus deemed to be inefficient. We discuss the various types of regulations and their effects later in this chapter. It is noteworthy that the findings in Proposition 4-1 are generally consistent with those obtained from the use of an inductive model with a CobbeDouglas-type matching function (see Yang and Yang, 2011 for details) with increasing returns-toscale.

4.2 An alternative method to obtain and analyse paretoefficient solutions Now we present an alternative method to obtain and analyse Pareto-efficient solutions, which enables better investigation of the effects of regulations (covered in the next subsection). For simplicity, we let BðQÞ ¼ f 1 ðQÞ, such that the Q function can be rewritten as: F ¼ BðQÞ bðwðQ; NÞ þ tÞ

(4.16)

where the average w (wðQ; NÞ) is a function of Q and N given by the vehicle conservation equation Eq. (2.28). This indicates that F can be written as a function of Q and N. Therefore, in the maximisation problems for P and S, we can treat Q and N (not F and N) as decision variables, and thus treat F and w as functions of Q and N. Treating Q and N as decision variables means that we can take the partial derivatives of both sides of Eq. (2.28) with respect to Q and N and obtain: vN v wþt ¼ 1 þ Qw0 vQ

(4.17)

92 Supply and Demand Management in Ride-Sourcing Markets

vN v 1 ¼ vN 1 þ Qw0

(4.18)

Then, the partial derivatives wðQ; NÞ with respect to Q and N are given by: vwðQ; NÞ vN v ðw þ tÞ 0 ¼ w0 ¼ w vQ  Qw0 þ 1 vQ

(4.19)

vwðQ; NÞ vN v w0 ¼ w0 ¼ vN vN 1 þ Qw0

(4.20)

The P-maximisation problem can be formulated as: max PðQ; NÞ ¼ Q½BðQÞ bðwðQ; NÞ þ tÞ CðNÞ

(4.21)

where Q and N are the two decision variables. The first-order conditions of this problem are   vPðQ; NÞ vwðQ; NÞ ¼ ½BðQÞ bðwðQ; NÞ þ tÞ þ Q B0 ðQÞ  b ¼ 0 (4.22) vQ vQ vPðQ; NÞ vwðQ; NÞ ¼ bQ C0 ðNÞ ¼ 0 vN vN Combining Eq. (4.22) and Eq. (4.23) gives rise to:      0 1 þ Qmo w0 C 0 Nmo mo ¼  bQmo wmo           wmo þ t B Qmo þ Qmo B0 Qmo ¼ b þ C0 Nmo

(4.23)

(4.24) (4.25)

Clearly, Eq. (4.24) has the same form as Eq. (2.57). In addition, Eq. (4.25) can be expressed as:            wmo þ t Qmo B0 Qmo (4.26) B Qmo b wmo þ t ¼ C0 Nmo        and B0 ðQÞ ¼ dC=dQ ¼ 1=ðdQ =dCÞ ¼ As B Qmo b wmo þ t ¼ Fmo 0 1=f ðCÞ, we can easily show that Eq. (4.25) and Eq. (4.58) also have the same form. Consequently, we can conclude that the first-order conditions given by the new method (Eqs. 4.24 and 4.25) are equivalent to the first-order conditions given by the original method (Eqs. (2.57) and (2.58)). That is, the two P-maximisation problems are equivalent to each other. Next, we examine the S maximisation problem, which can be formulated as: Z Q max SðQ; NÞ ¼ BðzÞdz bQ½wðQ; NÞ þ t Cp ðNÞ (4.27) 0

where Q and N are the two decision variables. The first-order conditions of this problem are vSðQ; NÞ vwðQ; NÞ ¼ BðQÞ b½wðQ; NÞ þ t  bQ ¼0 vQ vQ

(4.28)

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vSðQ; NÞ vwðQ; NÞ ¼ bQ Cp0 ðNÞ ¼ 0 vN vN Combining these two equations gives rise to:      0 1 þ Qso w0 Cp0 Nso so ¼  bQso wso          wso þ t B Qso ¼ b þ Cp0 Nso

93

(4.29)

(4.30) (4.31)

Clearly, Eq. (4.30) has the same form as Eq. (2.78). In addition, Eq. (4.25) can be expressed as:          (4.32) B Qso b wso þ t ¼ C 0 Nso wso þ t       , we can easily show that Eq. (4.31) and Eq. As B Qso b wso þt ¼ Fso (2.79) also have the same form. Then, we can conclude that the first-order conditions given by the new method (Eqs. 4.30 and 4.31) are equivalent to the first-order conditions given by the original method (Eqs. (2.78) and (2.79)). That is, the two S-maximisation problems are equivalent to each other. Finally, we examine the Pareto-efficient solutions, which are obtained by solving the bi-objective maximisation problem that aligns with the definition of S given by Eq. (4.27) and the definition of P given by Eq. (4.21), with Q and N as the decision variables. By repeating the process used in Section 2.4.3, we obtain the following Pareto-efficient solutions: Cp0 ðN  Þ þ xC 0 ðN  Þ  0 ðQ w þ 1Þ ¼  bQ w0 ð1 þ xÞ
 Cp0 ðN  Þ þ xC 0 ðN  Þ x Q B0 ðQ Þ ¼ b þ BðQ Þ þ ðw þ tÞ xþ1 ð1 þ xÞ

(4.33)

(4.34)

It can be seen that the Pareto-efficient solutions are linear combinations of the MO and SO solutions and are equivalent to the Pareto-efficient solutions obtained from the original method described Eqs. (2.95) and (2.96). Clearly, based on the assumption of Zha et al. (2016) that the supply is sufficient and r is homogeneous, we have the relationship EQ ¼ CðNÞ ¼ Cp ðNÞ ¼ cN; thus, the Pareto-efficient solutions become: cðQ w0 þ 1Þ ¼  bQ w0 F  ¼ cðw þ tÞ 

x Q B0 ðQ Þ xþ1

(4.35) (4.36)

which are equivalent to the results in Eq. (4.1) and Eq. (4.2). Therefore, by using the Pareto-efficient solutions obtained by this new method, we obtain the same analytical results as those presented in Proposition 4-1 and Lemma 4-1.

94 Supply and Demand Management in Ride-Sourcing Markets

4.3 Regulations We next examine the regulations that may be imposed by a government to induce a ride-sourcing platform to choose a predetermined Pareto-efficient solution (i.e., any solution along the Pareto-efficient frontier). We aim to identify which regulations lead to an efficient regulatory outcome, at which a pre-determined Pareto-efficient solution can be targeted. In all of the regulations considered below, we assume that a government is targeting Paretoefficient operating strategies associated with a trip fare F  , a vehicle fleet size N  , a realised w and a realised Q . As mentioned above, in the theoretical analyses we restrict our discussions to a scenario that has a sufficient supply and homogeneous r. In this case, we have EQ ¼ CðNÞ ¼ Cp ðNÞ ¼ cN, PS ¼ 0 and S equals the sum of P and consumer welfare. For the proofs in this sub-section, we alternate between using the original method presented in Chapter 2 and the new method presented in Section 4.2.

4.3.1 Price-cap regulation Price-cap regulation requires a ride-sourcing platform to set an F below or equal to a price ceiling determined by a government. For a targeted Paretoefficient solution ðF  ; N  Þ, a government sets a price ceiling (F  ), which is lower than the F at the MO. In this case, one can easily see that a platform will choose F ¼ F  , and its optimal strategy for maximising P is given by: max PðNÞ ¼ F  Q  CðNÞ

(4.37)

The market equilibrium described by Eq. (2.32) indicates that at a given F ¼ F  , Q (or N v ) can be implicitly represented as a univariate function of N, denoted by QðNÞ. This implies that N can also be written as a function of Q, denoted by NðQÞ. Eq. (2.36) shows that Q is a strictly increasing function of N for any given F in a normal regime, and thus we conclude that N is also an increasing function of Q for a given F ¼ F  , namely, vNðQÞ=vQjF¼F  > 0. Moreover, CðNÞ is increasing with N, and thus vCðNðQÞÞ=vQjF¼F  > 0. The P-maximisation problem therefore becomes max PðQÞ ¼ F  Q CðNðQÞÞ

(4.38)

where PðQÞ is a univariate function of Q consisting of two terms with opposite driving forces: F  Q, which increases with Q, and CðNðQÞÞ, which decreases with Q. If the targeted Pareto-efficient solution ðF  ; N  Þ equals the MO solution, the platform will straightforwardly choose ðF  ; N  Þ to maximise P. In contrast, if the targeted Pareto-efficient solution ðF  ; N  Þ is less than the MO solution, then vPðQÞ=vQ < 0 where Q ¼ Q , for the following reasons. First, vPðQÞ=vQ ¼ 0 where Q ¼ Q can be excluded, as it is not the MO solution. Second, if vPðQÞ=vQ > 0, then an increase in Q from Q ¼ Q will result in an

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increase in P. In addition, an increase in Q leads to an increase in CS, and thus S (the sum of CS and P) also increases. This implies that a deviation from the Pareto-efficient solution ðF  ; N  Þ can increase both P and S. However, this violates the definition of a Pareto-efficient solution, i.e., neither P nor S can be unilaterally improved without reducing the other (thus it is impossible that both can be simultaneously improved). As vPðQÞ=vQ < 0 where Q ¼ Q , a platform must choose a set of regulatory variables such that Q < Q to maximise P under price-cap regulation. This indicates that the targeted Pareto-efficient solution is not chosen by a platform subject to price-cap regulation. Moreover, as mentioned, NðQÞ is an increasing function of Q for a given F ¼ F  , and thus Q < Q implies that N < N  . These findings are summarised in the following proposition. Proposition 4e2. If the supply is sufficient and r is homogeneous, pricecap (i.e., F) regulation is inefficient and cannot induce a private ridesourcing platform to choose a predetermined Pareto-efficient solution unless this solution equals the MO solution. Specifically, if F is regulated to be lower than that at the MO, to maximise P a platform will choose a strategy such that both N and Q are less than they would be for the Paretoefficient solution.

4.3.2 Fleet size regulation Here we discuss minimum and maximum fleet size regulation, which sets an entry limitation to restrict N providing ride-sourcing services, e.g., by issuing a certain number of working permits. If the supply is sufficient and r is homogeneous, we show in Proposition 4-1 that N at the SO is greater than that at  > N  , and the optimal N decreases along the Paretothe MO, i.e., Nso mo efficient frontier from the SO to the MO. In this case, if a government targets a Pareto-efficient solution ðF  ; N  Þ, and only imposes a maximum fleet size regulation that requires N  N  (where N  is the regulated maximum fleet  , which is size), a platform will be unaffected and will simply choose N ¼ Nmo less than N  . In contrast, for a targeted Pareto-efficient solution ðQ ; N  Þ, a minimum fleet size regulation requires a platform to guarantee that N  N  . In this case, a platform’s optimal strategy for maximising P can be given by: max PðQ; NÞ ¼ FQ CðNÞ

(4.39)

s:t: N  N 

(4.40) N ,

If the optimal solution is achieved at N > this means that the constraint N  N  is non-binding and thus the optimal solution equals the MO. However, from Proposition 4-1, we know that N at the MO is the smallest N along the Pareto-efficient frontier, which is not consistent with the hypothesis that the

96 Supply and Demand Management in Ride-Sourcing Markets

optimal solution is achieved at N > N  . Therefore, we have N ¼ N  in Eq. (4.39), which can thus be rewritten as an unconstrained maximisation problem of Q, as follows: max PðQÞ ¼ ½BðQÞ bðwðQ; N  Þ þ tÞQ CðN  Þ ðF  ; E ; Q ;

(4.41)

NÞ

If the target is not the MO solution, we must have vPðQÞ=vQ < 0 where Q ¼ Q , as vPðQÞ=vQ > 0 at Q ¼ Q implies that a positive deviation of Q from Q ¼ Q leads to an increase in P, and also an increase in CS and thus S, which is inconsistent with the assumption that ðQ ; N  Þ is a Pareto-efficient solution. Therefore, we have the following proposition. Proposition 4e3. If the supply is sufficient and r is homogeneous: (1) a government cannot induce a ride-sourcing platform to choose a Paretoefficient solution by only setting a maximum N, as this will cause the platform to directly choose the MO; and (2) a government cannot induce a ride-sourcing platform to choose a Paretoefficient solution by only setting a minimum N. In this case, if N is regulated to be larger than that at the MO, to maximise P the platform will choose a strategy such that the values of N and Q are equal to and less than their respective Pareto-efficient solutions, namely, N ¼ N  and Q < Q . If CðNÞ exhibits increasing returns to scale, the relative magnitudes of N at the MO and at the SO are undetermined. Moreover, it is difficult to determine how the optimal N changes (i.e., whether it strictly increases or decreases) along the Pareto-efficient frontier. In this case, we cannot analytically infer whether maximum fleet size regulation can induce a platform to choose a predetermined Pareto-efficient solution.

4.3.3 Wage regulation Proposition 4-1 shows that E strictly increases along the Pareto-efficient frontier from the SO to the MO. Thus, for a targeted Pareto-efficient solution ðF  ; E ; Q ; N  Þ, setting a minimum E  E does not affect the decisions of a platform, as it will naturally increase E (and F and P) to maximise P, and these values will be greater than those for the targeted Pareto-efficient solution. In contrast, for a targeted Pareto-efficient solution ðF  ; E ; Q ; N  Þ, a government may require a platform to set E lower than the Pareto-efficient E, i.e., E  E . In this case, the platform will choose E ¼ E , and its optimal strategy for maximising P is given by: max P ¼ ðF  E ÞQ

(4.42)

where Q only depends on F, which indicates that F can also be written as a function of Q and thus P can be viewed as a function of Q, i.e., P ¼ PðQÞ. If

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the targeted Pareto-efficient solution ðF  ; E ; Q ; N  Þ equals the MO, the platform will straightforwardly choose ðF  ; E ; Q ; N  Þ to maximise P. In contrast, if the targeted Pareto-efficient solution ðF  ; E ; Q ; N  Þis not the MO as it has a lower required E, then we must have vPðQÞ=vQ < 0 at Q ¼ Q . This is because if vPðQÞ=vQ > 0 at Q ¼ Q , a platform can increase P by increasing Q to a value greater than Q . This increase in Q will also increase the CS, and thus increase S. This violates the characteristics of Pareto-efficient solution; in this case, that P and S cannot be simultaneously improved. Therefore, a platform will maximise P by choosing a Q < Q . From Proposition 4-1, this further indicates that a platform will choose an F < F  . In the scenario with sufficient supply and homogeneous r, the relationship E Q ¼ cN holds, and thus N < N  . This leads to the following proposition. Proposition 4e4. If the supply is sufficient and r is homogeneous, wage regulation is not Pareto-efficient, i.e., a government is unable to induce a platform to choose a targeted Pareto-efficient strategy by only regulating E. If E is regulated to be lower than that at the MO, a platform will maximise P by choosing a strategy such that Q, N, and F are all less than they would be for the Pareto-efficient solution.

4.3.4 Income regulation Income regulation requires that a platform ensures U is not less than a certain amount. If the supply is sufficient and r is homogeneous, then U always satisfies U ¼ EQ=N ¼ c, which is a constant. Thus, for a targeted Paretoefficient solution ðF  ; E ; Q ; N  Þ, the targeted drivers’ average income (U ) equals U at the MO. As a result, income regulation does not induce a platform to choose a targeted Pareto-efficient solution. Next, we examine the effects of income regulation in a scenario where r is heterogeneous and CðNÞ exhibits increasing returns to scale. Suppose that the targeted Pareto-efficient solution is ðF  ; E ; Q ; N  Þ and a platform is asked to guarantee that U  U ¼ E Q =N  . As the driver supply depends only on U, N equals N  . Accordingly, PS and the total E paid to all drivers (EQ ¼ U N  ) can be determined. Eq. (4.16) can then be used to identify the platform’s optimal strategy, as follows: max PðQÞ ¼ FQ EQ ¼ ½BðQÞ bðwðQ; N  Þ þ tÞQ þ U N  N

(4.43)



where and U are fixed and PðQÞ can be viewed as a univariate function of Q. Taking the partial derivative of PðQÞ with respect to Q affords:   vPðQÞ vwðQ; N  Þ  0 ¼ ½BðQÞ bðwðQ; N Þ þ tÞ þ Q B ðQÞ  b (4.44) vQ vQ If the targeted strategy ðF  ; E ; Q ; N  Þ is also the solution of the P-maximisation problem given in Eq. (4.21), then vPðQÞ=vQ ¼ 0, which

98 Supply and Demand Management in Ride-Sourcing Markets

satisfies the MO conditions in Eq. (4.22). Thus, as ðF  ; E ; Q ; N  Þ is a Pareto-efficient solution, it is also the MO. Conversely, if ðF  ; E ; Q ; N  Þ is not the MO, it is not the optimal solution to the P-maximisation problem (4.21), and thus vPðQÞ=vQs0. In this case, vPðQÞ=vQ < 0 where Q ¼ Q for the following reasons. (1) vPðQÞ=vQ ¼ 0 is not the MO and can thus be excluded; (2) If vPðQÞ=vQ > 0 where Q ¼ Q , P will increase if Q > Q . This will also increase CS, which is an increasing function of Q. However, PS only depends on income level R (which is enforced to be R ), and thus does not change. As a result, S increases, as it is the sum of P, CS, and PS. This is inconsistent with the Pareto-optimality ðF  ; E ; Q ; N  Þ. As a result, a platform will achieve P-maximisation by choosing a Q < Q . In summary, we have the following proposition. Proposition 4e5. If the supply is sufficient and drivers’ reservation rates are homogeneous, income regulation does not affect the decisions of the ride-sourcing platform. Similarly, if drivers’ reservation rates are heterogeneous, income regulation is unable to induce the platform to choose a targeted Pareto-efficient strategy. When the income is regulated to be higher than that at the MO, the platform maximises its profit by choosing a strategy that ensures the demand is smaller than it would be for the Pareto-efficient solution.

4.3.5 Commission regulation Here we show a government can regulate the commission charged by a platform to induce a platform to choose a predetermined Pareto-efficient solution. That is, for a targeted Pareto-efficient solution ðF  ; E ; Q ; N  Þ, a platform is required to set P less than the targeted commission (P*), where P ¼ F   E . Proposition 4-1 indicates that the platform chooses P ¼ P , and its optimal strategy for maximising P is given by: max P ¼ P Q ¼ ðF   E ÞQ F

 E

(4.45)

where is fixed and thus P is a strictly increasing function of Q. That is, the platform maximises Q under commission regulation. However, Q must satisfy Q  Q for the following reason. An increase of Q from Q ¼ Q will lead to an increase in P and CS and thus also increase S. This violates the property of the Pareto-efficient solution, at which the P and S cannot be simultaneously improved. In this case, the platform will always choose a maximum Q ¼ Q , (i.e., the predetermined Pareto-efficient strategy) to maximise P. To summarise, we have the following proposition. Proposition 4e6. If the supply is sufficient and r is homogeneous, a government can regulate the P charged by a platform to be less than that at the MO, thereby inducing a platform to choose a targeted Pareto-efficient solution.

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4.3.6 Commission ratio regulation Alternatively, a government can regulate the commission ratio (w), such that a platform can only charge a certain w for each trip, i.e., w ¼ ðF  EÞ=F ¼ 1 E=F. This means that for a targeted Pareto-efficient solution ðF  ; E ; Q ; N  Þ, a platform must set its w as w . Consequently, its optimal strategy for maximising P is given by: max P ¼ w FQ

(4.46)

s:t: ð1  w ÞFQ ¼ cN

(4.47)

where P depends on F and Q. Substituting Eq. (4.47) into Eq. (4.46) yields: max PðNÞ ¼

cw N 1  w

(4.48)

which indicates that under commission ratio regulation the P-maximisation problem is equivalent to maximising N. By viewing N as a function of F and Q, as determined in Eq. (4.47), and F as a function of N and Q, as shown in Eq. (4.16), we can obtain: N¼

ð1  w Þ Q½BðQÞ bðwðQ; NÞ þ tÞ c

(4.49)

Taking the partial derivative of N with respect to Q yields:

  ð1  w Þ vwðQ; NÞ ½BðQÞ bðwðQ; NÞ þ tÞ þ Q B0 ðQÞ  b vN c vQ   ¼ ð1  w Þ Qbw0 vQ 1þ c 1 þ Qw0 (4.50) If the targeted Pareto-efficient strategy ðF  ; E ; Q ; N  Þ happens to be a solution to the P-maximisation problem represented by Eq. (4.21), then the term in brackets in the numerator on the RHS of Eq. (3.50) equals zero, which corresponds to the MO conditions in Eq. (4.22). In addition, ðF  ; E ; Q ; N  Þ is a predetermined Pareto-efficient solution and thus satisfies the Paretoefficient optimality conditions in Eq. (4.35). These two observations allow us to conclude that ðF  ; E ; Q ; N  Þ is the MO. Similarly, it follows that if ðF  ; E ; Q ; N  Þ is not the MO, it cannot be the optimal solution of Eq. (4.21), i.e., ðF; E; Q; NÞs ðF  ; E ; Q ; N  Þ and vN=vQs0. If ðF  ; E ; Q ; N  Þ is not the MO, then vN=vQ < 0, for the following reasons. First, vN=vQ ¼ 0 can be excluded, as the solution is not the MO. Second, vN=vQ > 0 where Q ¼ Q implies that increasing Q can increase N under the condition stated in Eq. (4.47). As a result, P given by Eq. (4.48) will also increase. Moreover, increasing Q will directly increase CS and also S (which is the sum of P and CS). This is inconsistent with the Pareto-efficient

100 Supply and Demand Management in Ride-Sourcing Markets

optimality ðF  ; E ; Q ; N  Þ at which P and S cannot be simultaneously improved. Therefore, under commission ratio regulation, a ride-sourcing platform will choose a Q < Q and an N > N  . This leads to the following proposition. Proposition 4e7. If the supply is sufficient and r is homogeneous, commission ratio regulation is Pareto-inefficient, as it cannot induce a platform to choose a targeted Pareto-efficient strategy. Specifically, if w is regulated to be lower than that at the MO, the platform will maximise P by choosing a strategy such that N is greater and Q is less than they would be for the Paretoefficient solution.

4.3.7 Minimum utilisation rate regulation Utilization regulation requires a platform to ensure that its U is greater than or equal to a certain level. Proposition 4-1 shows that if the supply is sufficient and r is homogeneous, U  strictly decreases along the Pareto-efficient frontier  > U  . In this case, if a government targets a Pareto-efficient solution and Uso mo  ðF ; E Þ and requires U > U  , a platform will choose a strategy such that U ¼ U  . Therefore, as U ¼ Qt=N, the P-maximisation problem becomes:
 Qt max PðQÞ ¼ FQ C (4.51) U Using Eq. (4.16), P can be written as the following function of Q:
 Qt (4.52) P ¼ ½BðQÞ bðwðQ; U  Þ þ tÞQ C U where U  is treated as a constant and P can be regarded as a univariate function of Q. If the targeted Pareto-efficient solution ðF  ; E Þ associated with Q equals the MO, a platform will straightforwardly choose ðF  ; E Þ to maximise P. In contrast, if the targeted Pareto-efficient solution ðF  ; E Þ is not the MO, then vPðQÞ=vQ < 0 where Q ¼ Q . This is because: (1) vPðQÞ=vQ ¼ 0 is excluded, as the solution is not the MO; and (2) if vPðQÞ=vQ > 0 where Q ¼ Q , a platform can increase P by choosing Q > Q , which also increases CS and S. However, this violates the property of Pareto-efficient solutions, i.e., that P and S cannot be simultaneously improved. As a result, a platform will choose a Q < Q to maximise P. Given that U ¼ Qt=N, this means that a platform will implement an operating strategy in which N < N  . This leads to the following proposition. Proposition 4e8. If the supply is sufficient and r is homogeneous, minimum Utilization regulation is Pareto-inefficient, as it cannot induce a platform to choose a targeted Pareto-efficient strategy. Specifically, if U is regulated to be greater than that at the MO, a platform maximises P by choosing a strategy in which both N and Q are less than they would be for the Pareto-efficient solution.

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Accordingly, as EQ ¼ cN holds in a market with sufficient supply and homogeneous r, we have E ¼ cN=Q ¼ ct=U, and thus the maximum wage regulation E  E is equivalent to the minimum utilization regulation U  U  .

4.3.8 Demand regulation Demand regulation that is implemented by a government to achieve a targeted Pareto-efficient solution ðF  ; E ; Q ; N  Þ requires a ride-sourcing platform to make decisions such that the resulting realised Q is greater than a certain value, i.e., Q  Q . According to Proposition 4-1, Q decreases along the Pareto-efficient frontier from the SO to the MO, which means that under demand regulation a platform naturally chooses Q ¼ Q to maximise P. S is uniquely determined by the sum of P and CS, and the latter is uniquely determined by Q. Therefore, as the platform chooses Q ¼ Q , we must have      SðF; E; Q ; NÞ ¼ SðF ; E ; Q ; N Þ, which indicates that S where Q ¼ Q must be the targeted Pareto-efficient solution. Eq. (4.11) indicates that U is uniquely determined for a given Q ¼ Q , and as U ¼ Qt=N, this means that N is also uniquely determined as N  . Given Q and N  , the optimal F and E are thus equal to F  and E , respectively. To summarise, we have the following proposition: Proposition 4e9. If the supply is sufficient and r is homogeneous, a government can regulate the realised Q to be greater than that at the MO, thereby inducing a platform to choose a targeted Pareto-efficient solution.

4.3.9 Summary The outcomes of various regulatory regimes that can be implemented in attempts to achieve a targeted Pareto-efficient solution are summarised in the following table. As we have shown, only commission regulation and demand regulation are Pareto-efficient, i.e., can induce a platform to choose a targeted Pareto-efficient strategy (Table 4.1). In actual operations, commission regulation sets a maximum P that can be charged by a platform per ride and thus is probably more practical than demand regulation.

4.3.10 Numerical illustrations We next use a numerical example to demonstrate the outcomes of the regulations examined so far. The Q function is assumed to have the following negative exponential form: Q ¼ f ðF þ b $ ðw þ tÞÞ ¼ Q expfk $ ½F þ b $ ðw þ tÞg

(4.53)

where Q is the potential passenger demand and k is a parameter representing the Q sensitivity with respect to the generalised cost. We assume that

102 Supply and Demand Management in Ride-Sourcing Markets

TABLE 4.1 Summary of outcomes of various regulatory regimes. Regulatory regime

Choice of ride-sourcing platform

Price-cap F  F 

Q < Q  , N < N  Qmo

No

Minimum fleet-size N  N

Q < Q  , N ¼ N

No

Minimum wage E  E 

 < Q  , N ¼ N < N Q ¼ Qmo mo

No

Maximum wage E  Income R  1

R

Commission P 

P

Q< Q ¼

 Qmo

Q ¼

Q, Q,

N<
0, we have dyðxÞ=dx ¼ 1  ex > 0 and yð0Þ ¼ 0, and thus yðxÞ > 0 for all x > 0. Therefore, we have lDtA  1þ elDtA > 0 and we can see that Dt monotonically increases with DtA . Moreover, in the detour-unconstrained scenario, we have Dt ¼ 1=l, and thus vDt=vQ ¼ ðvl =vQÞ l2 < 0, which implies that Dt only depends on Q and monotonically decreases with Q as DtA /N. One can show that most of the properties presented hold for other assumptions regarding the distributions of e t (such as normal distribution and log-normal distribution assumptions). elDtA .

5.2.2 Pre-assigned pool-matching with meeting points Another strategy involves pre-assigned pool-matching between passengers with similar origins and destinations, and dispatching nearby idle vehicles to pick them up at a meeting point. This is also called a dynamic waiting strategy (as first presented by Yan et al., 2020) and is used in the Uber Express Pool service (Uber, 2019a). Specifically, two passengers are pool-matched if their origins are very close and so are their destinations, i.e., the distance between their pick-up locations and the distance between their drop-off locations are both within a certain walkable range or walking radius. These two passengers are then required to walk to the mid-point of their origins to be picked up and are dropped off at the mid-point of their destinations, such that they walk equal distances. The walking radius is usually short (up to a few hundred meters, e.g., 250 m in Uber), and thus, their walking time is negligible. This is a simplified RS strategy that is different from the en-route RS strategy presented in the previous subsection; the latter strategy is applied in UberPool (Uber, 2019b), in which a passenger is matched with a second passenger en-route (rather than ex-ante) and thus experiences an additional pick-up and/or drop-off during his/her trip.

112 Supply and Demand Management in Ride-Sourcing Markets

A pool-matching window is used in this matching strategy, within which passengers are accumulated for pool-matching that occurs at the end of the matching window (passengers are not pool-matched if there are no passengers within their walkable ranges). Then, a platform dispatches nearby vehicles to pick up pool-matched and un-pool-matched passengers, and pool-matched passengers are dropped off at the middle point of their destinations. We let f denote the length of the pool-matching window. As it is naturally expected that p depends on the number of passengers in the matching pool, p is proportional to the product of Q and f (Qf), i.e., p ¼ pðQfÞ

(5.9)

where the function pð $Þ is assumed to describe a saturation curve with the following properties: p0 > 0; p00 < 0; p ¼ 0 as Qf/0 and p ¼ 1 as Qf/ N. Based on these assumptions, we can obtain the following lemma. Lemma 5-1. For all Q  0, pðQ; fÞ increases with f, pðQ; 0Þ ¼ 0 and lim pðQ; fÞ ¼ 1; for all f  0, pðQ; fÞ increases with Q, pð0; fÞ ¼ 0 and

f/N

lim pðQ; fÞ ¼ 1.

Q/N

Ke et al. (2021) performed extensive experiments based on the actual ondemand shared mobility data in three cities (New York, USA; and Chengdu and Haikou (China)) to verify that the p function can be approximated by the following expression (a similar form was first used in the numerical) example of Yan et al. (2020), with a reasonably high goodness-of-fit (>95%): p ¼ 1 z expðgQfÞ

(5.10)

where g > 0 and 0 < z < 1. We can easily prove that Eq. (5.10) satisfies all of the statements in Lemma 5-1. Moreover, in the pre-assigned pool-matching scheme, passengers are required to walk to meeting points, which means that drivers only need to drive to a common meeting point to pick up and drop off passengers, i.e., drivers do not have to take extra-detours. In such scenarios, passengers’ Dt and drivers’ Dt can be assumed to be zero, but passengers experience a walking-time cost to reach a meeting spot and an additional w to be pool-matched.

5.2.3 Comparisons Both of these two pool-matching schemes are used in actual operations: for example, the en-route pool-matching scheme is used in UberPool, and the pre-assigned pool-matching scheme is used in Uber Express Pool. In the enroute pool-matching scheme, a platform leverages DtA and other decision variables (such as F and average wage per order E) to influence system efficiency. In the pre-assigned pool-matching scheme, a platform tunes f to balance the trade-off between p and the additional w required for poolmatching. The pre-assigned pool-matching scheme is easier for analytical

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113

derivations in modelling, as it obviates the need to examine the intricate relationships between p and detour times. In contrast, en-route pool-matching strategy-based equilibrium analyses are generally intractable, and theoretical managerial insights can be obtained only in the detour-unconstrained scenario. Nevertheless, numerical studies can be used to seek some policy insights, although these cannot be strictly proven. In the following chapters that examine RS services, we alternate between assuming that a platform adopts the en-route pool-matching mechanism or the pre-assigned pool-matching mechanism.

5.3 Equilibrium analyses In this section, we present a model that describes the equilibrium states in an RS market based on the en-route pool-matching mechanism. The model is applicable to both detour-constrained and detour-unconstrained scenarios, but for the sake of analytical tractability, the comparative static analyses are based on the simple detour-unconstrained scenario.

5.3.1 Supply and demand function In an RS market, the generalised cost of passengers opting for RS is F þ b$ ðw þtrs Þ, where trs ¼ t þ pDt, as given by Eq. (5.1). Similar to an NS market, Q can be written as a decreasing function of the generalised cost: Q ¼ f ðF þ b $ ðw þ t þ pDtÞÞ

(5.11)

where f 0 < 0. In particular, in the detour-unconstrained scenario, the average trip time for an RS service, t þ pDt, becomes t þ pDt. Unlike in an NS market, a vehicle in an RS market can be dispatched to an unmatched passenger or two matched passengers. In the former case, the number of vehicles in the pick-up phase and in-trip phase is ð1  pÞQ½t þ wðN v Þ. In the latter case, each vehicle corresponds to two passengers, and thus the number of vehicles used in the pick-up phase and in-trip phase can be estimated by 1 pQ½t þ Dt þ wðN v Þ, where Dt is the average driver detour time, i.e., the d d 2 extra-detour time a driver experiences when serving two passengers in comparison with the normal t the driver experiences when serving one passenger. Therefore, the vehicle conservation function in an RS market is given by 1 N ¼ N v þ pQ½t þ Dtd þ wðN v Þ þ ð1  pÞQ½t þ wðN v Þ 2

(5.12)

Next, we use realistic examples to illustrate Dtd and Dt, and how they are related. As shown in Fig. 5.2, there are two possible routing sequences in a shared ride. If a driver has already picked up passenger i and is dispatched to pick up a second passenger j, the driver has two routing sequences: (a) firstpickupelast-drop-off, i.e., the driver first drops off passenger j and then

114 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 5.2 Ride-pooling routing sequences. (A) First-pickupelast-drop-off. (B) Firstpickupefirst-drop-off. (C) Example 1. (D) Example 2.

drops off passenger i; (b) first-pickupefirst-drop-off, i.e., the driver first drops off passenger i and then drops off passenger j. Let ti and tj denote the normal trip time for passengers i and j without RS, and let t1ði;jÞ , t2ði;jÞ and t3ði;jÞ denote the trip time for the three consecutive segments in a ride shared by passenger i and j. In the first-pickupelast-drop-off sequence, the detour times of passengers i and j are t1ði;jÞ þ t2ði;jÞ þ t3ði;jÞ  ti and t2ði;jÞ  tj ¼ 0, respectively, whereas the detour time of drivers is the difference between the routing time   t1ði;jÞ þ t2ði;jÞ þ t3ði;jÞ and the average normal trip time ti þ tj 2. In the firstpickupefirst-drop-off sequence, the detour times of passengers i and j are t1ði;jÞ þ t2ði;jÞ  ti and t2ði;jÞ þ t3ði;jÞ  tj respectively, whereas the detour time   of drivers is given by t1ði;jÞ þ t2ði;jÞ þ t3ði;jÞ  ti þtj 2. Then, Dt and Dtd in this shared ride are estimated by: Dt ¼

t1ði;jÞ þ t2ði;jÞ  ti þ t2ði;jÞ þ t3ði;jÞ  tj 2

Dtd ¼ t1ði;jÞ þ t2ði;jÞ þ t3ði;jÞ 

ti þ tj 2

(5.13) (5.14)

Then, from Eqs (4.13) and (4.14) we obtain. Dtd  2Dt ¼

ti þ tj  t2ði;jÞ 2

(5.15)

where the first element on the right-hand side (RHS) is an estimate of the average normal t and the second element on the RHS is t for the second segment (the ‘shared’ segment) during the routing. This indicates that Dtd  2Dt is proportional to the difference between the average normal t and the average t for the shared segment. However, the latter is difficult to ascertain and depends on many factors, such as the algorithms used for matching in RS and network structures. Thus, it is intractable to identify the relationship between Dtd and Dt. For example, in the simplified case in which the directions of passengers i and j are the same, as shown in Fig. 5.2C and D below, there is

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115

  no Dt, and Dtd is given by t1ði;jÞ þ t2ði;jÞ þ t3ði;jÞ  ti þtj 2. In this case, if the origins and destinations of the two passengers are close to each other, Dtd z 0 (Fig. 5.2C); conversely, if the origins and destinations of the two passengers are far from each other, the shared segment is short and Dtd >> 0 (Fig. 5.2D). Overall, the relationship between Dtd and Dt is difficult to determine and requires further exploration from both theoretical and empirical perspectives. Nevertheless, it can generally be expected that Dtd increases with Dt. For simplicity, we let Dtd ¼ gDt in the following analysis, where g is an exogenous parameter.

5.3.2 Market equilibrium The equilibrium of an RS market can be determined by solving a system of simultaneous equations that consists of Eq. (5.11) and Eq. (5.12). In the detour-unconstrained scenario in which DtA /N, p/1, and Dt ¼ DtðQÞ with vDt=vQ < 0, all passengers are pool-matched, and thus the stationary equilibrium of the corresponding RS market can be simplified as the following system of nonlinear equations: Q ¼ f ðF þ b $ ðw þ t þ DtÞÞ

(5.16)

1 N ¼ N v þ Q½t þ gDt þ wðN v Þ 2

(5.17)

where Eq. (5.16) describes the Q curve and Eq. (5.17) describes the supply curve, and the intersection of these curves represents the equilibrium solution. As can be seen from the supply curve, Q cannot be written as an explicit function of N v in the absence of a specific form of Dt, which means it is intractable to investigate the marginal effects of decision variables on the endogenous variables, such as Q, N v and w. However, Ke et al. (2021) found that Dt is inversely proportional to Q by performing extensive experiments using actual mobility data from New York, Haikou and Chengdu. For analytical tractability, we use their findings and assume that Dt ¼ A=Q, where A is a parameter. This formula satisfies the properties of Dt in the probabilistic model described in Section 5.1.1.2. By combining this formula and Eq. (5.17), we can express Q as the following explicit function of N v :   1 v 2 N  gA  N 1 1 2 (5.18) N ¼ N v þ Q½t þ wðN v Þ þ gA5Q ¼ 2 2 t þ wðN v Þ Then, taking the partial derivative of Q with respect to N v yields: vQ  ðQw0 þ 2Þ ¼ vN v wþt

(5.19)

116 Supply and Demand Management in Ride-Sourcing Markets

Conversely, the partial derivative of N v with respect to Q is given by: vN v  ðw þ tÞ ¼ Qw0 þ 2 vQ

(5.20)

Clearly, the signs of vQ=vN v and vN v =vQ are indeterminate. If Qw0 þ 2 < 0, the market is said to be in a wild goose chase (WGC) regime, in which N v increases with Q; otherwise, the market is said to be in a normal regime, in which N v decreases with Q. Note that the conditions for a WGC regime in the NS and RS markets are Qw0 < 1 and Qw0 < 2, respectively. However, it is analytically difficult to compare these two split points in markets, as Q and w0 are endogenous variables that depend on many factors, such as the decision variables (F and N), and the values of g (an exogenous positive parameter representing the ratio of Dtd to Dt) and the area of the studied space (A). Accordingly, we examine a market equilibrium in an RS market, which can be determined by the following combination of Eq. (5.16) and Eq. (5.18) that represents an implicit equation of N v :   1 v 2 N  gA  N 2 ¼ f ðF þ b $ ðwðN v Þ þ tns þ DtÞÞ (5.21) Q¼ tns þ wðN v Þ Therefore, the market equilibrium is governed by the two key decision variables: F and N.

5.3.3 Comparative static effects of regulatory variables Next, we study the comparative static effects of regulatory variables, i.e., F and N, on the key endogenous variables, i.e., the number of idle vehicles, w, and Q. Taking the partial derivative of both sides of Eq. (5.21) with respect to F and N yields:  f 0 ðw þ tns Þ   vDt 0 0 ðQw þ 2Þ 1  bf þ f 0 bw0 ðw þ tns Þ vQ   0 vDt  2 1  bf vN v vQ     ¼ vQ vDt vN 0 0 bw0 ðtns þ wÞ  f 1  bf vN v vQ

vN v ¼ vF

(5.22)

(5.23)

Q can be written as an explicit function of N v , as given in Eq. (5.18), and thus we can derive the following partial derivatives of Q with respect to F and N: vQ vQ vN v ¼ ¼ vF vN v vF

f 0 ðQw0 þ 2Þ   vDt 0 0 þ f 0 bw0 ðw þ tns Þ ðQw þ 2Þ 1  bf vQ

(5.24)

Equilibrium analysis for ride-pooling services Chapter | 5

vQ 2 vQ vN v ¼ þ v ¼ vN tns þ w vN vN

117

2f 0 bw0   vDt ðQw0 þ 2Þ 1  bf 0 þ f 0 bw0 ðw þ tns Þ vQ (5.25)

As w is inversely proportional to N v , the partial derivatives of w with respect to F and N are given by  f 0 w0 ðw þ tns Þ   vDt þ f 0 bw0 ðw þ tns Þ ðQw0 þ 2Þ 1  bf 0 vQ   0 vDt 2 1  bf w0 v vw vQ 0 vN   ¼w ¼ vDt vN vN ðQw0 þ 2Þ 1  bf 0 þ f 0 bw0 ðw þ tns Þ vQ vw vN v ¼ w0 ¼ vF vF

(5.26)

(5.27)

In contrast to an NS market, in an RS market the signs of vQ=vN, vQ=vF, vN v =vN, vN v =vF, vw=vN and vw=vF are undetermined and dependent on the signs of Qw0 þ 2 and 1  bf 0vDt vQ . Fig. 5.3 illustrates the complicated relationships between the decision variables and endogenous variables in an NS market and RS market. Fig. 5.3A shows that Q and N v interact with each other. In addition, as discussed above, Q decreases with N v in the normal regime and increases with N v in the WGC regime. Moreover, w decreases with N v , and Q decreases with w. Therefore, the three endogenous variables e Q, N v and w e form an internal cycle leading to a market equilibrium, and the decision variables e F and N e influence the equilibrium via Q and N v . As shown in Fig. 5.3B, the RS market has a similar internal cycle to the NS market. However, a unit increase in Q has an additional indirect effect on the RS market: it decreases Dt, which thereby increases Q. Therefore, a decrease in F generates a larger marginal increase in Q in the RS market than in the NS market, and thus a platform operating in an RS market is unlikely to decrease F. It is interesting to find that the formulations of comparative static effects in the RS and NS markets are similar, aside from (a) the Qw0 þ 1 term being 0 þ 2 in the RS market, and (b) the RS market having an replaced by Qw  additional term

0  bf 0vDt vQ . The Qw þ 2 term indicates that the RS market

has a higher U than the NS market, as one vehicle transports two passenger requests in each ride in the detour-unconstrained scenario. The bf 0vDt vQ term represents the additional indirect effect on Q described above, which corresponds to the red-arrowed cycle in Fig. 5.3.

118 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 5.3 Relationships between decision and endogenous variables in two types of market. (A) Non-pooling market. (B) Ride-pooling market.

5.4 Market measures In this section, we examine the properties of the optimal solutions for RS services under three scenarios: (1) the MO scenario, in which a monopoly

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119

platform aims to maximise P; (2) the SO scenario, in which a platform aims to maximise S without a P constraint; and (3) the Pareto-efficient solution scenario, in which neither P nor S can be improved without reducing the other.

5.4.1 Monopoly optimum At the MO in an NS market, a ride-sourcing platform aims to maximise P by determining F and N. This is a typical market scenario that has been examined by researchers such as Zha et al. (2016) and Yang and Yang (2011). Specifically, Zha et al. (2016) argued that in this scenario a ride-sourcing platform behaves like a conventional taxi company, as it aims to maximise its revenue without possessing any vehicles by allowing drivers free entry into the market and considering that their supply of ride-sourcing services suffices unless it results in zero net earnings. Under these conditions, the revenuemaximising problem for a ride-sourcing platform is the same as the revenue-maximising problem for a monopoly street-hailing taxi market, as examined by Yang and Yang (2011). Therefore, the problem is formulated as described below. In the MO scenario in an RS market, the optimisation problem is similar to that in an NS market. The ride-sourcing platform receives F from passengers, pays E to drivers, and P is given by ðF EÞQ. As mentioned above, we assume that the supply is sufficient and r is homogeneous, and therefore drivers will enter the market until their net P equals zero, namely, EQ ¼ cN. Thus, the P-maximisation problem can be formulated as follows: maxPðF; EÞ ¼ ðF  EÞQ

(5.28)

s:t:EQ ¼ cN

(5.29)

which is equivalent to an unconstrained maximisation problem given by the following equation: maxPðF; NÞ ¼ FQ  cN

(5.30)

where Q is the solution of the market equilibrium in Eq. (5.21). The first-order conditions of Eq. (5.30) lead to  1   0 c Qrs wrs þ 2 ¼  bQrs w0 rs 2

(5.31)

 1  vDt Q Frs ¼ c wrs þ t þ bQrs   0rs 2 vQrs frs

(5.32)

where the subscript ‘rs’ specifies an RS market and the asterisk indicates optimality. Interestingly, the optimal pricing for an RS market, which is given by Eq. (5.32), also follows the Lerner formula, as the RHS consists of three terms: the average cost to a driver for transporting a passenger, where half of a

120 Supply and Demand Management in Ride-Sourcing Markets

 vehicle is required to transport each passenger



1c 2

wrs

þt

 

; an additional



term associated with Dt (bQrs vDt vQrs ); and a monopoly mark-up term  0     vDt  vDt  Qrs frs . As vQ < 0, bQrs vQ < 0; this implies that a decrease in F rs

rs

causes an increase in Q and a decrease in Dt, which further increases Q. That is, a unit decrease in F in an RS market also decreases Dt and thus typically attracts more passengers, whereas a unit decrease in F in an NS market does not have this effect. Therefore, a platform offering RS services has a stronger incentive to decrease F than a platform offering NS services. In addition,  v substituting w0 rs ¼ dwrs dNrs into Eq. (5.32) leads to     1 dwrs dwrs 1 c Qrs v þ 2 ¼ bQrs v 5 c$ dNrsv þ Qrs dwrs ¼ bQrs dwrs 2 2 dNrs dNrs (5.33) where the LHS indicates the marginal cost of operating vacant ride-sourcing v ) and vehicles in the pick-up phase (where each vehicle vehicles (c$dNrs corresponds to two passengers; c$12Qrs dwrs ), whereas the RHS represents the marginal pick-up time cost of passengers (bQrs dwrs ). In addition, from Eq.  0 (5.32) and the fact that w0 rs < 0, we can easily prove that Qrs wrs þ 2 > 0, which leads to the following lemma. Lemma 5-2. The MO in the RS market is always located in the normal regime rather than in the WGC regime.

5.4.2 Social optimum In an RS market under the detour-unconstrained scenario, the SO can be found by solving the following equation: Z Q max SðF; NÞ ¼ f 1 ðzÞdz b$ðw þ t þ DtÞQ  cN (5.34) 0

Its first-order conditions are given by  1   0 c Qrs wrs þ 2 ¼  bQrs w0 rs 2

(5.35)

 1  vDt Frs ¼ c wrs þ t þ bQrs  2 vQrs

(5.36)

Eq. (5.35) for the SO is the same as Eq. (5.31) for the MO. Eq. (5.36) states that the SO F in an RS market comprises two terms: the average cost of a driver for transporting a passenger (the first term) and an additional term associated with Dt (the second term). Eq. (5.36) is the same as Eq. (5.32), except it lacks the monopoly mark-up term of the latter. As mentioned above,

121

Equilibrium analysis for ride-pooling services Chapter | 5

the additional term associated with Dt is negative, which shows that a decrease in Dt due to an increase in Q will decrease the SO F. That is, a unit decrease in F appeals to more potential passengers in an RS market than in an NS market, and thus the SO F in an RS market is generally lower than that in an NS  dN v into Eq. (5.35) gives the ¼ dw market. In addition, substituting w0 rs rs rs following equation:       1 1    dwrs  dwrs v c Qrs v þ 2 ¼ bQrs v 5 c$ dNrs þ Qrs dwrs ¼ bQrs dwrs 2 2 dNrs dNrs (5.37) v ) which indicates that the marginal cost for operating vacant vehicles (c$ dNrs 1   and vehicles in the pick-up phase (c$2 Qrs dwrs ) equals the marginal pick-up time cost of passengers (bQrs dwrs ) at the SO. Moreover, the joint P of the platform and affiliated drivers at the SO is given by the following equation:

 2 vDt Pso ¼ Frs Qrs  cN ¼ cNrsv þ b Qrs wns and Dt > 0, we have Qrs < Qns . On the supply side, as v < N v . w is a decreasing and convex function with respect to N v , we have Nrs ns Then, given that the SO is located in the normal regime (as Q decreases with N v on the supply curve), we have: N  N v N  N v   rs >   ns v þ t w Nrsv þ tns w Nns ns

 v  If w Nrs þ tns  Dtd ¼ gDt , then:

  

   1 Qrs w Nrsv þ tns  gQrs Dt 5N Nrsv ¼ Qrs w Nrsv þ gDt þ tns 2  gQrs Dt

122 Supply and Demand Management in Ride-Sourcing Markets

Thus:

  1   2 N  gA  N v N Nrsv þ N  Nrsv  gQrs Dt 2      Qrs ¼ ¼ w Nrsv þ tns tns þ w Nrsv N  N v >  v  ns ¼ Qns w Nns þ tns

which contradicts the results on the demand side, i.e., Qrs < Qns . This implies   F  ) is false, and thus, we have F  < F  . that the initial assumption (Frs ns rs ns n This completes the proof.  v  þ tns  Dtd indicates that the sum of w and t is The condition w Nrs which generally holds in actual operations. greater than Dt  a shared ride,  dvin þ tns  Dtd is a sufficient but not a necessary condition, Moreover, w Nrs and we therefore expect that in most cases the SO F in an RS market is lower than that in an NS market. However, we cannot strictly prove that the MO F in an RS market is lower than that in an NS market, 0 as the magnitudes of the  monopoly mark-up in an NS market (Qns fns ) and that in an RS market (Qrs frs0 ) depend on the shape of the Q function f . Nevertheless, the following proposition can be made.   v þ t  Dt  and the relationship between Q Proposition 5-2. If w Nrs ns d and generalised cost is characterised by a negative exponential function (Q ¼ f ðCÞ ¼ QexpðkCÞ,) the MO F in an RS market is smaller than that in an NS market. Proof. The MO conditions of an NS market are given in Section 2.4.1.3. To allow a convenient comparison between the two types of market, we denote  , w , N v and Q . The dethe variables at the MO of an NS market as Fns ns ns ns rivative of Q with respect to generalised cost C is given by: f 0 ¼ kQexpðkCÞ ¼ kQ   Hence, Q=f 0 ¼ 1=k is a constant for any given Q. We first assume Frs  . Then, by comparing the optimal pricing formulas in an RS and NS market Fns 0 in view of Qrs frs0 ¼ Qns fns ¼ 1=k, we must have wrs > wns . On the    demand side, as Frs  Fns , wrs > wns , and Dt > 0, we have Qrs < Qns . On v < the supply side, as w is a decreasing and convex function of N v , we have Nrs v v Nns . As the SO is located in the normal regime (as Q decreases with N on the supply curve), we have:

N  N v N  N v   rs >   ns v v þ t w Nrs þ tns w Nns ns

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 v  þ tns  Dtd ¼ gDt , then we have: If w Nrs

  

   1 Qrs w Nrsv þ tns  gQrs Dt 5N Nrsv ¼ Qrs w Nrsv þ gDt þ tns 2  gQrs Dt Thus, we can obtain:   1   2 N  gA  N v N Nrsv þ N  Nrsv  gQrs Dt 2      Qrs ¼ ¼ tns þ w Nrsv w Nrsv þ tns N  N v >  v  ns ¼ Qns w Nns þ tns which is inconsistent with the results on the demand side, i.e., Qrs < Qns . This   F  ) is false, and thus we have F  < implies that the initial assumption (Frs ns rs . Fns This completes the proof.  vn The first condition (w Nrs þ tns  Dtd ) is the same as the condition in Proposition 5-1, which is generally expected, as mentioned. The second condition (that Q is proportional to a negative exponential function of the generalised cost) has been assumed in many previous studies (e.g., Yang and Yang, 2011). Therefore, the MO and SO F in an RS market should be lower than those in an NS market. This is intuitive: a unit decrease in F by a platform operating an RS service will attract more passengers than a unit decrease in F by a platform operating an NS service, and thus, a platform operating an RS service is more prone to decrease F to achieve an optimal P and S.

5.4.3 Pareto-efficient solutions Consider a bi-objective maximisation problem that aligns with S given by Eq. (5.34) and P given by Eq. (5.30), both of which are functions of F and N, as follows: ! SðF; NÞ (5.39) max ðF;NÞ ˛ U PðF; NÞ where U ¼ fðF; NÞ : F  0; N  0g. This bi-objective problem does not determine the combination of F and N to maximise either S or P (to locate the SO or MO, respectively); instead, it aims to identify a set of regulatory variables that determines the Pareto-efficient frontier of the two objectives. The MO and SO are the two endpoints of the Pareto-efficient frontier, and neither S nor P can be further increased without decreasing the other along the

124 Supply and Demand Management in Ride-Sourcing Markets

Pareto-efficient frontier, including at the two endpoints. To solve this problem, the following Lagrangian function is formed: Z Q LðF; NÞ ¼ f 1 ðzÞdz b$ðw þ tns þ DtÞQ cN þ x$½ðFQ  cNÞ (5.40) 0   ðF Q  cN Þ where x is a Lagrange multiplier, and F  and N  are the decision variables at optimality. The first-order conditions of LðF; NÞ are given by:  1   0 c Qrs wrs þ 2 ¼  bQrs w0 rs 2

(5.41)

 1  vDt xQrs Frs ¼ c wrs þ tns þ bQrs   2 vQrs ð1 þ xÞfrs0

(5.42)

Clearly, the pricing formula in Eq. (5.42) can be viewed as a linear combination of the pricing formulas for the MO and the SO. In addition, Eq. (5.41) leads to the following lemma. Lemma 5-4. The Pareto-efficient solutions in an RS market are always located in the normal regime rather than in the WGC regime.

5.5 Numerical illustrations In this section, a set of numerical experiments is conducted to demonstrate the effects of decision variables on the key endogenous variables, P (of the platform and affiliated drivers) and S. Both detour-unconstrained and constrained scenarios are examined.

5.5.1 Experimental settings The Q function is assumed to have the following negative exponential form: Q ¼ f ðF þ b $ ðw þ trs ÞÞ ¼ Q expfk $ ½F þ b $ ðw þ trs Þg

(5.43)

where Q is the potential passenger demand and k is a parameter representing the demand sensitivity with respect to the generalised cost. Throughout these numerical studies, we assume that Q ¼ 5:0  103 (trips/h), k ¼ 0:02 (1/HKD), b ¼ 60 (HKD/h), tns ¼ 0:4 (h), and c ¼ 50 (HKD/h). In addition, we assume that w is pffiffiffiffiffiffi inversely proportional to the square root of N v , i.e., w ¼ H= N v , where H is set as 5 h. Moreover, we assume that Dt follows an exponential distribution with l (based on the model proportional to Q that is presented in Appendix A).

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This implies that Dt is inversely proportional to Q in the detour-unconstrained scenario, which is consistent with the theoretical discussions. We set l ¼ 0:2Q, and thus Dt ¼ 5=Q in the detour-unconstrained scenario, and we also set g ¼ 2, i.e., Dtd y2Dt. These parameters are chosen for illustrative purposes and with partial reference to previous studies (e.g., Yang and Yang, 2011). In actual operations, the parameters of the proposed functions can be calibrated (e.g., Dt versus Q) and their properties identified using real data.

5.5.2 Detour-unconstrained scenario Here, we verify the theoretical findings with numerical examples obtained in the detour-unconstrained scenario (DtA / þ N). In this scenario, a platform has two key decision variables, F and N, in both an RS and an NS market. P and S are evaluated with different combinations of the two decision variables and are illustrated in two-dimensional contour maps depicted in Fig. 5.4. The optimal values of the contour maps in Fig. 5.4A and B correspond to the MO and the SO, respectively. Fig. 5.4A shows the iso-P contours and the MO of each type of market in a two-dimensional space, with N on the x-axis and F on the y-axis. Clearly, the optimal F and the optimal N for a monopoly in an RS market are lower than those in an NS market. This is because negative price elasticity means that a decrease in F in an RS market directly increases Q, which decreases Dt, which further increases Q (an additional indirect effect). This implies that in general, a decrease in F in an RS market increases passengers’ benefits and thus attracts more passengers than a decrease in F in an NS market. Therefore, a platform

FIGURE 5.4 Profit (of the platform and affiliated drivers) and social welfare in a twodimensional space of vehicle fleet size and trip fare. (A) Profit (103 HKD). (B) Social welfare (103 HKD).

126 Supply and Demand Management in Ride-Sourcing Markets

has a stronger incentive to reduce F (and has a lower optimal F) in an RS market than in an NS market. Fig. 5.4B depicts the iso-social-welfare contours and the SO of the two types of markets in two-dimensional space, with N on the x-axis and F on the y-axis. This reveals that F and N at the SO in an RS market are lower than those at the SO in an NS market. This is also attributable to the additional indirect effect caused by RS, which yields a larger CS with a unit decrease in F in the RS market than in the NS market.

5.5.3 Detour-constrained scenario Although it is difficult to theoretically identify the exact effects of DtA in the detour-constrained scenario, this section provides numerical examples to investigate the operating strategies of a ride-sourcing platform in such a scenario. A platform has three decision variables in this scenario: F, N and DtA . For illustrative purposes, we set N to 500 veh (a relatively low level of supply) and explore the contours of key endogenous variables (w, Q, p, and Dt), P (of the platform and its drivers), and S in a two-dimensional space with DtA on the x-axis and F on the y-axis. Fig. 5.5AeD show the contours of average w, Q, p and Dt, respectively. It is interesting to find that given a fixed N, Q first increases and then decreases with F, which is in contrast to the traditional wisdom that Q monotonically decreases with F. This is due to the fact that a market will collapse into a WGC if the supply is insufficient and/or (due to a low F) Q is extremely large. Therefore, as pointed out by Castillo et al. (2017), increasing F from an extremely low value can rescue a market from the WGC regime and increase realised Q. It can also be seen from Fig. 5.5A that w becomes very large at a very low F, which indicates that vehicles spend a long time picking up passengers, leading to the WGC regime. In addition, p increases with DtA , which is intuitive, but exhibits nonmonotonicity (first increasing and then decreasing) with respect to F. This is because in the normal regime (when F is high), an increase in F decreases Q, which decreases p. Conversely, in the WGC regime (when F is low), an increase in F increases Q, and thus increases p. In addition, Dt monotonically increases with DtA because increasing DtA tends to match more passengers with long detours. It can also be seen that Dt generally increases with F in the normal regime, as an increase in F decreases Q and thus leads to the pairing of passengers with longer Dt. Next, we discuss the MO and SO of DtA and F with different N: a low N ð500 veh) and a high N ð5; 000 veh). Fig. 5.6 shows the iso-contours of P

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FIGURE 5.5 Endogenous variables in a two-dimensional space of allowable detour time and trip fare. (A) Average pick-up time (h). (B) Passenger demand (102 trip/h). (C) Pool-matching probability. (D) Average actual detour time (h).

and S with a low N, with DtA on the x-axis and trip fare on the y-axis. If DtA ¼ 0, an RS market is reduced to an NS market; thus, the MO and SO F in an NS market with DtA ¼ 0 are denoted ‘MO for NS’ and ‘SO for NS’ on the y-axis in the figures. Clearly, the optimal F for an RS market is lower than that for an NS market, with either a P- or S-maximisation objective. This is because if the supply level is low, there is a large decrease in w in response to a unit increase in N v . In this case, an RS service with increased U can greatly reduce w. That is, a decrease in F causes a greater decrease in w in an RS market than in an NS market. Therefore, a platform in an RS market is more incentivised to reduce F than a platform in an NS market. In addition, an RS platform is incentivised to set a relatively large DtA to increase p to maximise P. Fig. 5.7 shows the iso-contours of P and S with a high N, together with the MO and SO, with DtA on the x-axis and trip fare on the y-axis. Interestingly, the MO of an RS market is very close to the MO of an NS market, with both at DtA ¼ 0.This is because the decrease in w in response to a unit increase in N v is small to negligible if N is high. However, pairing passenger requests increases Dt, which decreases Q. In this case, there is only a limited gain from

128 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 5.6 Profit (of the platform and its drivers) and social welfare in a two-dimensional space of allowable detour time and trip fare with a low level of supply (N ¼ 500 veh). (A) Profit (106 HKD). (B) Social welfare (106 HKD).

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FIGURE 5.7 Profit (of the platform and its drivers) and social welfare in a two-dimensional space of allowable detour time and trip fare with a high level of supply (N ¼ 5; 000 veh). (A) Profit (106 HKD). (B) Social welfare (106 HKD).

130 Supply and Demand Management in Ride-Sourcing Markets

the introduction of RS services, and thus a platform is less incentivised to match passenger requests, thereby resulting in a small DtA .

5.6 Conclusion This chapter investigates emerging on-demand RS services provided by a fleet of dedicated drivers affiliated with a ride-sourcing platform. A system of nonlinear equations is established to elucidate the complicated relationships between the platform decision variables (i.e., F, N, and DtA ) and the system’s endogenous variables (Q, p, and Dt) in a ride-sourcing market that provides on-demand RS services and a ride-sourcing market that does not. Based on the modelling framework, the effects of two decision variables (F and N) on P and S in the detour-unconstrained scenario are analysed theoretically. We prove that the MO, SO and Pareto-efficient solutions in an RS market are located in the normal regime rather than in the WGC regime. We also show that under some conditions F at the MO and SO in an RS market is lower than F at the MO and SO in an NS market. This is because a decrease in F not only directly increases Q, due to negative price elasticity, but also has some additional indirect effects, i.e., an increase in Q decreases Dt, which in turn increases Q. We use numerical experiments to further examine the effects of DtA and F on P and S at different N. We show that a platform tends to set a small DtA if N is sufficient and set a large DtA if N is insufficient. This chapter is based on one of our recent article (Ke et al., 2020).

Glossary of notation S trs F t b w Q N r p Dt DtA e t l f Dtd ti tj g

social welfare expected average trip time of passengers opting for an RS service average trip fare average trip time passengers have a homogeneous value of time average waiting time of passengers arrival rate of passengers vehicle fleet size drivers’ reservation rate pool-matching probability average actual detour time allowable detour time a random variable describes Dt between a pair of requests in a detour-unconstrained scenario a parameter of exponential distribution that depends on Q length of the pool-matching window average driver detour time normal trip time for passengers i without RS normal trip time for passengers j without RS an exogenous positive parameter representing the ratio of Dtd to Dt

Equilibrium analysis for ride-pooling services Chapter | 5 A Nv Nc wv Q k wrs Frs Qrs tns N v rs N v ns  Qns wns Fns

131

studied space number of vacant vehicles number of waiting passengers average idle/vacant time of drivers potential passenger demand a parameter representing the demand sensitivity with respect to the generalised cost average waiting time of passengers at optimal RS market average trip fare at optimal RS market arrival rate of passengers at optimal RS market average trip time at RS market number of vacant vehicles at optimal RS market number of vacant vehicles at optimal NS market arrival rate of passengers at optimal NS market average waiting time of passengers at optimal NS market average trip fare at optimal NS market

References Castillo, J.C., Knoepfle, D., Weyl, G., 2017. Surge pricing solves the wild goose chase. In: Proceedings of the 2017 ACM Conference on Economics and Computation. ACM, pp. 241e242. Chen, X.M., Zahiri, M., Zhang, S., 2017. Understanding ridesplitting behavior of on-demand ride services: an ensemble learning approach. Transportation Research Part C: Emerging Technologies 76, 51e70. Ke, J., Yang, H., Li, X., Wang, H., Ye, J., 2020. Pricing and equilibrium in on-demand ride-pooling markets. Transportation Research Part B: Methodological 139, 411e431. Ke, J., Zheng, Z., Yang, H., Ye, J., 2021. Data-driven analysis on matching probability, routing distance and detour distance in ride-pooling services. Transportation Research Part C: Emerging Technologies 124, 102922. Li, W., Cui, Z., Li, Y., Ban, X., 2019. Characterization of ridesplitting based on observed data: a case study of Chengdu, China. Transportation Research Part C: Emerging Technologies 100, 330e353. Shaheen, S., Cohen, A., Zohdy, I., 2016. Shared Mobility: Current Practices and Guiding Principles. U.S. Department of Transportation, Federal Highway Administration (Report No. FHWA-HOP-16-022). Tachet, R., Sagarra, O., Santi, P., Resta, G., Szell, M., Strogatz, S.H., Ratti, C., 2017. Scaling law of urban ride sharing. Scientific Reports 7, 1e6. Uber, 2019a. Uber ExpressPool. https://www.uber.com/us/en/ride/express-pool/. Uber, 2019b. UberPool. https://www.uber.com/us/en/ride/uberpool/. Wang, H., Yang, H., 2019. Ridesourcing systems: a review and framework. Transportation Research Part B: Methodological 129, 122e155. Yan, C., Zhu, H., Korolko, N., Woodard, D., 2020. Dynamic pricing and matching in ride-hailing platforms. Naval Research Logistics (NRL) 67 (8), 705e724. Yang, H., Yang, T., 2011. Equilibrium properties of taxi markets with search frictions. Transportation Research Part B: Methodological 45 (4), 696e713. Zha, L., Yin, Y., Yang, H., 2016. Economic analysis of ride-sourcing markets. Transportation Research Part C: Emerging Technologies 71, 249e266.

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Chapter 6

Ride-pooling services and traffic congestion Jintao Ke1 and Hai Yang2 1

Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China; Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China 2

6.1 Introduction Most previous studies have assumed that vehicles travel at a constant speed and thus ignore the effects of traffic congestion (caused by ride-sourcing vehicles and regular private cars, i.e., background traffic) on market equilibria. The trip fare (F) and vehicle fleet size (N) are two important decision variables in ride-sourcing markets, as F influences the arrival rate of passengers (Q), and N affects the availability of vehicles, which governs the average waiting time (w) and thus affects Q. In the absence of traffic congestion, the average trip time (t) is regarded as constant (or an exogenous parameter). However, in the presence of traffic congestion, an increase in N adds additional vehicle miles to a road network and thus increases the level of traffic congestion, which further influences t and thus affect passengers’ mode choices, meaning that t is no longer constant but determined exogenously. Accordingly, platform operators may take into account the effect of traffic congestion by adjusting their operating strategies (e.g., F and N) to maximise profit (P) or social welfare (S). In addition, the monopoly optimum (MO) and social optimum (SO) of a ride-sourcing market may differ in the presence and absence of traffic congestion. Therefore, as traffic congestion is common, especially in metropolitan areas, incorporating its effects into the modelling of ride-sourcing markets will assist transportation network companies (TNCs) and governments to better manage supply and demand by applying appropriate market operating strategies. In this chapter, we extend the model presented in Chapter 2 to delineate the characteristics of a ride-sourcing market in the presence of traffic congestion. Specifically, we use a macroscopic fundamental diagram (MFD) to depict the relationship between the average speed of vehicles in a market and the densities of ride-sourcing vehicles and background traffic. Similar to analyses in Supply and Demand Management in Ride-Sourcing Markets. https://doi.org/10.1016/B978-0-443-18937-1.00010-3 Copyright © 2023 Elsevier Inc. All rights reserved.

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previous chapters, we consider that a platform can leverage two key decision variables, F and N, to maximise P or S in response to the traffic congestion externality caused by ride-sourcing vehicles and background traffic. We further discuss the MO, SO, and second-best solutions of the ride-sourcing market in the presence of traffic congestion. A few assumptions and model settings warrant mention. First, we use the first-come-first-served (FCFS) model as an example for derivations and note that analyses based on other models can be performed in a similar manner. Second, we assume that the supply is sufficient and r is homogeneous. Third, we focus on equilibrium analyses and operating strategy designs of a ride-sourcing market without a ride-splitting (RS) service, i.e., a non-ridesplitting (NS) market, in the presence of traffic congestion, in which each ride-sourcing vehicle transports only one passenger in each ride.

6.2 Equilibrium analyses 6.2.1 Demand function We consider a city in which passengers who do not own a private car can choose between two transportation modes da ride-sourcing service and public transit d according to these modes’ generalised costs. We use an FCFS dispatching mechanism, in which a platform immediately assigns the nearest idle vehicle to a passenger upon receiving the passenger’s request. This mechanism means that passengers experience a non-negligible w and t, which form part of their generalised cost and affect their willingness to use ride-sourcing services. We denote the average trip and pick-up distances as L and Lp , respectively, and the average network speed as v. In contrast to previous analyses that have considered traffic congestion, we also consider the average
trip and pick-up times, which are endogenously given by t ¼ L=v and w ¼ Lp v, respectively. L is assumed to be a constant, whereas Lp is assumed to be a convex and decreasing function of N v in a stationary equilibrium state and has the following properties: Lp ¼ Lp ðN v Þ,

LP 0 ¼ d Lp dN v < 0, L00 p ¼ d2 Lp dðN v Þ2 > 0, Lp /0 as N v /N, and Lp / N as N v /0. DðF; t; wÞ represents Q for ride-sourcing (and the number of passengers who request a trip per hour or the passenger arrival rate, as before) and it is a function of F, t and w. We further assume that D is a decreasing function of the generalised cost of a ride-sourcing service and has the following form:    L þ Lp ðN v Þ D ¼ f ðF þ b $ ðt þ wÞÞ ¼ f F þ b $ (6.1) v where f 0 < 0, and b is passengers’ value of time and is assumed to be a given constant. This demand function and its properties have been used in many previous studies of conventional taxi markets and emerging ride-sourcing markets (e.g., Yang et al., 2005; Zha et al., 2016). Without loss of

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generality, we assume that b is identical in the pick-up and in-trip phases. Our model can be extended to consider a different b in each phase for future research.

6.2.2 Speed function In contrast to most previous studies that did not consider traffic congestion effects, we assume that v is a function of ride-sourcing N, as follows: v ¼ vðNÞ

(6.2)

Note that v also depends on other exogenous variables, such as the arrival rate of regular private car users (Qn ), the total network length (L), and a speededensity relationship. For illustrative purposes, we consider a linear traffic-flow model expressed as v ¼ a  bk, where a and b are positive parameters, and k represents density and is given by: k¼

ðN n þ NÞ L

(6.3)

where N n represents the number of normal/regular private vehicles. Without loss of generality, we assume that regular private car users have the same L as ride-sourcing passengers. Thus, N n is given by: N n ¼ Qn

L v

(6.4)

By combining Eqs. (6.3) and (6.4) with the linear traffic-flow model, we obtain the following two closed-form solutions of the equilibrium speed in the normal regime (vnor ) and hypercongested flow regime (vcon ): ffi   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 aL  bN þ aL  bN  4LbQn L (6.5) vnor ¼ 2L ffi   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 aL  bN  aL  bN  4LbQn L vcon ¼ (6.6) 2L These solutions lead to the following lemma. Lemma 6-1. An NS market operating according to a linear traffic-flow model of the form v ¼ a  bk has two market equilibrium solutions (i.e., one solution in the normal flow regime and the other solution in the hypercongested flow regime) if sffiffiffiffiffiffiffiffiffiffiffi aL Qn LL 2 (6.7) N < Nmax ¼ b b and one market equilibrium solution if N ¼ Nmax .

136 Supply and Demand Management in Ride-Sourcing Markets

Proof. The equilibrium is given by   L Qn þ N L v 5vL þ bQn ¼ aL  bN v¼ab v L   2 n 5Lv  aL  bN v þ bQ L ¼ 0 ffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2   aL  bN þ aL  bN  4LbQn L 5v ¼ 2L which yields: vnor ¼

vcon ¼

ffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2   aL  bN þ aL  bN  4LbQn L 2L q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi    2 aL  bN  aL  bN  4LbQn L 2L

The existence of the solutions requires 

aL  bN

2

aL 2  4bQn LL  05N  b

sffiffiffiffiffiffiffiffiffiffiffi Qn LL b

(6.8)

(6.9)

(6.10)

(6.11)

It can also be found that vvnor bvnor ffi0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 vN aL  bN  4LbQn L

(6.13)

which indicate that v monotonically decreases with N in the normal flow regime and monotonically increases with N in the congested flow regime. This completes the proof. This lemma reveals that the number of ride-sourcing vehicles admitted to a network should be less than a certain threshold (Nmax ) that is inversely proportional to the arrival rate of private cars (Qn ). Moreover, we prove that vnor (vcon) decreases (increases) with N, as illustrated in the numerical example below. Example 1. Consider a ride-sourcing market operating in a network with L ¼ 50 km, and in which speed and density follow the linear traffic-flow
model v ¼ a  bk, where a ¼ 60 km=h and b ¼ 0:3 km2 h. L is assumed to be 10 km. Fig. 6.1 demonstrates the effect of N on traffic states (v and t), given three levels of Qn , i.e., 0:5; 1:0; and 1:5  104 trip=h. It can be shown

Ride-pooling services and traffic congestion Chapter | 6

FIGURE 6.1 Illustrative traffic states for ride-sourcing with various N.

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138 Supply and Demand Management in Ride-Sourcing Markets

that a feasible N ( 0, Q decreases with N v or decreases with Q, whereas if v þ QL0 p < 0, Q increases with N v or N v increases with Q. The latter scenario represents the wild goose chase (WGC) regime, which was first observed by Castillo et al. (2017) and is a form of market failure in which there is an extremely low N v and thus many vehicle hours are wasted in the pick-up phase (as the matching radius is extremely large). The WGC regime is a unique characteristic of a ride-sourcing market that involves distant online matching between drivers and passengers. In contrast, a street-hailing taxi market involves a physical meeting between drivers and passengers and thus the pick-up phase is negligible; this means that the supply function is simply N ¼ N v þ Qt, and Q is always a decreasing function of N v , i.e., the market always lies in a normal regime. Nv

6.2.4 Equilibrium solution The market equilibrium of an NS market occurs where D exactly equals Q supplied by N, namely, Q ¼ D. Based on Eqs. (6.1) and (6.15), we have     1 N  Nv v vðNÞ ¼ f F þ b $ L þ Lp ðN Þ (6.18) vðNÞ L þ Lp ðN v Þ Eq. (6.18) can be viewed as an implicit equation of endogenous variable N v for a given F and N, which are the two decision variables leveraged by a ridesourcing platform to influence market equilibrium. Taking the partial derivatives of the two sides of Eq. (6.18) with respect to F and N gives vN v f 0 ¼ 0 L0 p vF v þ QL p þ f 0b L þ Lp v   v0 v N  Nv 0 0 þ v þ f b L þ L p vN v L þ Lp L þ Lp v2 ¼ 0 1 v þ QLp vN f 0 bLp 0 þ v L þ Lp

(6.19)

(6.20)

140 Supply and Demand Management in Ride-Sourcing Markets

The partial derivatives of Q (or D) with respect to F and N can be simplified as:   f 0 v þ QLp 0 vQ ¼ (6.21)  L þ Lp vF  v þ QLp 0 þ f 0 bLp 0 v L þ Lp 0 v f 0 bLp 0  f 0 b vQ v ¼  L þ Lp vN v þ QLp 0 þ f 0 bLp 0 v

(6.22)

In a normal regime, where v þ QLp 0 > 0, it is easy to find that N v monotonically increases with F (vN v =vF > 0) and Q monotonically decreases with F (vQ=vF < 0). However, the signs of vN v =vF and vQ=vF are indeterminable in the WGC regime, where v þ QLp 0 < 0. Similarly, in the normal regime, the sign of vQ=vN is indeterminable as it contains an  
additional term associated with traffic congestion effects ( bf 0 v0 L þLp v), the sign of which is different from that of the term that represents the effects of N on pickup time and thus on Q ( f 0 bLp 0 ). This suggests that an increase in N has opposite effects: it increases Q by increasing the availability of vehicles (and thus decreases the pick-up time), but also decreases Q by increasing levels of traffic congestion (and thus increases w and t). In summary, the marginal change in the value of Q caused by a unit increase in N can be either positive or negative, depending on the relative magnitude of the two opposite effects of N. Specifically, with a relatively low level of congestion, Q should increase with N; in contrast, with a relatively high level of congestion, Q may decrease with N.

6.3 Market measures This section investigates the optimal operating strategies in three scenarios: (a) an MO scenario, in which a platform maximises P in the absence of regulations; (b) an SO scenario, in which a platform maximises S regardless of P; and (c) the Pareto-efficient solution scenario, in which neither P nor S can be improved without reducing the other.

6.3.1 Monopoly optimum Recall that E is the wage the platform pays to a ride-sourcing driver for each ride service, and c is the unit time operating cost of one ride-sourcing vehicle. Again, we adopt the assumption made by Zha et al. (2016): the supply is sufficient and r is homogeneous, such that drivers will continue entering the market until their net earnings reach zero, i.e., EQ  cN ¼ 0. Therefore, the

Ride-pooling services and traffic congestion Chapter | 6

141

optimal operating strategies of a monopoly platform can be formulated as the following optimisation equations: max p ¼ FQ  EQ

(6.23)

s:t: EQ  cN ¼ 0

(6.24)

F;F d

Clearly, Eqs. (5.23) and (5.24) are equivalent to the following equation: max p ¼ FQ  cN F;N

(6.25)

which implies that a monopoly ride-sourcing platform in an NS market attempts to maximise its P by altering F and N. The first-order necessary conditions for this are given by:   Lp 0 Lp 0 L þ Lp 0 þ bQ c 1þQ v (6.26) ¼  bQ v v v2   L þ Lp L þ Lp 2 bQ Q v0  0   (6.27) F¼c f v v v þ QLp 0 Eq. (6.27) is the MO pricing formula in an NS market, which is akin to a Lerner formula (Lerner, 1934), given its three terms. The first term  
(c L þLp v) represents the average time cost incurred by a driver serving a passenger in the in-trip phase and pick-up phase, which depends on v the level of congestion. The second term represents a positive congestion externality, which implies that considering the traffic congestion effect incentivises a platform to increase F. The third term is the monopoly markup, which indicates the market power of a monopoly platform to set an F above its equilibrium marginal cost.

6.3.2 Social optimum Next, we investigate which combinations of F and N maximise S in an NS market. S is equal to the sum of consumer surplus (CS) and platform profit (P) (as the provider surplus PS ¼ 0 when the drivers’ reservation rate r is homogeneous), and thus the problem is given by the following equation:   Z Q L þ Lp 1  cN (6.28) max S ¼ f ðzÞdz  Qb F;N v 0 The first-order necessary conditions are given by the following equations:   Lp 0 Lp 0 L þ Lp 0 c 1þQ þ bQ v (6.29) ¼  bQ v v v2   L þ Lp L þ Lp 2 bQ  v0  (6.30) F¼c v v 0 v þ QLp

142 Supply and Demand Management in Ride-Sourcing Markets

Eq. (6.29) can be written as the following differential equation: 

ðb þ cÞQ dLp L þ Lp 0 ¼ c  bQ v v dN v v2

(6.31)

 (6.31)  is the sum of the marginal costs of drivers  Theleft-hand side of Eq. cQdLp bQdLp and passengers in response to a unit increase in N v . The vdN v vdI right-hand side (RHS) of Eq. (6.31) consists of c andan additional term LþL associated with the level of traffic congestion bQ v2 p v0 . In the absence of traffic congestion, the additional term equals zero, and thus the decision variables are chosen such that the sum of the marginal costs of drivers and passengers in the pick-up phase is equal to a constant value of c. In contrast, in the presence of traffic congestion, the decision variables are chosen such that the sum of the marginal costs of drivers and passengers in the pick-up phase is greater than c. Moreover, as the additional term equals zero if the congestion effect is not considered, Eq. (6.31) becomes   Lp 0 Lp 0 (6.32) c 1þQ ¼  bQ v v As Lp 0 < 0, we therefore have v þ QLp 0 > 0, which implies that in markets that have no congestion, the SO is always located in the normal regime. However, in markets with congestion, the
additional term is not zero and has a negative sign, whereas the term bQLp 0 v has a positive sign, and thus it may be that v þ QLp 0 > 0, in which case the SO is always located in the WGC regime. Eq. (6.30) describes the optimal pricing formula at the SO: the first term on the RHS represents the marginal cost per trip, and the second term on the RHS represents the congestion externality that depends on v. The marginal change in the congestion externality with respect to N is given by v0 ¼ vv=vN, and if 
 congestion can be ignored, vv=vNy0, which means that F ¼ c L þLp v at the SO. However, if v0 ¼ vv=vN < 0, v is always less than the free-flow speed in the presence of traffic congestion, and if v þ QLp 0 > 0 in the normal regime, the following proposition can be made. Proposition 6-1. The optimal F at the SO of an NS market in a normal regime in the presence of traffic congestion is always higher than that in the same market in the absence of traffic congestion. The SO in the absence of traffic congestion always leads to a negative P, as FQ  cN ¼ cN v < 0, whereas the SO in the presence of traffic congestion does not necessarily lead to a negative P. From   L þ Lp 2 bQ  v0 Q  0 (6.33) FQ  cN ¼ cN v  v v þ QLp 0 it can be seen that the condition for a positive P at the SO is given by:

Ride-pooling services and traffic congestion Chapter | 6



v þ QLp 0 v ¼ jv j  cN bQ2 0

0

v



v L þ Lp

143

2 >0

(6.34)

This indicates that the profitable first-best SO occurs in a severely congested market in which the entry of additional ride-sourcing vehicles results in a large marginal congestion effect.

6.3.3 Pareto-efficient solutions Consider a bi-objective maximisation problem that aligns with S given by Eq. (6.28) and P given by Eq. (6.25), both of which are functions of F and N, as follows: ! SðF; NÞ max (6.35) ðF;NÞ ˛ U PðF; NÞ where U ¼ fðF; NÞ : F  0; N  0g. This biobjective problem does not determine the combination of F and N required to maximise S or P (which lead to the SO and the MO, respectively), but instead aims to identify a set of regulatory variables that determine the Pareto-efficient frontier of S and P. The MO and the SO are the two endpoints of the Pareto-efficient frontier, which means that neither S nor P can be further improved along the Paretoefficient frontier without reducing the other, including at the two endpoints. To solve this problem, the following Lagrangian function is formed:   Z Q L þ Lp 1 cN þ x$½ðFQ  cNÞ ðF  Q  cN  Þ f ðzÞdz Qb LðF; NÞ ¼ v 0 (6.36) where x is a Lagrange multiplier, and F  and N  are the decision variables at optimality. The first-order necessary conditions of this problem are given by:   Lp 0 Lp 0 L þ Lp 0 þ bQ v (6.37) c 1þQ ¼  bQ v v v2   L þ Lp L þ Lp 2 bQ x Q  v0   (6.38) F¼c 1 þ x f0 v v v þ QL0p Clearly, the optimal pricing formula in the Pareto-efficient frontier given by Eq. (6.38) is a linear combination of the pricing formulas for the MO and the SO in an NS market. Moreover, Eq. (6.37) has the same form as Eqs. (6.26) and (6.29). We assume that Lp is given by the specific function Lp ¼ Rm ðN v Þa , where Rm and a are positive parameters, which satisfies all of the inherent properties of R. Then, by substituting Lp ¼ Rm ðN v Þa and Lp 0 ¼
aLp N v into Eq. (6.37), we obtain:

144 Supply and Demand Management in Ride-Sourcing Markets

  L þ Lp 0 1 v c  bQ N v ¼ aðb þ cÞQLp 2 v v

(6.39)

The U of ride-sourcing vehicles is given by the ratio of the number of intrip vehicles (QL=v) to N. Thus, N v is given by:     Q Lp þ L QL 1 Lp v ¼ 1 N ¼N  (6.40) v v U L Substituting Eq. (6.40) into Eq. (6.39) yields:   L þ Lp 0   v c  bQ 1 v2   1 L Lp ¼ L þ Lp 0 U aðb þ cÞ þ c  bQ v v2

(6.41)

In the absence of traffic congestion, Eq. (6.41) can be reduced to   c 1 1 L (6.42) R¼ aðb þ cÞ þ c U Eq. (6.42) indicates that in an NS market without congestion effects, Lp is inversely proportional to U at the Pareto frontier, with a coefficient that is a constant. Eq. (6.41) indicates that the same relationship exists in an NS market with congestion effects, but that the coefficient depends on the level of traffic congestion.

6.4 Conclusion This chapter investigates the stationary equilibrium state of an NS ridesourcing market in the presence of traffic congestion. An MFD model is used to describe the relationship between v and the densities of vehicles. We examine the MO and SO of an NS market with congestion, which reveals that the SO may be sustainable, with a positive P, if the traffic congestion is severe. In contrast, in a ride-sourcing market in the absence of traffic congestion, the SO is always unsustainable. We also show that the F at the SO in a market with traffic congestion is always higher than it is at the SO in a market without traffic congestion. This is because the presence of traffic congestion affects both the t and pick-up time of passengers, which means that a government alleviates traffic congestion by simultaneously increasing F and decreasing N to maximise S. This demonstrates that traffic congestion must be considered in the planning and operation of ride-sourcing markets, especially in congested areas. However, the model presented in this chapter is only adapted to a ridesourcing market without RS. In the next chapter, we examine the market equilibrium properties of an RS market and compare these with the outcomes presented in this current chapter. This chapter is based on one of our recent articles (Ke et al., 2020).

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145

Glossary of notation S F t b w Q N r P L Lp v L Qn Nn vnor vcon Nv

social welfare average trip fare average trip time passengers have a homogeneous value of time average waiting time of passengers arrival rate of passengers vehicle fleet size drivers’ reservation rate profit average trip distances average pick-up distances average network speed total network length the arrival rate of regular private car users number of normal/regular private vehicles equilibrium speed in the normal regime equilibrium speed in the hypercongested flow regime number of vacant vehicles

References Castillo, J.C., Knoepfle, D., Weyl, G., June 2017. Surge pricing solves the wild goose chase. In: Proceedings of the 2017 ACM Conference on Economics and Computation, pp. 241e242. Ke, J., Yang, H., Li, X., Wang, H., Ye, J., 2020. Pricing and equilibrium in on-demand ride-pooling markets. Transportation Research B: Methodological 139, 411e431. Lerner, A.P., 1934. The concept of monopoly and the measurement of monopoly power. The Review of Economic Studies 1, 157e175. Yang, H., Ye, M., Tang, W.H., Wong, S.C., 2005. Regulating taxi services in the presence of congestion externalities. Transportation Research A, Policy and Practice 39 (1), 17e40. Zha, L., Yin, Y., Yang, H., 2016. Economic analysis of ride-sourcing markets. Transportation Research C: Emerging Technologies 71, 249e266.

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Chapter 7

Equilibrium analysis for ridepooling services in the presence of traffic congestion Jintao Ke Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China

7.1 Introduction The emergence of ride-sourcing services has led to various debates, with one of the most contentious concerning the potential effects of ride-sourcing services on traffic congestion. Advocates of ride-sourcing services maintain that such services complement existing modes in transportation systems and decrease car ownership, thereby reducing traffic congestion. Moreover, transportation network companies (TNCs) can execute more efficient matching between drivers and passengers than typical taxi-hailing services, which improves vehicle utilisation (by reducing the time drivers spend searching for passengers on streets), thereby further reducing traffic congestion. These claims are supported by a study by Li et al. (2016), whose analysis of datasets from Uber and the Urban Mobility Report revealed that the entry of Uber services has significantly reduced traffic congestion in urban areas of the United States. However, critics claim that as TNCs provide more convenient, cheaper and comfortable ride services, they increase vehicle traffic by attracting those passengers who would otherwise use space-efficient modes such walking, public transit or biking. For example, a recent consultancy report (Schaller Consulting, 2018) stated that TNCs added 5.7 billion miles of driving per year in the metropolitan areas of nine cities in the United States (Boston, Chicago, Los Angeles, Miami, New York, Philadelphia, San Francisco, Seattle and Washington, DC). Recently, TNCs have launched on-demand ride-splitting (RS) services to improve vehicle utilisation rates as these services are regarded as an effective way to achieve various societal benefits, such as alleviating traffic congestion and decreasing fuel consumption. For example, it was reported that by 2022 Lyft aims to have 50% of its rides being shared (Schaller Consulting, 2018). Whether RS services attract passengers from more space-efficient modes, as mentioned above, Supply and Demand Management in Ride-Sourcing Markets. https://doi.org/10.1016/B978-0-443-18937-1.00008-5 Copyright © 2023 Elsevier Inc. All rights reserved.

147

148 Supply and Demand Management in Ride-Sourcing Markets

has been examined empirically (Alexander and Gonza´lez, 2015; Chen et al., 2020; Li et al., 2016; Schaller Consulting, 2018; Zheng et al., 2019), but none of these studies’ findings were equivocal, partly due to their dependency on demand patterns, supply patterns and urban transportation network topologies. In another approach, Ke et al. (2020) used a theoretical model to examine the aggregate effects of RS on levels of traffic congestion, which revealed that RS services generated a winewin situation for ride-sourcing passengers and private car users. However, their model examined the effects of a platform’s equilibrium operating strategies by varying the level of exogenous demand. In other work, Yang et al. (2005) proposed a model to describe the equilibrium state of a taxi market in the presence of traffic congestion. However, their model cannot be directly adapted to a ride-sourcing market because such a market involves vehicles being dispatched via a platform to pick up passengers, in addition to empty vehicle relocating, which differs from a traditional taxi market based on drivers with empty vehicles cruising around looking for passengers. The extent to which RS reduces congestion relies on a few key factors. One key factor for a successful RS programme is the passenger demand (D) for RS services. As D increases, a platform can match more passengers and achieve a higher poolmatching probability (p). This means that fewer vehicles are required to serve a given number of passengers if D is high, which decreases traffic congestion and the average trip time (t) of ride-sourcing passengers and private car users. Consequently, if D is high, RS services are more economical (due to a decreased t) and environmentally friendly (due to decreased traffic congestion) than non-ridesplitting (NS) services. Furthermore, a certain high D can mean that RS services generate a winewin situation for ride-sourcing passengers and private car users. Thus, it is important to identify the D required to generate such an optimal result. Another key factor for a successful RS programme to reduce congestion is the pool-matching strategy that is used, such as what length of the poolmatching window (f) is selected. This is because at the end of a poolmatching window, passengers opting for RS services are either matched or not matched with another passenger. Thus, the pool-matching window greatly affects p and the time cost of passengers. If f is large, then p is high, which reduces traffic congestion and passengers’ t. However, a large f also directly increases the average waiting time (w). Therefore, the magnitude of f has positive or negative effects on the time cost of passengers and thereby influences passengers’ choice between an NS and an RS service. It is thus critical to determine the optimal f to minimise passengers’ time cost and to identify the range of f that generates a winewin situation for a given D. To address the above aspects, this chapter develops a theoretical model to study the equilibria of ride-sourcing markets with RS services in the presence of traffic congestion. A platform operating RS services affects the market equilibrium by leveraging its trip fare (F), vehicle fleet size (N), and f, which governs the trade-offs between p (directly affecting the utilisation rate of drivers U) and the additional w of pool-matched passengers. The effects of

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149

these decision variables on the equilibria of markets with RS are examined via theoretical analyses and numerical studies. We then discuss and compare the MO, SO, and the second-best solutions of ride-sourcing markets with and without RS in the presence of traffic congestion. Consequently, our findings will assist city managers and ride-sourcing platform operators to develop ondemand RS services to decrease traffic congestion.

7.2 Equilibrium analyses In this section, we extend the model presented in Chapter 6 to an RS market with congestion effects. In this case, a pool-matching mechanism is used to match pairs of passengers who select RS services. For analytical tractability, we use the pre-assigned pool-matching mechanism described in Section 5.1.2 of Chapter 5, as this mechanism simplifies the RS procedure by pool-matching two passengers with close origins and close destinations. For simplicity, we also assume that each ride-sourcing vehicle can serve a maximum of two passenger requests in each ride.

7.2.1 Vehicle conservation In an RS market using the pre-assigned pool-matching mechanism, there are two types of in-trip vehicles: (a) vehicles occupied by only one un-poolmatched passenger; and (b) vehicles occupied by two pool-matched passengers. In terms of p and passenger demand (Q), the service rate of ride-sourcing vehicles (mðQ; fÞ), i.e., the number of vehicles dispatched to serve RS passengers per hour, is given by:   
1 1 (7.1) mðQ; fÞ ¼ Q pðQfÞ þ ½1 pðQfÞ ¼ Q 1  pðQfÞ 2 2 where the first term in brackets shows that each passenger occupies half a vehicle if he/she is pool-matched, whereas the second-term in brackets shows that each un-pool-matched passenger occupies a whole vehicle. We denote the partial derivatives of m with respect to Q and f as m1 and m2 , respectively. At market equilibrium, f is a decision variable determined by the platform, whereas Q is an endogenous variable that depends on f. Thus, we have vmðQ; fÞ 1 1 ¼ m1 ¼ 1  pðQfÞ  p0 Qf vQ 2 2   vmðQ; fÞ vQ 1 1 0 1 vQ ¼ m1 þ m2 1  pðQfÞ  p Qf  Q2 p0 ¼ vf vf 2 2 2 vf

(7.2) (7.3)

where the first term on the right-hand side (RHS) of Eq. (7.2) represents the indirect effect of f on mðQ; fÞ due to variations in Q, and the second term on

150 Supply and Demand Management in Ride-Sourcing Markets

 the RHS of Eq. (7.3), i.e., m2 ¼ Q2 p0 2, represents the direct effect of f on mðQ; fÞ, which is always negative.1 By applying the specific formula for p given in Eq. (5.10), we obtain the following lemma. Lemma 7-1. If p is given by Eq. (5.10): m ðQ; fÞ increases with Q for all f  0; the marginal increase in mðQ; fÞ due to a unit increase in Q is less than 1, i.e., vmðQ; fÞ=vQ < 1; and the elasticity of m ðQ; fÞ with respect to Q vmðQ;fÞ

Q is less than 1, i.e., εm Q ¼ m < 1. vQ Proof. Taking the partial derivative of MðQ; fÞ with respect to q gives:

vMðQ; fÞ 1 ¼ ½1 þ z expðgQfÞ zgQfexpðgQfÞ vQ 2

(7.4)

If we let eðxÞ ¼ 1 þ expðxÞ xexpðxÞ, where x  0, then the first-order derivative of eðxÞ is equal to e0 ðxÞ ¼ ðx 2ÞexpðxÞ. Clearly, e0 ðxÞ < 0 if xe½0; 2Þ and e0 ðxÞ > 0 if xe½2; þNÞ, which implies that eðxÞ reaches its global minimum at x ¼ 2. As eð2Þz0:86 > 0, we can conclude that eðxÞ is always positive in the range of xε½0; þNÞ and thus vMðQ; fÞ=vQ > 0, indicating that MðQ; fÞ always increases with Q. Also, as eð0Þ ¼ 2 and eð þNÞ ¼ 1, eðxÞ ¼ 1 þ z < 2. This implies that vVðQ; fÞ=vQ < 1. In addition, the elasticity satisfies   1 1 1  pðQfÞ  p0 Qf Q vmðQ; fÞ Q 2 2   0, which indicates a normal regime, whereas Q increases with N v or N v increases with Q, which indicates the wild goose chase (WGC) regime. The fact that both of these regimes are possible is due to the non-monotonic properties of Q as a function of N v , as indicated in Eq. (7.8): an increase in N v decreases N  N v and thus decreases Q and m; in addition, an increase in N v decreases Lp ðN v Þ and thus increases Q and m.

7.2.4 Equilibrium solution Similar to an NS market, the market equilibrium of an RS market can be achieved when D given by Eq. (7.6) equals the Q supplied by N working drivers, i.e., Q ¼ D, which is equivalent to

Equilibrium analysis for ride-pooling services Chapter | 7

mðQ; fÞ ¼ mðD; fÞ

153

(7.13)

or     

1 N  Nv v þH ;f vðNÞ ¼ m f F þ b $ L þ Lp ðN Þ vðNÞ L þ Lp ðN v Þ

(7.14)

which can be regarded as an implicit equation of the endogenous variable N v , given the decision variables of F, N and f. Taking the partial derivatives of both sides of Eq. (7.14) with respect to the three decision variables yields: vN v m01 f 0 ¼ 0 vF v þ mLp 1 þ m01 f 0 b L0p v L þ Lp

(7.15)



v0 v N  Nv 0 þ v þ m01 f 0 b L þ Lp 2 L þ Lp L þ Lp vN v ¼ 0 v þ mL vN 1 p m01 f 0 bL0p þ v L þ Lp

(7.16)

1 2 0 vH Q p  m01 f 0 b vN v 2 vf ¼ 0 v þ mL vf 1 p þ m01 f 0 bL0p v L þ Lp

(7.17)

v

Based on these results, the partial derivatives of Q (or D) with respect to F, N and f are given by:

0 0 f v þ mL p vQ ¼ (7.18)

L þ Lp vF v þ mL0p þ m01 f 0 bL0p v vQ ¼ vN

L þ Lp 0 v f 0 bL0p  f 0 b v

L þ Lp v þ mL0p þ m01 f 0 bL0p v

(7.19)

154 Supply and Demand Management in Ride-Sourcing Markets



vH 1 L0p L þ Lp þ Q 2 p0 f b vþ vf 2 v v þ mL0p vQ ¼ L þ Lp vf ðv þ mR0 Þ þ m01 f 0 bL0p v 0

!

mL0p

(7.20)

It can be seen that in the normal regime (with v þ mL0p > 0), N v strictly increases with F, and Q strictly decreases with F. In contrast, in the WGC regime (with v þ mL0p > 0), these two monotonicity properties do not necessarily hold. In addition, the sign of vQ=vN is indeterminate, even in the normal regime, as a unit increase in N increases Q by increasing the availability of vehicles (i.e., decreasing the pick-up time), but decreases Q by increasing congestion. Moreover, Eq. (7.20) describes the aggregate effect of f on Q in two opposite ways. On the one hand, a large f directly increases the additional w for pool-matching (H); on the other hand, a large f increases p, which further decreases mðQ; fÞ, thereby releasing more idle vehicles . and thus decreasing the w of passengers (which is reflected by the term L0p v). The first

effect suppresses passenger demand, while the second effect induces passenger demand. As a result, the sign of vQ=vf depends on the relative magnitudes of these two effects.

7.3 Market measures This section investigates the optimal operating strategies in an RS market with traffic congestion in three scenarios: (a) an MO scenario, in which a platform maximises P in the absence of regulations; (b) an SO scenario, in which a platform aims to maximise S regardless of P; and (c) a Pareto-efficient solution scenario, in which neither P nor S can be improved without reducing the other.

7.3.1 Monopoly optimum (MO) The optimal operating strategies of a monopoly platform in an RS market can be found by solving the following optimisation problem: max P ¼ FQ  cN

F;N;f

(7.21)

The first-order conditions of this problem with respect to F, N and f, after some arrangements, are given by:   L0p L0p L þ Lp 0 c 1þm v (7.22) ¼  bQ þ bQ v v v2

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155

  L þ Lp 0 bQm01 L þ Lp 2 0 Q

m1  F¼c v  0 f v v v þ mL0

(7.23)

L0p L þ Lp vH 1 1 dLp vN v ¼  Q2 p0 m2 ¼  vf 2 v dN v vm v v þ mL0p

(7.24)

p

Eq. (7.23) describes the MO pricing formula and has three major terms, akin to the Lerner formula. The first term represents the effect of the average time cost of drivers in the in-trip phase and pick-up phase for each trip. From Lemma 7-1, we have 0 < m01 ¼ vmðQ; fÞ=vQ < 1, and thus we find that the marginal cost per trip in an RS market is less than that in the NS market. This is because each driver can serve two passengers in each ride (in the case of successful pool-matching) in an RS market, and thus the marginal cost per trip is reduced to m01 . The second term represents the externality caused by traffic congestion, which is positive. This implies that a platform is incentivised to increase F in the presence of traffic congestion. In Eq. (7.24), the partial derivative vH=vf indicates the marginal increase in w for pool-matching due to a unit increase in f, which further increases passengers’ generalised cost and reduces Q. In contrast, the  RHS of Eq. (7.22) represents the marginal decrease in pick-up time cost (Lp v), which further decreases passengers’ generalised cost and increases Q and is due to the direct effect of a unit increase in f. This implies that f is chosen such that the marginal negative and positive effects of a unit increase in f on Q are identical. Furthermore, as seen from Eq. (7.21), f influences P only via Q (i.e., not via F and N), and thus f is chosen such that a unit increase in f has identical marginal positive and negative effects on P.

7.3.2 Social optimum (SO) In an RS market, the problem of maximising S takes the following form with the three decision variables of F, N and f:

  Z Q L þ Lp 1 þ H  cN (7.25) f ðzÞdz Qb maxS ¼ F;N;f v 0 The first-order necessary conditions are given by:   L0p L0p L þ Lp 0 v c 1þm ¼  bQ þ bQ v v v2   L þ Lp 0 bQm01 L þ Lp 2 0

m1  v F¼c v v v þ mL0 p

(7.26)

(7.27)

156 Supply and Demand Management in Ride-Sourcing Markets

L0p L þ Lp vH 1 1 dLp vN v ¼  Q 2 p0 m2 ¼ 0 vf 2 v dN v vm v v þ mLp

(7.28)

Eq. (7.27) describes the optimal pricing formula at the SO. The first term on the RHS is the marginal cost in the in-trip and pick-up phases per ride service (which is less than that in an NS market), while the second term is the congestion externality associated with v. Furthermore, Eq. (7.28) shows that the SO f is achieved where the marginal increase  in w for pool-matching (vH=vf) is equal to the marginal decrease in Lp v due to the direct effect of a unit increase in f. This also implies that at the SO, the f is chosen such that the marginal positive and negative effects on Q caused by a unit increase in f are identical. In addition, as v0 ¼ vv=vN < 0, v is always less than the free-flow speed in the presence of traffic congestion, and as v þ mL0p > 0 in the normal regime, we make the following proposition. Proposition 7-1. In a normal regime, the optimal F at the SO of an RS market in the presence of traffic congestion is always higher than that at the SO of an RS market in the absence of traffic congestion. Similar to an NS market, P at the SO is given by:     L þ Lp bQ2 m1 L þ Lp 2 0

m1 Q  N  FQ cN ¼ c v (7.29) v v v þ mL0p From Lemma 7-1, we have εm Q ¼ m1 Q=m < 1, which indicates that m1 Q < m. Therefore, P in the absence of traffic congestion satisfies:     L þ Lp L þ Lp m1 Q  N < c m  N ¼ cN v < 0 (7.30) FQ cN ¼ c v v This indicates that P at the SO in an RS market without congestion is always negative. However, this does not necessarily hold in an RS market with congestion, and the condition required to achieve a non-negative P at the SO is given by:     L þ Lp bQ2 m1 L þ Lp 2 0

FQ cN ¼ c m1 Q  N  v 0 (7.31) v v v þ mL0p Equivalently, 

 L þ Lp 0 0 m1 Q v ¼ jv j  c N  v



v þ mL0p  bQ2 m1

v L þ Lp

2 >0

(7.32)

which implies that the profitable first-best SO emerges in a highly congested market in which there is a sufficiently large marginal congestion effect caused by an additional increase in N.

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7.3.3 Pareto-efficient solutions Consider a bi-objective maximisation problem that aligns with S given by Eq. (7.25) and P given by Eq. (7.21), both of which are functions of F, N and f, as follows: ! SðF; N; fÞ max (7.33) ðF;N; fÞ ˛ U PðF; N; fÞ where U ¼ fðF; N; fÞ : F  0; N  0; f  0g. This bi-objective problem does not determine the combinations of F, N and f to maximise S or P (which lead to the SO and the MO, respectively), but instead aims to identify a set of regulatory variables that determine the Pareto-efficient frontier of S and P. The MO and the SO are the two endpoints of the Pareto-efficient frontier, and neither S nor P can be further improved without reducing the other along the Pareto-efficient frontier, including at the two endpoints. To solve this problem, the following Lagrangian function is formed

  Z Q L þ Lp 1 þ H cN þ x$½ðFQ  cNÞ f ðzÞdz Qb LðF; NÞ ¼ v 0 ðF  Q  cN  Þ

(7.34)

where x is a Lagrange multiplier, and F  and N  are the decision variables at optimality. The first-order necessary conditions of this problem are given by:   L0p L0p L þ Lp 0 v (7.35) c 1þm ¼  bQ þ bQ v v v2   L þ Lp 0 bQm01 L þ Lp 2 0 x Q

m1  v  (7.36) F¼c 1 þ x f0 v v v þ mL0 p

L0p L þ Lp vH 1 1 dLp vI ¼  Q2 p0 m2 ¼ vf 2 v dI vm v v þ mL0p

(7.37)

Similarly, the optimal pricing formula in Eq. (7.36) is a linear combination of the pricing formulas at the MO and the SO. As the optimal f at both the MO and SO in an RS market follows the same formula (given by Eq. 7.37), we conclude that the sum of the marginal decrease in w and t costs is always equal to the marginal increase in the additional w cost of passengers for poolmatching that results from a unit increase in f at the Pareto-efficient frontier. This is because f affects P and S only via Q (not via F and N). As a result, the marginal positive or negative effects of an increase in f on P or S are directly governed by the marginal positive or negative effects of an increase in f on Q, or equivalently, the generalised cost.

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Furthermore, Eq. (7.35) has the same formulation as Eq. (7.22) at the MO and Eq. (7.26) at the SO. Thus, by assuming that Lp ¼ Rm ðN v Þa , Eq. (7.35) can be transformed to   L þ Lp 0 1 (7.38) c  bQ v N v ¼ aLp ðbQ þ cmÞ v v2 If we define the U in an RS market as the ratio of the number of in-trip vehicles (mL=v) to N, N v can be written as:

  m Lp þ L mL 1 Lp v 1 ¼ N ¼N  (7.39) v U L v Substituting Eq. (7.39) into Eq. (7.38) yields:   L þ Lp 0   v c  bQ 1 v2    1 L Lp ¼  Q L þ Lp 0 U a b þ c þ c  bQ v m v2

(7.40)

In the absence of traffic congestion, Eq. (6.40) can be reduced to   c 1 1 Lp ¼ 0 1 L (7.41) U 1 B C a @b þ cA þ c 1  12 pðQfÞ Eq. (7.41) shows that in an RS market without congestion, Lp is inversely proportional to U and decreases with pðQfÞ at the Pareto frontier. It can also be found that pðQfÞ is inversely proportional to Lp . This is because as more passengers are pool-matched, fewer vehicles are required for serving a given number of passengers, and thus more idle vehicles are released, thereby decreasing Lp . Eq. (7.41) also indicates that Lp depends on the level of traffic congestion.

7.4 Numerical studies In this section, we provide a numerical example to illustrate the effects of the decision variables in an RS and NS market on the market equilibrium, P, and S. We assume that the relationship between v and density follows a linear traffic-flow model described by v ¼ a  bk, where a ¼ 60 km=h, b ¼  0:3 km2 h and the Q function is Q ¼ f ðCÞ ¼ Q expðk $ CÞ

(7.42)

where C is the generalised cost, which equals F þ b$ðt þ wÞ in the NS market and F þ b$ðt þ w þ HÞ in the RS market; and k is an exogenous parameter

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representing the sensitivity of Q to cost. This specific Q function satisfies the stated assumptions of Q  Q and f 0 < 0. Throughout the numerical studies, we assume that the potential Q ¼ 1:0  104 (trips/h), k ¼ 0:03 (1/HKD), b ¼ 60 (HKD/h), L ¼ 10 (km), c ¼ 50 (HKD/h) and L ¼ 50 (km). We then assume pffiffiffiffiffiffi that Lp is inversely proportional to the square root of N v , i.e., Lp ¼ h= N v , where h ¼ 100 ðkmÞ. In addition, we assume that Q is uniform and thus H ¼ f=2. These parameters are selected for illustrative purposes; one may calibrate relevant parameters as required for actual operations, using real-world data.

7.4.1 Equilibrium outcomes Fig. 7.1 demonstrates the demand curves (dashed line) and supply curves (solid lines) and their intersections (i.e., the equilibrium outcomes) vs. N v in an RS market (blue lines) and an NS market (red lines) for different cases. The NS market has f ¼ 0 ðhÞ, while the RS market has f ¼ 0:05 ðhÞ. In case (a), with F ¼ 0 and N ¼ 1000 ðvehÞ, it can be seen that both markets fall into the WGC regime (an increase in the segment of the supply function with respect to N v ), indicating that most of the vehicle hours are wasted on picking up passengers, rendering the system inefficient. In case (b), with F ¼ 20 ðHKDÞ and N ¼ 1000 ðvehÞ, the equilibrium of the RS market moves to a normal regime

FIGURE 7.1 Various equilibrium cases of non-pooling and ride-pooling markets. (A) Case. (B) Case. (C) Case.

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(a decrease in the segment of the supply function with respect to N v ), whereas the equilibrium of the NS market remains in the WGC regime. In case (c), with F ¼ 80 ðHKDÞ and N ¼ 300 ðvehÞ, both markets achieve equilibrium in a normal regime. These results imply that different combinations of decision variables may result in WGC or normal outcomes in RS and NS markets. Case (b) shows that given the same F and N, an RS market may be able to avoid a WGC regime, whereas an NS market may not. We further examine the differences between the feasible sets of decision variables that can enable the RS and NS markets to avoid the WGC regime. We solve the equilibria of the two markets and identify whether each equilibrium is in a WGC or normal regime under various combinations of F and N. Fig. 7.2 demonstrates the WGC and normal outcomes in a two-dimensional space of N and F for various values of f ð0; 0:05; 0:1 ðhÞÞ, with Fig. 7.2A showing the results in the absence of traffic congestion (with a constant v equal to the free-flow speed) and Fig. 7.2B showing the results in the presence of traffic congestion. It can be seen that the RS market (with f ¼ 0:05; 0:1 ðhÞ) has a smaller WGC area than the NS market (with f ¼ 0 ðhÞ). This implies that an RS market can more often avoid the WGC regime than an NS market, and thus shows that an RS market provides more scope for efficient platform operations (i.e., enables a platform to explore the ability of more combinations of decision variables to generate desirable outcomes) than an NS market.

7.4.2 Optimal operating strategies (non-pooling market) This subsection illustrates the effects of the decision variables at a market equilibrium. Fig. 7.3 displays P, S, t and w in a two-dimensional space of F and N in an NS market (or an RS market with no matching window). The results for an uncongested market (in the absence of traffic congestion and with a constant free-flow v ¼ a) and a congested market (in the presence of traffic congestion) are displayed. The green dashededotted lines and black dashededotted lines indicate the boundaries between WGC and normal regions in congested and uncongested markets, respectively. It can be seen that

FIGURE 7.2 Phase diagrams of WGC and non-WGC (normal) outcomes in terms of vehicle fleet size and trip fare with various lengths of matching window (f). (A) In the absence of congestion. (B) In the presence of congestion.

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FIGURE 7.3 Phase diagrams of platform profit, social welfare, average trip time and pick-up time in terms of vehicle fleet size and trip fare in a non-pooling market (i.e., matching window length ¼ 0) in the absence and presence of traffic congestion. (A) Platform profit ( 104 HKD) (B) social welfare (104 HKD). (C) Average trip time (0:1 h). (D) Average pick-up time (h).

the MO and the SO in both the congested and uncongested markets are located in their respective normal regions. Moreover, the MO F in the congested market (the red dot in Fig. 7.3A) is higher than that in the uncongested market (the blue dot in Fig. 7.3B), which implies that a platform is prone to increase F to maximise its P if traffic congestion has non-negligible effects on passengers’ t and w. This observation is consistent with the theoretical findings in Proposition 7-1. Fig. 7.3C shows that t does not depend on F but increases with N. In addition, Fig. 7.3D shows that w decreases with N and F in both congested and uncongested markets. The former occurs because a large N directly increases the availability of idle vehicles, whereas the latter occurs because an increase in F reduces D and thus releases more idle vehicles. It can also be seen that under a given combination of F and N, the congested market has a longer pick-up time than the uncongested market, due to the presence of traffic congestion.

7.4.3 Optimal operating strategies (ride-pooling market) Fig. 7.4 demonstrates the effects of F and N on P and S in an RS market with f ¼ 0:1 ðhÞ. The trends visible in Fig. 7.4 are similar to those visible in Fig. 7.3: the market without congestion has a lower SO F than the market with

162 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 7.4 Phase diagrams of platform profit, social welfare, average trip time and pick-up time in terms of vehicle fleet size and trip fare in a ride-pooling market with a matchingwindow length ¼ 0:1 h in the absence and presence of traffic congestion. (A) Platform profit (104 HKD). (B) Social welfare (104 HKD). (C) Average trip time (0:1 h). (D) Average pick-up time (h).

congestion. This is because traffic congestion causes a platform or a government to charge a higher F to offset the negative effects of congestion and thus achieve an optimal P or S. In addition, the congested market has a smaller positive P region (i.e., fewer feasible combinations of F and N) than the uncongested market. This indicates that a platform that does not consider the effects of traffic congestion may overestimate its feasible sets of decision variables. Moreover, F at the SO in each of the regimes in the RS market is generally lower than F at the SO in each of the regimes in the NS market. This is because a large f increases passengers’ generalised cost, and thus a platform must decrease F to compensate passengers and attract sufficient D.

7.4.4 Effects of matching window We next compare the effects of F and f on P and S with different N: a low N (500 veh; Fig. 7.5) and a high N ð1; 800 veh; Fig. 7.6). Interestingly, the MO and SO occur at a relatively large f with a low N (Fig. 7.5) and at a relatively small f at a high N (Fig. 7.6). This is because an increase in f has two major

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FIGURE 7.5 Phase diagrams of platform profit and social welfare in terms of matching-window length and trip fare in a ride-pooling market with a small fleet size (N ¼ 500 veh). (A) Profit. (B) Social welfare.

FIGURE 7.6 Phase diagrams of platform profit and social welfare in terms of matching-window length and trip fare in a ride-pooling market with a large fleet size (N ¼ 1; 800 veh). (A) Profit. (B) Social welfare.

effects: (a) it increases H (this effect is steady, as H is linearly proportional to f); and (b) it increases p, and thus releases more idle vehicles, which consequently reduces w. Due to the fact that w is a decreasing and convex function of N v , the second effect decreases with N v (which positively depends on N). Therefore, at a low N, the second effect dominates, i.e., a unit increase in f significantly decreases w, which means a platform is strongly incentivised to set a relatively large f. Conversely, at a high N, there are sufficient idle vehicles, such that a unit increase f only marginally decreases pick-up time and also deters some passengers by increasing their w for pool-matching, and thus a platform must impose a relatively small f in these cases. If N is sufficiently large, a platform may set f to zero, thereby creating an NS service.

7.5 Conclusion and remarks This chapter investigates the stationary equilibrium state of an RS market in the presence of traffic congestion, and how its equilibrium state depends on

164 Supply and Demand Management in Ride-Sourcing Markets

operating strategies in terms of F, N, and f. Our model can characterise the effects of traffic congestion on both t and pick-up times of passengers, which affect passengers’ willingness to choose ride-sourcing services. We find that a feasible operating strategy may lead to two equilibrium solutions: a normal (non-WGC) outcome or an inefficient (WGC) outcome. We then ascertain the conditions for the MO and the SO for maximising P and S in these markets, respectively, in the absence and in the presence of traffic congestion. We prove that the F at the MO and SO in markets with congestion is always higher than the F at the MO and SO in markets without congestion. In addition, we use numerical studies to illustrate these theoretical findings and obtain some useful managerial insights. For example, a platform can maximise its P or S by imposing a relatively large f at a low N and a relatively small f at a high N.

Glossary of notation S F t b w Q N r P L Lp v L Qn Nn vnor vcon Nv

social welfare average trip fare average trip time passengers have a homogeneous value of time average waiting time of passengers arrival rate of passengers vehicle fleet size drivers’ reservation rate profit average trip distances average pick-up distances average network speed total network length the arrival rate of regular private car users number of normal/regular private vehicles equilibrium speed in the normal regime equilibrium speed in the hyper-congested flow regime number of vacant vehicles

References Alexander, L.P., Gonza´lez, M.C., 2015. Assessing the impact of real-time ridesharing on urban traffic using mobile phone data. Proceedings of UrbComp 15, 1e9. Chen, X.M., Zheng, H., Ke, J., Yang, H., 2020. Dynamic optimization strategies for on-demand ride services platform: surge pricing, commission rate, and incentives. Transportation Research B: Methodological 138, 23e45. Ke, J., Yang, H., Li, X., Wang, H., Ye, J., 2020. Pricing and equilibrium in on-demand ride-pooling markets. Transportation Research Part B: Methodological 139, 411e431. Li, Z., Hong, Y., Zhang, Z., 2016. Do ridesharing services affect traffic congestion? An empirical study of Uber entry. Social Science Research Network 2002, 1e29. Schaller Consulting, 2018. The New Automobility: Lyft, Uber and the Future of American Cities. http://www.schallerconsult.com/rideservices/automobility.pdf.

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Yang, H., Ye, M., Tang, W.H., Wong, S.C., 2005. Regulating taxi services in the presence of congestion externalities. Transportation Research A, Policy and Practice 39 (1), 17e40. Zheng, H., Chen, X., Chen, X., 2019. How does on-demand ridesplitting influence vehicle use and purchase willingness? A case study in Hangzhou, China. IEEE Intelligent Transportation Systems Magazine 11 (3), 143e157.

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Chapter 8

Revisiting government regulations for ride-sourcing services under traffic congestion Jintao Ke1, Xinwei Li2, Hai Yang3 and Yafeng Yin4 1

Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China; School of Economics and Management, Beihang University, Beijing, China; 3Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China; 4Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI, United States 2

8.1 Introduction Chapter 4 presents some interesting theoretical effects of government regulations by assuming that drivers have a homogeneous reservation rate (r) and that traffic congestion does not influence platforms’ decisions. In actual operations, a platform may adjust its operating strategies to offset the negative effects of traffic congestion, which means that the presence of traffic congestion may influence the effects of government regulations. Drivers are also more likely to have a heterogeneous r, which means that some drivers are more willing than others to join a ride-sourcing platform. A given effective E (i.e., earnings per hour) only attracts drivers whose r is lower than the E available for providing on-demand ride services. Clearly, this heterogeneity directly affects driver supply and also indirectly affects platform operations and government regulations. In this chapter, we present a more generalised model that takes into account the heterogeneity of drivers’ r and traffic congestion externality. We systematically analyse a few important regulatory schemes by examining whether they can achieve a targeted Pareto-efficient outcome and by determining their effects on a platform’s decisions and the realised passenger demand for RS services (D) and supply. We perform these analyses under several market scenarios: with drivers with homogeneous or heterogeneous r and under mild or heavy traffic congestion. This delineates the effects of drivers’ Supply and Demand Management in Ride-Sourcing Markets. https://doi.org/10.1016/B978-0-443-18937-1.00011-5 Copyright © 2023 Elsevier Inc. All rights reserved.

167

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heterogeneity and traffic congestion externality on the performance of various regulatory schemes. Consequently, we are able to offer some interesting and novel managerial insights, which include but are not limited to the following: l

l

l

l

In the market scenarios with drivers with homogeneous r and no/light traffic congestion, a government can only induce a platform to choose the targeted Pareto-efficient strategy by regulating the commission (P) and service level. If drivers have heterogeneous r, commission regulation remains effective but service-level regulation does not. In the presence of traffic congestion, neither commission regulation nor service level regulation is Pareto-efficient. If drivers have homogeneous r, the optimal fare (F) decreases and the optimal wage per order (E) increases from the monopoly optimum (MO) to the social optimum (SO) along the Pareto-efficient frontier. In contrast, if drivers have heterogeneous r, the optimal E decreases from the MO to the SO along the Pareto-efficient frontier. This property affects the outcome of government regulations in different market scenarios; i.e., only minimum wage regulation can affect a platform’s decisions if drivers have homogeneous r, while only maximum wage regulation can affect a platform’s decisions if drivers have heterogeneous r. Heavy traffic congestion may reverse the trends of key variables, such as F, E and fleet size (N), along the Pareto-efficient frontier from the MO to the SO. This substantially affects a government’s policies: with light traffic congestion, a government tends to use a price cap or a minimum N provision to force a platform to increase the consumer surplus (CS) and provider surplus (PS); in contrast, with heavy traffic congestion, a government attempts to mitigate the negative effects of traffic congestion externality by capping N or setting a maximum E. Under some regulations, such as a minimum per-hour income regulation, a platform may recruit only a subset of drivers who are willing to participate, i.e., perform driver rationing. If no regulation is imposed, a platform will recruit all drivers who are willing to participate, i.e., not perform driver rationing.

In the following section, we first establish a more generalised model that carefully considers drivers’ heterogeneity and traffic congestion externality and then present the major results under different market scenarios, e.g., with light and heavy traffic congestion. Finally, we summarise the main insights.

8.2 Theoretical analyses This section first presents a model that delineates the intriguing relationship between the endogenous variables and decision variables of a non-ride-splitting ride-sourcing market, which serves as a foundation for our analyses of the effects of government regulations. Consider a market where passengers can

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select either a ride-sourcing service offered by a monopoly ride-sourcing platform or other transportation modes (such as a public transit service). F, E, the arrival rate of passengers (Q) and N are defined as before. We assume that a platform adopts a first-come-first-served (FCFS) matching mechanism with an infinite matching radius, such that a passenger is immediately matched with the nearest idle driver after the passenger submits an order, no matter how far away the passenger is from the driver. This matching mechanism was first adopted for a ride-sourcing service by Castillo et al. (2017) and in subsequent studies (e.g., Ke et al., 2020), while a similar ‘first-dispatch’ protocol is used in taxi-dispatching systems. The w consists of two components: the matching time (the time from an order being submitted to when it is confirmed) and the pick-up time (the time taken by a driver to pick up a passenger after the passenger’s order is confirmed online). This instant matching mechanism results in near-negligible matching time, but each passenger experiences a non-negligible average waiting time (w) and trip time (t). We acknowledge that some platforms may adopt a batch-matching mechanism with a finite matching radius in actual operations. However, the models developed for batch matching with a finite matching radius (Xu et al., 2020; Yang et al., 2020) are generally too complex for in-depth theoretical analysis. Thus, for analytical tractability, we establish our model based on the FCFS matching mechanism.
Here, t and average pick-up time are given by t ¼ L=v and w ¼ Lp v, respectively, where L and Lp refers to the average trip and pickup distances respectively. Lp is generally regarded as a convex and decreasing function of
N v and has the following properties: Lp ¼ Lp ðN v Þ, L0p ¼ dLp dN v < 0, L00p ¼
d2 Lp dðN v Þ2 > 0, Lp /0 as N v /N, and Lp /N as N v /0. A widely used
pick-up distance function is Lp f1 ðN v Þ2 , which was first proposed by Daganzo (1978) and satisfies all of these properties. Diao et al. (2021) found that the entrance of transportation network companies (TNCs) into markets leads to increased road congestion in terms of intensity (þ0.9%) and duration (þ4.5%). Erhardt et al. (2019) claimed that TNCs are the biggest contributor to growing traffic congestion in San Francisco, using a combination of data scraped from Uber and Lyft and observed t data. Their research indicated that from 2010 to 2016, the weekday vehicle hours of delay in San Francisco increased by 62%, compared with 22% in a counterfactual 2016 scenario without TNCs. To characterise traffic congestion effects, v is assumed to be a decreasing function of N, as follows: v ¼ vðNÞ:

(8.1)

v also depends on other exogenous variables, such as the arrival rate of regular private car users (normal vehicle demand) Qn , L, and a speededensity relationship.

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On the demand side, all passengers are homogeneous in b and heterogeneous in their reservation price (i.e., willingness to pay) for a ride. Thus, reservation price is continuously distributed across potential passengers with a cumulative distribution function Gd ð$Þ and G0d > 0. The aggregated Q is thus governed by F, w, and t, as follows:     L þ Lp ðN v Þ Q ¼ Q$ð1 Gd ðF þ bðw þ tÞÞÞ ¼ Q$ 1 Gd F þ b (8.2) vðNÞ Q represents the potential Q and is exogenously given, the term Fþ where   L þ Lp ðN v Þ b represents the average generalised cost per ride (full ride price), vðNÞ and the Q function indicates that Q decreases with the average generalised cost. Moreover, in a stationary equilibrium state, each vehicle is in one of the following three phases: (1) the idle phase (waiting for online matching); (2) the pick-up phase (en route to pick up an online-matched passenger); or (3) the in-trip phase (delivering a passenger to his/her destination). Thus, the following vehicle-conservation equation should hold: N ¼ N v þ Qw þ Qt ¼ N v þ Q

Lp ðN v Þ L þQ vðNÞ vðNÞ

(8.3)

which indicates that N equals the sum of N v , the number of vehicles in the pick-up phase (Q L ). (Q vðNÞ

Lp ðN v Þ ), vðNÞ

and the number of vehicles in the in-trip phase

According to Little’s law, the number of vehicles in the pick-up and

in-trip phases in any stationary equilibrium state equals the arrival rate of vehicles (which is the same as the arrival rate of Q) and the w and t. On the supply side, drivers decide whether to enter the market and provide ride-sourcing services depending on the average income per unit time for providing these services and r (the opportunity cost for other job options). The average income per driver per hour (U) is given by:

U¼

EQ N

(8.4)

where EQ is the total income of N drivers per unit time. As shown by Yang et al. (2005), drivers’ average income is generally a decreasing function of N. There are two competing primary theories of how ride-sourcing drivers choose their working hours. The neoclassical theory states that drivers work longer hours when their E rate is higher, which represents a positive hourly E rate elasticity of working hours (Farber, 2005, 2015). In contrast, the incometargeting theory states that drivers exit the market once they reach their target

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income level, which represents a negative labour-supply elasticity (Camerer et al., 1997; Farber, 2015). Both theories can be explained by empirical analyses of different datasets. Recently, Angrist (2017), Sheldon (2016), and Chen and Sheldon (2016) have identified positive labour-supply elasticity in a real working-driver dataset from Uber. Similarly, Sun et al. (2019) found a positive labour-supply elasticity in a high-resolution DiDi dataset. In our proposed model, the supply side adopts a neoclassical approach, as a higher average revenue rate corresponds to a larger labour supply. Suppose the aggregate labour supply has a continuous distribution of r across potential drivers with a cumulative distribution function Gs ð$Þ. If the distribution r is a one-point distribution, this implies that the ride-sourcing drivers have a homogeneous r, which we denote c. If the supply is sufficient, as assumed by Zha et al. (2016), at the market equilibrium drivers will continue entering the market until U decreases to c, i.e., U ¼ EQ N ¼ c. A potential ride-sourcing driver is thus assumed to enter the market if U is larger than his/her r, i.e., U ¼ EQ=N  r. As mentioned, a monopoly platform may perform driver rationing, i.e., choose only a subset of those drivers who want to enter the ride-sourcing market. Thus, we denote a as the rationing factor, i.e., the percentage of drivers recruited from those drivers who want to enter the ride-sourcing market. Hence, the aggregate supply function and the total PS are given by   EQ N ¼ aNGs ðUÞ ¼ aNGs (8.5) N and Z

N aN

PS ¼ aN 0

  Z N aN EQ  r dGs ðrÞ ¼ EQ  aN rdGs ðrÞ N 0

(8.6)

respectively, where a ˛ ð0; 1 and N is the potential driver supply. In particular, if a ¼ 1, this model reduces to a market scenario in which a platform recruits all drivers willing to participate, as has been shown in many previous studies (Castillo et al., 2017; Yu et al., 2020). For simplicity, we let   N CðN; aÞ ¼ NG1 (8.7) s aN Z

N aN

Cp ðN; aÞ ¼ aN

rdGs ðrÞ

(8.8)

0

where CðN; aÞ represents the cost of recruiting N from the labour market at equilibrium and equals drivers’ total revenue (EQ); and Cp ðN; aÞ is the cumulative sum of these drivers’ r. Both CðN; aÞ and Cp ðN; aÞ are strictly increasing functions of N, and CðN; aÞ exhibits increasing returns to scale for a given a and any increasing function Gs ð$Þ.

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In summary, the stationary equilibrium state of a ride-sourcing market can be characterised by a system of non-linear equations (Eqs. 8.1e8.8). These contain three independent variables and seven undetermined variables (F, E, Q, N N v , U and a). Therefore, the equilibrium can be solved by determining any three of these undetermined variables. Although F and E are the direct decision variables of a platform, the platform can also affect the market equilibrium by choosing two indirect decision variables (Q and N). In the following optimisation problem for maximising P or S, or finding Paretoefficient solutions, we treat Q, N, and a as decision variables, and F, E, U and N v as endogenous variables solved at market  equilibrium. For the con1 Q ; then, from the Q funcvenience of exposition, we let BðQÞ ¼ G1 d Q

tion, F can be written as:

  Lp ðQ; NÞ L þ F ¼ BðQÞ bðwðQ; NÞ þ tÞ ¼ BðQÞ b vðNÞ vðNÞ

(8.9)

where wðQ; NÞ and Lp ðQ; NÞ are regarded as functions of Q and N, and implicitly given by Eq. (8.3). Clearly, Eq. (8.9) can also be regarded as a formula for F as a function of Q and N. Thus, by combining Eqs. (8.4), (8.5) and (8.7), E can be easily given by CðN; aÞ=Q.

8.2.1 Monopoly optimum We first consider the MO where a ride-sourcing platform chooses a combination of decision variables to maximise profit (P) per unit time, i.e., P ¼ ðF EÞQ, where F  E is the net P of each completed order. Without loss of generality, we do not consider the operations cost of a platform. As mentioned, both F and E can be solved by the equilibrium conditions if we treat Q, N and a as decision variables. Thus, the P-maximisation problem can be formulated as:    L þ Lp ðQ; NÞ maxPðQ; N; aÞ ¼ ðF  EÞQ ¼ Q BðQÞ b CðN; aÞ vðNÞ (8.10) where Lp ðQ; NÞ is implicitly solved by Eq. (8.3). The first-order conditions of this problem are    vPðQ; N; aÞ L þ Lp ðQ; NÞ ¼ BðQÞ b vQ vðNÞ   (8.11) vLp ðQ; NÞ 1 þ Q B0 ðQÞ  b ¼0 vQ vðNÞ

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  vLp ðQ; NÞ vðNÞ v0 ðNÞ L þ Lp ðQ; NÞ vCðN; aÞ vN ¼0  v2 ðNÞ vN (8.12)    0 vPðQ; N; aÞ vCðN; aÞ N2 N ¼ ¼ 2 >0 (8.13) G1 s va va aN a N

vPðQ; N; aÞ ¼  bQ vN

Eq. (8.17) implies that amo ¼ 1. Then, combining Eq. (8.15) and Eq. (8.16) with amo ¼ 1 gives rise to      L0 L0 L þ Lp;mo 0    0   p;mo    p;mo     v Nmo C Nmo ; 1 1 þ Qmo ¼ bQ þ bQ mo mo    v Nmo v Nmo v2 Nmo (8.14)         L þ L0   p;mo B Qmo þ Qmo B0 Qmo ¼¼ b þ C0 Nmo ;1  v Nmo (8.15)     2 L þ L0 bQmo v0 Nmo p;mo        þ Qmo L0 v Nmo v Nmo p;mo    vCðN;1Þ    0 where C0 Nmo ; 1 ¼ vN and Qmo , Lp;mo , L0 p;mo , and B Qmo are the  N¼Nmo

passenger demand, the average pick-up distance, the derivative of average pick-up distance with respect to the number of idle vehicles and the derivative of the inverse-Q function at the MO, respectively. The first-order conditions in Eqs. (8.18) and (8.19) can be obtained by solving the P-maximisation problem that takes any two of the six undetermined variables (F, E, Q, N N v and U) as decision variables with amo ¼ 1 (including but not limited to the combination of Q and N). This indicates that without government regulations, a platform will not ration the willing-to-join drivers.

8.2.2 Social optimum Next, we consider the SO maximisation problem of choosing the best combination of Q, N, and a to maximise social welfare (S), which is given by

  Z Q L þ Lp ðQ; NÞ SðQ; N; aÞ ¼ BðzÞdz  F þ b Q þ PSðQ; N; aÞ vðNÞ 0 L þ PðQ; N; aÞ  bQn vðNÞ (8.16) where SðQ; NÞ equals the sum of CS (the sum of the first and second term on the right-hand side (RHS) of Eq. (8.20)), PðQ; N; aÞ, and PSðQ; N; aÞ, minus

174 Supply and Demand Management in Ride-Sourcing Markets L ; to measure the delay cost of the travel cost of regular private cars (bQn vðNÞ

background traffic caused by the increase in traffic congestion). By substituting Eqs. (8.6) and (8.14) into Eq. (8.20), the S-maximisation problem can be formally written as   Z Q L þ Lp ðQ; NÞ L maxSðQ; N; aÞ ¼ BðzÞdz  Qb Cp ðN; aÞ  bQn vðNÞ vðNÞ 0 (8.17) where Lp ðQ; NÞ is an implicit function of Q and N, determined by Eq. (8.3). The first-order conditions of the S-maximisation problem are   vSðQ; N; aÞ L þ Lp ðQ; NÞ vLp ðQ; NÞ 1 ¼ BðQÞ b ¼ 0 (8.18)  bQ vQ vðNÞ vQ vðNÞ   vLp ðQ; NÞ vðNÞ v0 ðNÞ L þ Lp ðQ; NÞ vN v2 ðNÞ (8.19) 0 vCp ðN; aÞ n v ðNÞ þ bQ L 2  ¼0 va v ðNÞ  Z N Z N  aN aN vSðQ; N; aÞ EQ EQ ¼ N  r dGs ðrÞ > 0 (8.20) rdGs ðrÞ ¼ N va a N 0 0 vSðQ; N; aÞ ¼  bQ vN

Eq. (8.24) implies the SO rationing factor is aso ¼ 1. Then, by combining Eqs. (8.22) and (8.23) with aso ¼ 1, we obtain      L0 L0 L þ Lp;so    p;so p;so Cp0 Nso ; 1 1 þ Qso    ¼ bQso    þ bQso 2    v0 Nso v Nso v Nso v Nso    (8.21)   0 0 L   p;so  n v Nso  þ 1 þ Qso bQ L 2   v Nso v Nso      L þ Lp;so     L þ Lp;so 2 bQso

v0 Nso      B Qso ¼ b þ Cp0 Nso ;1   v Nso v Nso v þ Q L0 so p;so

L þ Lp;so  3   v Nso    vCp ðN;1Þ 0 where Cp Nso ; 1 ¼ vN

  bQn Lv0 Nso (8.22)

 N¼Nso

  0 , and Qso , Lp;so , L0 p;so , and B Qso denote

the passenger demand, the average pick-up distance, the derivative of average pick-up distance with respect to the number of idle vehicles and the derivative of the inverse-demand function at the SO, respectively. Similarly,

Revisiting government regulations for ride-sourcing services Chapter | 8

175

the first-order conditions in Eqs. (8.25) and (8.26) can be obtained by solving the S-maximisation problem that takes any two of the six undetermined variables (F, E, Q, N, N v and R) as decision variables with aso ¼ 1 (including but not limited to the combination of Q and N).

8.2.3 Pareto-efficient solutions A government and a monopoly platform have different objectives, which are in partial conflict: a government aims to maximise S, whereas a platform aims to maximise P. We are thus interested in the Pareto-efficient frontier, along which neither a platform nor a government can improve how well it achieves its aim without decreasing how well the other does so. The set of Pareto-efficient solutions can be obtained by solving a biobjective maximisation problem that aligns with S given by Eq. (8.21) and P given by Eq. (8.14), both of which are functions of Q, N, and a, as follows: ! SðQ; N; aÞ max (8.23) ðQ;N;aÞ ˛ U PðQ; N; aÞ where U ¼ fðQ; N; aÞ : Q  0; N  0; a ˛ ð0; 1g. This bi-objective problem does not determine the combination of Q, N and a for maximising either S or P (which lead to the SO and the MO, respectively), but instead aims to identify a set of decision variables that determine the Pareto-efficient frontier of S and P. The MO and the SO are the two endpoints of the Pareto-efficient frontier, and neither S nor P can be further improved without reducing the other along the Pareto-efficient frontier, including at the two endpoints. If ðQ ; N  ; a Þ is a Pareto-efficient solution, then it must solve the following maximisation problem (Geoffrion, 1967; Yang and Yang, 2011):   L þ Lp ðQ; NÞ L Cp ðN; aÞ  bQn vðNÞ vðNÞ

ZQ maxSðQ; N; aÞ ¼

BðzÞdz Qb 0

(8.24) subject to

   L þ Lp ðQ; NÞ PðQ; N; aÞ ¼ Q BðQÞ b CðN; aÞ  Q vðNÞ    L þ Lp ðQ ; N  Þ BðQ Þ b CðN  ; a Þ vðN  Þ

(8.25)

176 Supply and Demand Management in Ride-Sourcing Markets

To solve this constrained maximisation problem, we form the following Lagrange function:   Z Q L þ Lp ðQ; NÞ LðQ; N; aÞ ¼ BðzÞdz  Qb Cp ðN; aÞ vðNÞ 0     L þ Lp ðQ; NÞ n L þ x Q BðQÞ b bQ vðNÞ vðNÞ (8.26)     L þ Lp ðQ ; N  Þ   CðN; aÞ  Q BðQ Þ b vðN  Þ  CðN  ; a Þ where x  0 is a Lagrange multiplier. Then, the first-order conditions of the Lagrange problem can be written as vL vS vP ¼ 00 þ x ¼0 vQ vQ vQ

(8.27)

vL vS vP ¼ 00 þ x ¼0 vN vN vN

(8.28)

vL vS vP ¼ þx >0 va va va

(8.29)

From Eq. (8.29), we can obtain the Pareto-efficient rationing factor a ¼ 1, which enables us to obtain the following Pareto-efficient solutions:   Cp0 ðN  ; 1Þ þ xC0 ðN  ; 1Þ L0 p 1 þ Q 1þx vðN  Þ   L0 L þ Lp 0  L0 1 L p p   þ bQ v ðN Þ þ 1 þ Q bQn 2  v0 ðN  Þ 1þx v ðN Þ vðN  Þ v2 ðN  Þ vðN  Þ (8.30)   Cp0 ðN  ; 1Þ þ xC 0 ðN  ; 1Þ L þ Lp x Q B0 ðQ Þ ¼ b þ BðQ Þ þ 1þx 1þx vðN  Þ   L þ Lp 2 bQ

  (8.31) vðN Þ vðN  Þ þ Q L0 p ! 1 L þ Lp n þ bQ L v0 ðN  Þ 1 þ x v3 ðN  Þ 0 ðN  ; 1Þ ¼ vCp ðN;1Þ where C0 ðN  ; 1Þ ¼ vCðN;1Þ , C , and Q , Lp , L0 p p vN vN ¼ bQ

N¼N 

N¼N 

and B0 ðQ Þ represent the passenger demand, the average pick-up distance, the

Revisiting government regulations for ride-sourcing services Chapter | 8

177

derivative of average pick-up distance with respect to the number of idle vehicles and the derivative of the inverse-Q function along the Pareto-efficient frontier, respectively. It can be seen that these Pareto-efficient solutions are linear combinations of the MO and SO. In addition, it can be found that Eq. (8.30) always holds along the Pareto-efficient frontier (independent of the Lagrange multiplier) and has the same form as Eq. (8.14) at the MO and Eq. (8.21) at the SO. In addition, by reorganising Eq. (8.31) . and combining the result with the definition F  ¼ BðQ Þ b

L þLp

vðN  Þ , we obtain

Cp0 ðN  Þ þ xC0 ðN  Þ L þ Lp x Q B0 ðQ Þ  1þx vðN  Þ x þ 1 !   L þ Lp 2 bQ 1 L þ Lp n

þ bQ L v0 ðN  Þ  3 ðN  Þ 1 þ x vðN  Þ v   0 vðN Þ þ Q L F ¼

(8.32)

p

which is the pricing formula along the Pareto-efficient frontier. The first term on the RHS of this formula (

Cp0 ðN  ÞþxC 0 ðN  Þ LþLp ) 1þx vðN  Þ

represents the marginal w and

t cost of a driver serving a passenger at a solution along the Pareto-efficient frontier. The term Q B0 ðQ Þ represents the monopoly mark-up, which reflects the market power of a monopoly platform to distort F from its socially efficient level. The multiplier x measures the relative position of a specific Pareto-efficient solution along the Pareto-efficient frontier that connects the MO and the SO. If x ¼ 0, the pricing formula in Eq. (8.32) becomes an SO pricing formula; if x/N, the pricing formula in Eq. (8.32) becomes an MO pricing formula. In addition, the fact that a ¼ 1 indicates that a platform should recruit all of the drivers who intend to enter the ride-sourcing market along the Pareto-efficient frontier. Theoretically, we can prove the following proposition. Proposition 8.1. When r is heterogeneous, there is no traffic-congestion effect and a platform does not implement driver rationing, and income regulation cannot induce the platform to choose a targeted Pareto-efficient strategy. If U is regulated, a platform will maximise P by choosing a strategy such that Q is less than that at the Pareto-efficient solution. We acknowledge that due to the complexity of the model, some propositions can only be proven in markets with heterogeneous r and without traffic congestion, whereas some outcomes of government regulations can only be proven in markets with homogeneous r and without traffic congestion and driver rationing. Nevertheless, we provide extensive numerical examples in the next section to examine the regulatory outcomes of all of the abovementioned regulations in several situations, including a market with no/light traffic congestion and homogeneous drivers, and markets with no/light/heavy traffic congestion and heterogeneous r. The above analytical findings are derived for scenarios in which a platform does not implement driver rationing; we

178 Supply and Demand Management in Ride-Sourcing Markets

numerically analyse the influence of driver rationing on the effects of these regulations in the next section. Moreover, we highlight the similarities and differences between these regulatory outcomes in different situations and offer rational explanations.

8.3 Numerical studies In this section, we conduct numerical studies to examine the regulatory outcomes of the regulations discussed so far under market scenarios for which theoretical results cannot be obtained, namely, scenarios (1) with heterogeneous drivers and no traffic congestion or (2) with heterogeneous drivers and traffic congestion. Numerical studies for a ride-sourcing market with homogeneous drivers and no traffic congestion are provided in Chapter 3.

8.3.1 Settings The Q function is assumed to have the following form:   Z FþbðwþtÞ k ke dx Q ¼ Q$ð1 Gd ðF þ bðw þ tÞÞÞ ¼ Q$ 1  0

(8.33)

¼ Qexpfk $ ½F þ b $ ðw þ tÞg where Q is the potential passenger demand, and the distribution of passengers’ reservation price follows an exponential distribution with a mean 1= k. In this numerical example, we assume Q ¼ 1:0  104 (trips/h), k ¼ 0:06 (1/$), and b ¼ 27:69 ($/h). To characterise the traffic congestion effects, we employ a

linear speededensity function, v ¼ a  bk, where k ¼ ðN LþNÞ ¼ QLvL þ NL is the traffic density, and a and b are two positive parameters. Here, we restrict our discussions to the normal flow regime, and thus vðNÞ is given by ffi   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 aL  bN þ aL  bN  4LbQn L (8.34) vðNÞ ¼ 2L   mph$mi2 where the parameter values are set as a ¼ 25ðmphÞ, b ¼  0:07 , veh n

n

L ¼ 3:808 (mi) and L ¼ 25 (mi). Lp is assumed to be inversely proportional to

the square root of the number of N v , i.e., Lp ¼ AðN v Þ1=2 , where A is set as v 1=2

Þ

L and w ¼ AðN 83.07 (mi). In addition, we have t ¼ vðNÞ vðNÞ . The free-flow v in

the absence of traffic congestion is set as 13:6 mph. On the supply side, r is assumed to be uniformly distributed in the interval ½cl ; ch 1 , such that N satisfies

179

Revisiting government regulations for ride-sourcing services Chapter | 8

EQ    cl EQ N ¼ NGs ¼NN N ch  cl

(8.35)

where N is the number of potential drivers. We assume cl ¼ 3:968 ($= h), ch ¼ 99:20 ($/h) and N ¼ 4:0  103 (vehicles/h). Note that the values of k, b, L, v, c, cl and ch are based on actual mobility data collected in Chicago (please refer to Zhang and Nie, 2021). Other parameters are chosen for illustrative purposes, with partial reference to previous studies (e.g., Ke et al., 2020; Vignon et al., 2021; Yang and Yang, 2011). In actual operations, one may calibrate the parameters of the proposed functions and identify their properties using real data.

8.3.2 Market with drivers with heterogeneous reservation rates and no traffic congestion The regulatory outcomes of different regulations introduced in the last section, with drivers with heterogeneous r and without driver rationing, are demonstrated in Fig. 8.1 and Table 8.1. The green and blue circles indicate the SO and the MO, respectively. The solid black line indicates the projection of the Pareto-efficient frontier on different axes (e.g., F vs. N in Fig. 8.1A). The solid grey line indicates the regulation set by a government, which allows the platform to make decisions in the unshaded space (i.e., the platform cannot choose a strategy from the shaded space), and the red dot indicates the projection of the strategy chosen by the platform under the corresponding regulation on the two-dimensional space. Fig. 8.1A shows that the platform is only allowed to choose a strategy below the binding curve in terms of the targeted Pareto-efficient F (the solid grey line). Under this regulation, the platform eventually chooses the red circle, at which point the P contour is tangential to the binding curve. It can be seen that the red circle is not coincident with the targeted Pareto-efficient solution (the black dot) and has a smaller N. Thus, the maximum trip fare regulation is not Pareto-efficient. Fig. 8.1B indicates that the fleet size regulation that requires the platform to set an N less than a targeted Pareto-efficient N (the solid grey line) does not affect the decisions of the platform. As the MO inherently has a lower N than the SO, the platform will directly choose the MO under such a regulation. In contrast, Fig. 8.1C shows that if the government sets a minimum N, the platform will set an N equal to the targeted N but will set an F higher than the targeted F, which leads to a non-Pareto-efficient outcome. Interestingly, in scenarios where drivers have homogeneous r, the E is  > E  (Fig. 8.1D), whereas in higher at the MO than that at the SO, i.e., Emo so

180 Supply and Demand Management in Ride-Sourcing Markets

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

FIGURE 8.1 Regulatory outcomes of various regulations in a market with drivers with heterogeneous reservation rates; (A) maximum fare regulation; (B) maximum fleet-size regulation; (C) minimum fleet-size regulation; (D) minimum wage regulation; (E) maximum wage regulation; (F) minimum utilisation regulation; (G) maximum commission regulation; (H) minimum demand regulation; (I) minimum income regulation.  < E (Fig. 8.1E). This the scenario where drivers have heterogeneous r, Emo so is because the PS is no longer zero when drivers have heterogeneous r, and PS increases with E, which increases the optimal E at the socially efficient level. Due to the different relationships between E and other factors at the MO and the SO, the regulatory outcomes of minimum wage regulation and maximum wage regulation in the scenario with drivers with homogeneous r and in the scenario with drivers with heterogeneous r are also different. From Fig. 8.1D, we can see that the minimum wage regulation affects the decisions of the platform, although it cannot induce the platform to choose the targeted Pareto-

Effect on effective demand/supply

Pareto-efficient?

No/light congestion

Heavy congestion

No/light congestion

Heavy congestion

Price-cap regulation F  F 

Q # < Q  , N# < N

 > Q, Q # ¼ Qmo  > N N # ¼ Nmo

No

No

Maximum fleet-size regulation N  N 

 < Q, Q # ¼ Qmo  < N N # ¼ Nmo

Q # < Q  , N# ¼ N

No

No

Minimum fleet-size regulation N  N 

Q # < Q  , N# ¼ N

 > Q, Q # ¼ Qmo  > N N # ¼ Nmo

No

No

Minimum wage regulation E  E 

Q # < Q  , N# < N

 > Q, Q # ¼ Qmo  > N N # ¼ Nmo

No

No

Maximum wage regulation E  E 

 < Q, Q # ¼ Qmo  < N N # ¼ Nmo

Q # > Q  , N# > N

No

No

Commission regulation P  P 

Q # ¼ Q  , N# ¼ N

 > Q, Q # ¼ Qmo  > N N # ¼ Nmo

Yes

No

Utilisation rate regulation U  U 

Q # < Q  , N# < N

 > Q, Q # ¼ Qmo  > N N # ¼ Nmo

No

No

Demand regulation Q  Q 

Q # ¼ Q  , N# < N

 > Q, Q # ¼ Qmo  > N N # ¼ Nmo

No

No

Income regulation R  R 

Q # < Q  , N# ¼ N

 > Q, Q # ¼ Qmo #  N ¼ Nmo > N 

No

No

181

Regulatory regime

Revisiting government regulations for ride-sourcing services Chapter | 8

TABLE 8.1 Summary of the effects of regulation on a market with no driver rationing and with drivers with heterogeneous reservation rates.

182 Supply and Demand Management in Ride-Sourcing Markets

efficient solution. In particular, the platform can maximise P by choosing a higher F than the targeted Pareto-efficient F. Fig. 8.1E indicates that the maximum wage regulation does not affect the decision of the platform, and thus it straightforwardly chooses the MO. These effects are in contrast to the effects of regulation in market scenarios with drivers with homogeneous r. Fig. 8.1F reveals that the minimum utilisation rate (U) regulation imposed in New York is also not Pareto-efficient, as the platform chooses a strategy leading to a lower Q than the Pareto-efficient target. Fig. 8.1G shows that the government can achieve its target by simply imposing a maximum commission regulation, under which the platform’s best strategy is coincident with the targeted Pareto-efficient solution. Fig. 8.1H demonstrates that the minimum service level (Q) regulation is not Pareto-efficient, as it induces the platform to achieve a larger U than the targeted Pareto-efficient U. As U ¼ Qt=N, Fig. 8.1H also implies that under the minimum service level regulation, the platform will choose a relatively low N. The findings presented in Proposition 8.1 on income regulation are validated with numerical experiments, the results of which are depicted in Fig. 8.1I. It can be seen that the platform will choose a strategy leading to a higher F than the targeted Pareto-efficient F and an U equal to the targeted Pareto-efficient U. If the platform is required to guarantee U, it will increase F to protect P, which will decrease passengers’ benefits.

8.3.3 Market with drivers with heterogeneous reservation rates and traffic congestion One of our central concerns in the numerical example is the effect of traffic congestion on the ride-sourcing market equilibrium. Fig. 8.2A depicts how P and S respond to the arrival rate of background traffic (which is a measure of the level of congestion). Fig. 8.2BeI plots the changes in the MO and the SO with respect to different levels of background traffic, in terms of Q, N, F, E, U, U and passengers’ generalised cost. From Fig. 8.2A, we can see that P decreases with increasing congestion. This is because the increased congestion increases passengers’ travel cost, which leads to a decrease in Q (as shown in Fig. 8.2B and I). To compensate for this P decrease, the platform increases F, P and E to attract more drivers without reducing P per order at the MO (as shown by the blue line in Fig. 8.2DeF). Fig. 8.2G and H demonstrate that U and U decrease due to increasing congestion. This is because traffic congestion increases the pick-up time (w). Fig. 8.2C and G illustrate that as U decreases, N also decreases. As shown in Fig. 8.2A, S at the SO (the red dashed line) changes more dramatically than P at the MO (the blue line); this is because S also includes the travel cost of regular private cars, such that changes in the level of congestion contribute more to S at the SO. In addition, Fig. 8.2BeI reveals that

Revisiting government regulations for ride-sourcing services Chapter | 8

(a)

(d)

(g)

(b)

(c)

(e)

(f)

(h)

183

(i)

FIGURE 8.2 Effects of the level of congestion Qn ; (A) variation of profit and social welfare; (B) variation of passenger demand; (C) variation of vehicle fleet size; (D) variation of average trip fare; (E) variation of average wage; (F) variation of average commission; (G) variation of average driver’s income; (H) variation of utilisation rate; (I) variation of passenger’s generalised cost.

the level of traffic congestion significantly affects the relative magnitude of key endogenous variables, such as Q, N, F, E and U, at the MO and the SO. Specifically, if the congestion level is low (e.g., Qn ¼ 5:0  103 (vehicles/h)), Q, N, E, U and U are higher at the SO than at the MO (i.e., the red dashed line is above the blue line), whereas F and P are lower at the SO than at the MO (i.e., the blue line is above the red line). This is consistent with the findings in Section 6.2 in the absence of traffic congestion and implies that a low level of congestion does not alter the outcomes of regulations. The reverse is true if the congestion level is high (e.g., Qn ¼ 1:28  104 (vehicles/h)): Q, N, E, U, and U are lower at the SO than at the MO, whereas F and P are higher at the SO than those at the MO. These opposite trends occur because if the congestion level is low, the effect of traffic congestion externality is small and thus has no

184 Supply and Demand Management in Ride-Sourcing Markets

effect on the relationships between the variables at the MO and the SO. In contrast, if the congestion level is high, the SO changes more dramatically than the MO, and thus the relationships between the variables at the SO and the MO also change dramatically. Without considering driver rationing, we set Qn to 1:28  104 (vehicles/h) and depict in Fig. 8.3, for a scenario with a high level of congestion and in which drivers have heterogeneous r, the outcomes of the various regulations introduced in section 8.1. To avoid unnecessary duplication, we do not depict

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

FIGURE 8.3 Outcomes of various regulations in a market with drivers with heterogeneous reservation rates and a high level of congestion; (A) maximum fare regulation; (B) maximum fleetsize regulation; (C) minimum fleet-size regulation; (D) minimum wage regulation; (E) maximum wage regulation; (F) minimum utilisation regulation; (G) maximum commission regulation; (H) minimum demand regulation; (I) minimum income regulation.

Revisiting government regulations for ride-sourcing services Chapter | 8

185

the analogous results for a low level of congestion, as the results are similar to those of the no-congestion situation, which has been discussed in Section 8.1. As shown in Fig. 8.3A, CeD, FeI, Q, N, E, U and U at the MO are greater than those at the SO, while F and P at the MO are less than those at the SO. This implies that the platform straightforwardly chooses the MO under regulations of maximum F, minimum N, minimum E, minimum U, maximum P, minimum Q or minimum U. Fig. 8.3B shows that although maximum fleet size regulation causes the platform to choose an N equal to the targeted N, it also causes the platform to set a higher F to protect its P, which leads to a nonPareto-efficient outcome. Fig. 8.3E indicates that the maximum wage regulation is also not Pareto-efficient; as the MO E is higher than the SO E, and P decreases with F around the targeted Pareto-efficient solution, the platform will choose a strategy leading to a lower F.

8.3.4 Effects of driver rationing In the previous sections, the outcomes of regulation are analysed under the assumption that N matches the labour supply curve, that is, the platform does not ration drivers. However, as mentioned, a monopoly ride-sourcing platform may ration drivers to maximise P. Thus, if an implemented regulation is nonbinding, such as maximum N and maximum wage regulation in a scenario with drivers with heterogeneous r and a low level of congestion, a platform will set the MO rationing factor as amo ¼ 1 (see subsection 8.2.1), which means that all drivers who wish to enter the ride-sourcing market are recruited. However, if a binding regulation is implemented, such as minimum income regulation in a scenario with drivers with heterogeneous r and without congestion externalities, an excess supply of drivers may be generated, which means that only some drivers who wish to enter a ride-sourcing market will be recruited. To explore this, the outcomes of different binding regulations in scenarios with light traffic congestion, drivers with heterogeneous r and in the presence of driver rationing are depicted in Fig. 8.4. Each point in the contours indicates the amo required to obtain the maximum P for the corresponding x and y values. It can be seen that Fig. 8.4AeD, and H are the same as Fig. 8.1A, C, D, F, and H, respectively, which indicates that if the two fixed decision variables are ðF; NÞ or ðQ; UÞ, the MO for the platform is where amo ¼ 1. As shown in Figs. 8.4C and 8.1D, if E is low (e.g., E < 16 ð$ =rideÞ), the platform has the same P irrespective of driver rationing, whereas if E is high (e.g., E > 18 ð$ =rideÞ) and F is low (e.g., F < 17 ð$ =rideÞ) the platform P is higher under the driver rationing scenario. This implies that when P is small, the platform can reduce its P loss by decreasing the number of trips by hiring fewer drivers, i.e., amo s1. Figs. 8.4E and 8.1G show that the feasible solution area is larger if the platform rations drivers. This is because the feasible domain is a ˛ ð0; 1 in Fig. 8.4E but a ¼ 1 in Fig. 8.1G, and this larger feasible

186 Supply and Demand Management in Ride-Sourcing Markets

(a)

(b)

(c)

(d)

(e)

(f)

(g)

FIGURE 8.4 Outcomes of various regulations in a market with driver rationing and drivers with heterogeneous reservation rates. (A) Maximum fare regulation; (B) minimum fleet-size regulation; (C) minimum wage regulation; (D) minimum utilisation regulation; (E) maximum commission regulation; (F) minimum demand regulation; (G) minimum income regulation.

domain of a enlarges the feasible solution area in Fig. 8.4E. In addition, the resulting P under binding minimum income regulation is larger in Fig. 8.4G than in Fig. 8.1I. This is because such regulation creates an excess supply of drivers, leading to relatively high recruitment costs for the platform, which responds by only selecting some drivers from the total supply of willing drivers. For this reason, under minimum income regulation, amo s1; e.g., in this numerical example, amo ¼ 0:776:

8.3.5 Summary and discussion Table 8.1 summarises the outcomes of various regulations in scenarios with drivers with heterogeneous r. We can see that the effects of some regulations

Revisiting government regulations for ride-sourcing services Chapter | 8

187

are influenced by drivers’ heterogeneity with respect to r. First, in the scenario with light congestion, only by imposing commission regulation can a government achieve its Pareto-efficient target. Second, service level regulation is no longer Pareto-efficient. Third, in the MO, drivers’ heterogeneity of r causes the opposite trend of E relative to that in the SO, as minimum wage regulation affects the platform’s decisions in markets with drivers with heterogeneous r (although this regulation remains unable to yield a Pareto-efficient outcome), whereas it does not affect the platform’s decisions in markets with drivers with homogeneous r. Fourth, if minimum income regulations are imposed, the platform will increase F beyond the Pareto-efficient solution to protect its P. In this case, the CS will be reduced by the increased F. Finally, light traffic congestion does not alter the effects of various regulations. Table 8.1 also shows that in scenarios with heavy congestion, none of the regulations can reach their respective Pareto-efficient targets. This indicates that in heavy traffic congestion, regular vehicles have a significant effect on traffic flow and ride-sourcing Q, such that regulating the ride-sourcing market is not sufficient to achieve optimal ride-sourcing performance. Furthermore, in a market with heavy traffic congestion, price-cap, minimum N, minimum per-order E, maximum P, minimum U, minimum service level (Q) and minimum income regulations have no effect, as the platform straightforwardly chooses the MO under these traffic conditions. This occurs because heavy traffic congestion significantly influences the CS, the PS and the delay cost of background traffic vehicles, which reverses the trend in the key endogenous variables from the MO to the SO. This shows that a government should tailor its regulatory schemes according to the levels of traffic congestion in a city. For example, in cities such as New York, the traffic congestion is heavy and thus N and U at the SO are less than those at the MO; thus, a government should cap N and/or set a minimum U to alleviate congestion and enhance S. However, in suburban areas with light traffic congestion, the optimal N and U at the SO are greater than those at the MO; thus, a government should encourage a platform to recruit more drivers to increase the CS and the PS. It is noteworthy that in a market with moderate or heavy congestion, regulating P alone will not achieve a Pareto-efficient outcome. As suggested by Vignon et al. (2021), a P cap must be supplemented by levying congestion tolls on ride-sourcing vehicles or congestion fees on other users. In fact, congestion pricing, and U and price caps, are implemented in New York. We acknowledge that our study focuses on the effects of a single regulatory approach; it would also be interesting to examine the effects of multiple regulatory approaches on a system and a platform’s decisions. It is logical to posit that a government should use multiple regulations to increase its probability of achieving a Pareto-efficient target. Nonetheless, such an approach will also increase administrative costs. By comparing Fig. 8.4 with Fig. 8.1, it can be seen that although the effects of regulation are similar irrespective of the use of driver rationing, different

188 Supply and Demand Management in Ride-Sourcing Markets

rationing factors are used. That is, under minimum income regulation, the platform will recruit only 77:6% drivers who wish to enter the ride-sourcing market. This implies that such income regulation may not protect all drivers’ incomes, as some willing drivers will not be employed.

8.4 Conclusion This chapter first discusses the properties of the Pareto-efficient frontier that connects the SO and the MO and then investigates the effects of various regulations, such as price-cap regulation, fleet size regulation, wage regulation, minimum utilization rate regulation, commission regulation, and service level (demand) regulation. We also examine the decisions of a platform under various regulatory schemes and compare the resulting realised Q and supply with the targeted Pareto-efficient solutions. For example, we prove that fleet size regulation will cause a platform to decrease both supply and Q away from the Pareto-efficient frontier, resulting in a non-Pareto-efficient outcome. In contrast, commission regulation will directly induce a platform to choose the targeted Paretoefficient strategy. In addition, we consider market scenarios with traffic congestion and drivers with heterogeneous r and find that these substantially influence the effects of various regulations. For example, the effects of wage regulation and fleet size regulation in scenarios with drivers with homogeneous r are opposite to their effects in scenarios with drivers with heterogeneous r. Similarly, service level (demand) regulation is effective in a scenario with drivers with homogeneous r, but ineffective in a scenario with drivers with heterogeneous r. We also find that by requiring a platform to ensure a minimum U, a government cannot achieve a desirable Pareto-efficient outcome; instead, a platform will increase F to protect its P, which decreases the benefits to passengers. Moreover, a high level of traffic congestion may reverse the trend of optimal F, E and N along the Pareto-efficient frontier and thus influence the effectiveness of regulatory schemes. This means that cities with heavy traffic congestion should apply maximum fleet size regulations to reduce traffic congestion, whereas cities with light traffic congestion should encourage platforms to recruit more drivers to increase the CS and PS. This chapter is based on one of our recent articles (Ke et al., 2021).

Glossary of notation S F t b w Q N

social welfare average trip fare average trip time passengers have a homogeneous value of time average waiting time of passengers arrival rate of passengers vehicle fleet size

Revisiting government regulations for ride-sourcing services Chapter | 8 r P L Lp v P x Q F N w Q N U Emo Eso

189

drivers’ reservation rate profit average trip distances average pick-up distances average network speed commission per ride Lagrange multiplier passenger demand at Pareto-efficient solutions trip fare at Pareto-efficient solutions vehicle fleet size at Pareto-efficient solutions average waiting time of passengers at Pareto-efficient solutions potential passenger demand number of potential drivers utilisation rate optimal wage at the MO resulting wage at the SO

References Angrist, J.D., Caldwell, S., Hall, J.V., 2017. Uber vs. Taxi: A Driver’s Eye View (Report No. w23891). National Bureau of Economic Research. Camerer, C., Babcock, L., Loewenstein, G., Thaler, R., 1997. Labor supply of New York City cabdrivers: one day at a time. The Quarterly Journal of Economics 112 (2), 407e441. Castillo, J.C., Knoepfle, D., Weyl, G., 2017. Surge pricing solves the wild goose chase. In: Proceedings of the 2017 ACM Conference on Economics and Computation. ACM, pp. 241e242. Chen, M.K., Sheldon, M., July 2016. Dynamic pricing in a labor market: surge pricing and flexible work on the uber platform. In: Proceedings of the 2016 ACM Conference on Economics and Computation, p. 455. Daganzo, C.F., 1978. An approximate analytic model of many-to-many demand responsive transportation systems. Transportation Research 12 (5), 325e333. Diao, M., Kong, H., Zhao, J., 2021. Impacts of transportation network companies on urban mobility. Nature Sustainability 1e7. Erhardt, G.D., Roy, S., Cooper, D., Sana, B., Chen, M., Castiglione, J., 2019. Do transportation network companies decrease or increase congestion? Science Advances 5 (5), eaau2670. Farber, H.S., 2005. Is tomorrow another day? The labor supply of New York City cabdrivers. Journal of Political Economy 113 (1), 46e82. Farber, H.S., 2015. Why you can’t find a taxi in the rain and other labor supply lessons from cab drivers. The Quarterly Journal of Economics 130 (4), 1975e2026. Geoffrion, A.M., 1967. Solving bicriterion mathematical programs. Operations Research 15 (1), 39e54. Ke, J., Yang, H., Li, X., Wang, H., Ye, J., 2020. Pricing and equilibrium in on-demand ridesplitting markets. Transportation Research Part B: Methodological 139, 411e431. Ke, J., Li, X., Yang, H., Yin, Y., 2021. Pareto-efficient solutions and regulations of congested ridesourcing markets with heterogeneous demand and supply. Transportation Research Part E: Logistics and Transportation Review 154, 102483. Sheldon, M., 2016. Income targeting and the ridesharing market 56, 1457131797556. (Unpublished Manuscript). Available at: https://static1.squarespace.com/static/56500157e4b0cb70 6005352d.

190 Supply and Demand Management in Ride-Sourcing Markets Sun, H., Wang, H., Wan, Z., 2019. Model and analysis of labor supply for ride-sharing platforms in the presence of sample self-selection and endogeneity. Transportation Research Part B: Methodological 125, 76e93. Vignon, D.A., Yin, Y., Ke, J., 2021. Regulating ridesourcing services with product differentiation and congestion externality. Transportation Research Part C: Emerging Technologies 127, 103088. Xu, Z., Yin, Y., Ye, J., 2020. On the supply curve of ride-hailing systems. Transportation Research Part B: Methodological 132, 29e43. Yang, H., Qin, X., Ke, J., Ye, J., 2020. Optimizing matching time interval and matching radius in on-demand ride-sourcing markets. Transportation Research Part B: Methodological 131, 84e105. Yang, H., Yang, T., 2011. Equilibrium properties of taxi markets with search frictions. Transportation Research Part B: Methodological 45 (4), 696e713. Yang, H., Ye, M., Tang, W.H., Wong, S.C., 2005. Regulating taxi services in the presence of congestion externalities. Transportation Research Part A: Policy and Practice 39 (1), 17e40. Yu, J.J., Tang, C.S., Max Shen, Z.J., Chen, X.M., 2020. A balancing act of regulating on-demand ride services. Management Science 66 (7), 2975e2992. Zha, L., Yin, Y., Yang, H., 2016. Economic analysis of ride-sourcing markets. Transportation Research Part C: Emerging Technologies 71, 249e266. Zhang, K., Nie, Y.M., 2021. To pool or not to pool: equilibrium, pricing and regulation. Transportation Research Part B: Methodological 151, 59e90.

Chapter 9

Third-party platform integration in ride-sourcing markets Yaqian Zhou4, Jintao Ke2, Hai Yang1 and Hai Wang3 1 Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China; 2Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China; 3School of Computing and Information Systems, Singapore Management University, Bras Basah, Singapore; 4School of Automation, Chongqing University, Chongqing, China

9.1 Background Over the past few years, the worldwide growth in ride-sourcing services has disruptively reshaped the way people travel and substantially affected the traditional taxi industry and multimodal transportation system. Multiple ridesourcing companies now co-exist and compete with each other in many local markets, as the entry permits in these markets are less restrictive than those in other markets. For example, Didi and Uber were fierce competitors in China until the end of 2016, when Uber China was acquired by Didi. Moreover, although Didi now has the largest share of the ride-sourcing market in China, its dominance is continuously challenged by new rivals, such as Meituan, which has expanded its ride-sourcing services to several cities in eastern China. By 2019, there was a competition between ride-sourcing companies in countries all over the world, including between Uber and Lyft in the United States; Grab and Go-Jek in Southeastern Asia; Uber and Bolt in Europe; Uber and Didi in Australia; Careem and Uber in the Middle East; Ola and Uber in India; and 99 Taxi, Cabify and Uber in Brazil (Wang and Yang, 2019). Competition is a double-edged sword for ride-sourcing markets. On the one hand, as in other service markets, competition between platforms prevents a monopolistic platform from greedily maximising its own profit by distorting trip fares from an efficient level. On the other hand, competition between platforms leads to market fragmentation and increases matching friction. Specifically, the matching of demand with supply in ride-sourcing markets generally exhibits increasing returns to scale; that is, the efficiency of Supply and Demand Management in Ride-Sourcing Markets. https://doi.org/10.1016/B978-0-443-18937-1.00005-X Copyright © 2023 Elsevier Inc. All rights reserved.

191

192 Supply and Demand Management in Ride-Sourcing Markets

matching increases as the number of passengers and drivers involved in one platform increases. Therefore, demand splitting between multiple independent platforms may result in a certain level of market inefficiency. The novel business model known as platform integration that has recently emerged can prevent market fragmentation from occurring in the presence of platform competition. In this model, third-party companies denoted ‘integrators’ integrate the ride-sourcing services offered by multiple companies into one application (app) interface. For example, Baidu Map integrates the services of Didi, and some smaller ride-sourcing companies into its app and offers its users a wide variety of ride-service options (Song, 2019). Without platform integration, a passenger typically requests a ride via a single platform and can only be matched with an idle driver affiliated with this platform. In contrast, with platform integration, a passenger can request a ride via an integrator and can be matched with an idle driver affiliated with any of its integrated ride-sourcing platforms (e.g., Didi or Meituan). Accordingly, compared with the absence of platform integration, in the presence of platform integration, the accessible supply level for a passenger is higher and the average pick-up distance/time is lower. Platform integration is different from a multi-homing situation, which is when drivers work for more than one of several competing platforms to maximise their earnings and/or passengers use multiple apps to quickly receive responses for their ride requests. Despite these benefits, multi-homing can be inconvenient for drivers and passengers. First, drivers must continually switch between various mobile phones or apps and must turn off all but the relevant platform’s app after they are matched with passengers from this platform. Second, once drivers are dispatched to passengers on a given platform, the passengers must cancel requests that they have made via other platforms. The above-described inefficiencies of multi-homing situations are overcome by a novel business model created by third-party platform integrators, which have recently emerged as integrators of the ride services offered by multiple independent ride-sourcing platforms. On the demand side, integrators enable passengers to use a single app to request and receive services from any one of the integrators’ participating platforms. On the supply side, drivers are affiliated with and are paid by one ride-sourcing platform. In actual operations, passengers have the freedom to request services from the full set of ridesourcing platforms collected by an integrator (denoted ‘full platform integration’) or from a subset of these platforms (denoted ‘partial platform integration’). In this chapter, we restrict our analysis to full platform integration; specifically, for analytical tractability, we analyse a ride-sourcing market with full platform integration in which passengers and drivers are single-homing. Table 9.1 illustrates all of the possible scenarios for a ridesourcing market and the scenario studied in this chapter. Platform integration differs from simply merging competing platforms. In a market scenario with platform integration, participating platforms retain the freedom to independently determine their trip fares and other operating

Third-party platform integration in ride-sourcing markets Chapter | 9

193

TABLE 9.1 All of the possible scenarios for a ride-sourcing market and the scenario studied in this chapter (U). Passenger demand Singlehoming Platform integration

Multihoming

Driver supply Singlehoming

Multihoming

None Partial Full

U

U

strategies, while also sharing some information (e.g., the locations of idle drivers) with the integrator. By contrast, in a market scenario with platform merging, some of the platforms are combined into a single platform to acquire a competitive advantage and thus no longer make individual pricing and operating decisions. In an extreme case, all of the platforms are merged and thus the market becomes a monopoly. In summary, platform integration (1) maintains the competition between platforms and thereby prevents one platform from making excessive profits and (2) reduces market fragmentation. This chapter explores the effect of platform integration on platforms’ operating strategies and the resulting passenger demand, platform profit, and social welfare. Platform integration is expected to improve system efficiency by increasing market thickness. However, competing and integrated platforms may nevertheless adjust their operating strategies (e.g., trip fare), which may in turn affect the market equilibrium and system efficiency. These represent complex feedback loops that must be thoroughly characterised with tractable mathematical models. Accordingly, we devise mathematical models to describe a ride-sourcing market with multiple competing platforms and compare the vehicle utilisation rate, realised demand, profit and social welfare in two market scenarios: (1) a competitive market without platform integration; and (2) a competitive market with platform integration. We determine that platform integration increases the total realised demand and social welfare at the Nash equilibrium (NE) and the social optimum (SO) but that this does not necessarily increase profit. Furthermore, we show that the market with platform integration generally achieves greater social welfare than the market without platform integration. We also conduct numerical studies to investigate the effects of the commission fee charged by an integrator to ride-sourcing platforms on the market measures in a general mixed market, which is a market in which some passengers access platforms’ ride-sourcing services via an integrator and other passengers do so directly via platforms. The managerial

194 Supply and Demand Management in Ride-Sourcing Markets

insights obtained from these theoretical and numerical studies will assist ridesourcing platforms, integrators and governments to design operating strategies that maximise platform profit and social welfare.

9.2 Market equilibrium and optimal strategies In this section, we first present a model to describe the equilibrium of a market without platform integration and then extend the model to describe the equilibrium of a market with platform integration. We then study the optimal strategies for maximising platform profit and social welfare, namely, the NE and the SO. The model is based on some assumptions and simplifications. First, we model the ride-sourcing market at a stationary equilibrium without considering system dynamics. At this equilibrium, passengers choose between making a ride request directly via platforms or via the integrator based on the equilibrium costs of each. Second, we use an aggregate model in which passengers are homogeneous and make their transport mode choices based on the average trip fare, average ride time and average waiting time, following the traditions of previous studies (e.g., Zha et al., 2016). Third, we do not consider traffic congestion externality and surge pricing. Fourth, for analytical tractability, we assume that the vehicle fleet size is exogenously given to examine whether platform integration has differential effects on platforms of various sizes (e.g., whether small-scale platforms benefit more than large-scale platforms from platform integration). Fifth, we assume that drivers transport only one passenger, i.e., we do not consider shared rides. In a ride-sourcing service, a passenger’s total waiting time can be separated into two periods: a matching time, which represents the period from the ride request being received by the service to when a driver is assigned to the passenger, and a pick-up time, which represents the period from the driver being assigned to the passenger to when the driver picks up the passenger. Most studies have examined either the matching time or the pick-up time to obtain analytical results. In this chapter, we adopt the assumption of firstcome-first-served (FCFS) matching made by Castillo et al. (2017), which has been widely used for describing ride-sourcing markets (Castillo et al., 2017; Ke et al., 2020). FCFS matching means that passengers who make an order are immediately matched with their nearest idle driver, regardless of the distance between a passenger and this driver. In this case, passengers have a negligible matching time, so the major component of their pick-up time is their waiting time. Moreover, we assume that the matching pool has no more than one waiting passenger, so the average pick-up time depends only on the number of idle drivers. To check the robustness of our findings, we also conduct numerical studies (presented in Appendix 9.J.) in which we consider the effects of a more general matching mechanism that was adopted by Zha et al. (2018). This

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195

mechanism captures the aggregate matching of waiting for passengers with idle drivers and the meeting of matched drivers and passengers.

9.2.1 Market without platform integration We consider a ride-sourcing market with I competing platforms (i ¼ 1; 2; /; I), a group of passengers, and a group of drivers affiliated with only one specific platform. In the market without platform integration, a driver works for only one platform, and a passenger sends a ride request for a given trip to only one platform. Let Q ¼ Qf ðCÞ represent the demand function, where Q is the maximum possible demand (i.e., the potential arrival rate of passengers), f ð $Þ is a decreasing function of a generalised cost C, and Q is the realised passenger demand (i.e., the actual arrival rate of passengers in the ride-sourcing market, regardless of which platform they choose). Thus, the
  inverse demand function (denoted by BðQÞ) is given by BðQÞ ¼ f 1 Q Q . P Let qi  0 denote the realised demand for platform i. Accordingly, Q ¼ i qi . Passengers are assumed to be homogeneous in terms of their value of time b (). Each platform i chooses the trip fare Fi to maximise its profit, and  do not
we consider advanced pricing features, such as surge pricing. Let W Niv denote the waiting and pick-up times of passengers opting for platform i, which depend on the number of idle vehicles Niv on the platform. With reference to Castillo et al. (2017), we assume that each platform implements an FCFS matching mechanism and dispatches idle vehicles to passengers immediately after they make a request.
 In this case, the waiting time for matching is negligible and thus W Niv is dominated by the pick-up time. Let Ni and Niv denote the fleet size and number of idle vehicles in equilibrium for the platform i, respectively. Similarly, let N and N v denote the total fleet size and total number of idle vehicles P v in equilibrium in the market, respectively (i.e., N ¼ P v i Ni and N ¼ i Ni ). Let T denote the average trip time, which is assumed to be equal for all of the platforms. A user equilibrium is reached when passengers do not prefer one platform over another, as the generalised costs (comprising the monetary cost and time cost) of all ride-sourcing platforms are equal. In equilibrium, the vehicle
fleet size
for  each platform equals the sum of the numbers of vacant vehicles Niv þqi W Niv and occupied vehicles (qi T). Vacant vehicles are idle vehicles that are available (Niv ) and those en route to
 pick up matched passengers (qi W Niv ). Thus, we have the following equations:

 (9.1) BðQÞ ¼ Fi þ b T þ W Niv ; ci;

196 Supply and Demand Management in Ride-Sourcing Markets



 Ni ¼ Niv þ qi T þ W Niv ; ci; Q¼

I X

(9.2)

qi ;

(9.3)

Niv ;

(9.4)

Ni :

(9.5)

i¼1

Nv ¼

I X i¼1

N¼

I X i¼1

Let c denote the average operating cost of a vehicle per time unit. In the market without platform integration, each platform decides its trip fare Fi . The ne ne optimal profit of each platform at the NE, denoted as Pne 1 ; P2 ; /; PI , can be ne obtained using a set of non-negative qi that solves the following optimisation problem for each platform i (hereinafter, variables with a superscript ‘ne’ denote the optimal solutions at the NE).


  cNi (9.6) maxPi ¼ qi Fi cNi ¼ qi BðQÞ b T þ W Niv qi 0

s.t. Eqs. (9.1)e(9.3). After substituting Fi into the objective function using the inverse demand function (Eq. 9.1), searching for the optimal passenger demand of each platform qi is equivalent to searching for the optimal fare with the
Fi . neTogether ne can be ; q ; /; q vehicle capacity constraint (Eq. 9.2), the solutions qne I 1 2 obtained using the following first-order conditions: !  dNiv  ne ne 0 ne BðQ Þ þ qi B ðQ Þ þ b (9.7) qne ¼ 0; ci; dqi qi ¼qne i i

 dNiv  ne ne 0 ne  0; ci; BðQ Þ þ qi B ðQ Þ þ b dqi qi ¼qne

(9.8)

i

where Qne ¼

I X

ne qne i ; qi  0:

(9.9)

i¼1

The SO scheme represents an ideal scenario in which a social planner has full control of all of the ride-sourcing platforms and aims to maximise the social welfare (denoted as S) by determining an appropriate value for the trip fares. After substituting Fi into the objective function below by using the

Third-party platform integration in ride-sourcing markets Chapter | 9

197

inverse demand function (Eq. 9.1), the unconstrained welfare-maximising problem can be formulated as follows: Z Q I I X X maxS ¼ BðxÞdx  qi BðQÞ þ qi Fi  cN qi 0

0

Z ¼

i¼1 Q

BðxÞdx 

0

s.t. Eqs. (9.1)e(9.3),where and

I P i¼1

I X

i¼1




qi b T þ W




Niv



(9.10)  cN;

i¼1

RQ 0

BðxÞdx 

I P

qi BðQÞ is the consumer surplus

i¼1


 so so denote the optimal qi Fi  cN is the total profit. Let qso 1 ; q2 ; /; qI

solution to Eq. (9.10) (hereinafter, the superscript ‘so’ indicates the SO), which can be obtained using the following first-order conditions: !  v dN (9.11) BðQso Þ þ b i  qso ¼ 0; ci; dqi qi ¼qso i i

BðQso Þ þ b

dNiv dqi

jqi ¼qso  0; ci; i

(9.12)

where Qso ¼

I X

so qso i ; qi  0:

(9.13)

i¼1

Taking the first-order and second-order derivatives of both sides of Eq. (9.2) with respect to qi yields: dNiv T þ Wi ¼ ; ci; (9.14) dqi 1 þ qi Wi0   v 0 00 dNi ðT þ Wi Þ 2Wi þ qi Wi d2 Niv dqi ¼ ; ci; (9.15)
2 2 0 dqi 1 þ qi W i
 where, for simplicity, Wi and Wi0 represent þ and dW Niv dNiv , respectively.   The signs of dNiv dqi and d 2 Niv dq2i are uncertain. If 1 þ qi Wi0 < 0, N v increases with demand, which is an inefficient market outcome termed a wild goose chase (WGC) (Castillo et al., 2017). In such a case, the number of idle drivers is extremely low, which forces the platform to match passengers with distant drivers. The following lemma excludes this outcome. Lemma 9-1. The optimal solutions at the NE and the SO in the market without platform integration are not located in the WGC region.

198 Supply and Demand Management in Ride-Sourcing Markets

See Appendix 9.A. for the proof.  Lemma 9-1 clearly specifies that 1 þ qi Wi0 > 0, implying that dNiv dqi <  0 and d2 Niv dq2i < 0, which indicates the existence and/or uniqueness of the optimisation solutions in the market without platform integration. Theorem 9-1. An optimal solution at the NE and a unique optimal solution at the SO exist in the market without platform integration. See Appendix 9.B. for the proof. The uniqueness of this Nash game is generally not guaranteed. Nevertheless, the optimal solutions at the NE in the market without platform integration are unique, as shown in Section 9.4.

9.2.2 Market with platform integration We consider an alternative scenario with an integrator that enables passengers to simultaneously access all I competing platforms. Hereinafter, we use a tilde (‘w’) to denote the counterparts of variables with platform integration. e consists of two groups of passengers: those who Realised passenger demand Q e1 ) and choose to make an order via the integrator (with an arrival rate of Q e e e e those who do not (with an arrival rate of Q2 ) (i.e., Q ¼ Q1 þ Q2 ). Accordingly, the realised demand qei for each platform consists of the demand for the platform via the integrator (denoted by qei1 ) and the direct demand for the I I P P e1 ¼ e2 ¼ platform (denoted by qei2 ). Then, Q qei1 and Q qei2 . Requests that i¼1

i¼1

passengers make via the integrator are assumed to be assigned to the ridesourcing platforms depending on their availability of idle vehicles and the corresponding pick-up distances, without discrimination between platforms. Specifically, when passengers make a request through the integrator, the integrator assigns the passengers to the nearest idle drivers regardless of the v platform to which the drivers are affiliated. Let Ne denote the total number of v v idle vehicles and Nei the total number of idle vehicles on platform i; thus, Ne ¼ P v Nei should be met. The sum of waiting and pick-up times of passengers i


v making a request via the integrator is W Ne , whereas that of passengers
v making a direct request to platform i is W Nei . If platforms are free to re-optimise their trip fares after platform integration e i denote the trip fare and is introduced, then a new NE may form. Let Fei and u  e1 ), respectively. e i ¼ qei1 Q market share of platform i in the integrator (i.e., u

199

Third-party platform integration in ride-sourcing markets Chapter | 9

Then, the expected trip fare offered by platform i via the integrator is

I P

e i Fei . u

i¼1

Let es denote the commission set by the integrator,1 where es  0 indicates that passengers pay a fee to the integrator and es < 0 means that the integrator offers some compensation to attract passengers. In this case, the monetary cost of passengers who make a request via the integrator can be approximated by I P e i Fei þ es. Let Te denote the average trip time in the market with platform u i¼1

integration, which is assumed to be equal for all platforms. The generalised equilibrium trip costs of passengers who make a request via the integrator, denoted as Ce1 , and those who make a direct request to platform i, denoted as Ce2 , are given by: !! I X v e i Fei þ b Te þ W Ne Ce1 ¼ u þ es (9.16) i¼1

and

!! Ce2 ¼ Fei þ b Te þ W

v Nei

.

(9.17)

Suppose that all passengers choose the mode that minimises their trip cost at equilibrium. Without loss of generality, let k denote ( the ) index for the platform charging the highest trip fare (i.e., Fek ¼ max Fei ; ci) and let j denote the index for the platform charging the lowest trip fare (i.e., Fej ¼ ( ) es < Fej  Fek ; then min Fei ; ci). If Ce1  Ce2
v bW Nej ,

then

1. At present, platform integrators such as Baidu and Gaode do not impose commission charges on passengers but indirectly benefit from passengers’ patronage of their free services. Here, for generality, we suppose that a positive, zero, or negative commission charge is imposed on passengers rather than on platforms.

200 Supply and Demand Management in Ride-Sourcing Markets

Ce1  Ce2 >

I P

! e i Fei þ bTe þ es  Ce2 > Fej þ bTe þ bW u

i¼1

v Nej

 Ce2 ¼ 0,

which indicates that the generalised trip cost of passengers who make a request via the integrator is always higher than that of the passengers who make a direct request to a platform; hence, no passengers choose the integrator. If es is sufficiently small (e.g., less than a threshold es1 ), such that Ce1 < Ce2 always holds, then all passengers make a request via the integrator; if es is sufficiently large (e.g., greater than a threshold es2 ), such that Ce1 > Ce2 always holds, then no passengers make a request via the integrator; and if es is of intermediate magnitude (e.g., between es1 and es2 ) with Ce1 ¼ Ce2 , passengers have no preference between making a request via the integrator or directly to I independent platforms. es1 and es2 depend on many factors, including exogenous variables (e.g., vehicle fleet size), platform decisions (i.e., trip fare), the inverse demand function Bð $Þ, and waiting time Wð $Þ. It is possible that there exists no non-negative s1 , such that all passengers make a request via the integrator when the vehicle supply is sufficiently large and/or the vehicle fleet sizes vary greatly between platforms. Based on the above discussions, we derive the following equations: ! ! e ¼ min Ce1 ; Ce2 ; B Q (9.18) ! qei1 Ce2  Ce1

 0; ci;

(9.19)

 0; ci;

(9.20)

! qei2 Ce1  Ce2

where Eqs. (9.19) and (9.20) are constraints that indicate qei1 ¼ 0 if Ce1 > Ce2 (i.e., no passengers make a request via the integrator) and qei2 ¼ 0 if Ce2 > Ce1 (i.e., all passengers make  a request via the integrator). At equilibrium, the number of rides

qei1 ; qei2

that platform i serves should also meet the

following vehicle conservation constraint: !! Ni ¼

v Nei

þ qei1 Te þ W Ne

v

þ qei2 Te þ W

!! v Nei

; ci

(9.21)

With platform integration, the supply (i.e., idle vehicle supply) of the competing platforms is managed in a single matching pool without discrimination, and the integrator assigns passengers to the nearest idle drivers. In this case, the realised demand qei1 of each platform in the integrator should be

Third-party platform integration in ride-sourcing markets Chapter | 9

201

v proportional to its number of idle vehicles Nei , or equivalently, it should be representable as v qei1 Nei ¼ v ; ci; j. qej1 Nej

(9.22)

We make the following two mild assumptions for further analysis. Assumption 9-1. The average trip times on each platform in a platform e integrator and not in a platform integrator are equal (i.e., T ¼ T). This assumption is realistic because vehicles operate on the same road network under the same traffic conditions, irrespective of the platform on which they are located. Assumption 9e2. (a) The waiting and pick-up time function Wð $Þ is convex, strictly decreasing with N v , and continuously differentiable. (b) The inverse demand function Bð $Þ is convex, strictly decreasing with the realised passenger demand, and continuously differentiable; and Q$ BðQÞ is concave for Q > 0. (c) The total realised demand at the NE and the SO is positive (i.e., Qne > 0; ene > 0; Q eso > 0). Qso > 0; Q Assumption 9-2-(a) means that the waiting and pick-up times monotonically decrease N v . However, the decreasing slope (representing the marginal decrease caused by a unit increase in N v ) decreases further as N v increases. Assumption 9-2-(b) ensures the existence and uniqueness of optimal solutions at either the NE or the SO and is often adopted for demand functions in revenue management. Assumption 9-2-(c) prevents the existence epm ¼ 0; or Q ewm ¼ 0. of unrealistic cases Qpm ¼ 0; Qwm ¼ 0; Q If commission fee es charged by the integrator is less than es1 , then the passengersdall of whom make requests via the integratordpay equal exI P e i Fei ) and have an equal waiting and pick-up pected trip fares (given by u i¼1 ! v time (given by W Ne ). Thus, Eqs. (9.18)e(9.22) reduce to the following equations:

!! ! I X v e ¼ e i Fei þ b Te þ W Ne u þ es; B Q i¼1

(9.23)

202 Supply and Demand Management in Ride-Sourcing Markets

!! v v Ni ¼ Nei þ qei Te þ W Ne

v Ne ¼

I X

;

ci;

(9.24)

v Nei ;

(9.25)

qei ;

(9.26)

i¼1

e¼ Q

I X i¼1

v qei Nei ¼ ; ci; j. qej Nevj

We

!!!,

v v Nei þe qi Te þW Ne

(9.27)

further

v Nei ¼

obtain

!!!,

v v Nej þe qj Te þW Ne

v Nej

from

the vehicle conservation constraint (Eq. , 9.24) and the ,demand split rule for the integrator (Eq. 9.27). Consequently, Ni

v Nei ¼ Nj

v Nej dthat is, the N v on

each platformdis proportional to each platform’s vehicle fleet size, which leads to the following relationship: Ni v v Nei ¼ Ne ; ci. N

(9.28)

If es lies between es1 and es2 , then passengers who make a request via the integrator and those who make a direct request with platform i should have the same generalised trip costs at equilibrium. Therefore, Eqs. (9.16)e(9.18) are equivalent to the following equation: ! !! !! I X v v e e e e e e e e i Fi þ b T þ W N B Q ¼ u þ es ¼ F i þ b T þ W N i . i¼1

(9.29) If es is greater than es2 , then no passengers make a request via the integrator due to its high commission charge. Accordingly, the market is exactly equivalent to that without platform integration, which is analysed in the previous section. In the market with platform integration, each platform i chooses a value of e F i to maximise its own profit (denoted as Pei ) at the NE.2 If es < es1 , all trip fare 2. Here, for generality, we suppose that each platform can re-optimise its fare after joining the integrator. In practice, platforms may not do so. Thus, we also perform numerical studies to examine the case with unchanged fares in the presence of platform integration.

Third-party platform integration in ride-sourcing markets Chapter | 9

! sets

Fe1 ; Fe2 ; /; FeI

that fail to satisfy Fei ¼ Fej ¼

I P

203

e i Fei ; ci; j are not u

i¼1

equilibrium solutions because the platform that charges a lower fee is incentivised to increase its fee to reap greater profits. Specifically, all of the plate to achieve a stable market forms must set an equal trip fare (denoted as F) e equilibrium. Thus, after substituting F i into the objective function below using I P e i Fei ; ci; j, the inverse demand function (Eq. 9.23) and using Fei ¼ Fej ¼ u i¼1

the NE when es < es1 and with platform integration can be formulated as follows: ! !! ! v e b Te þ W Ne maxPei ¼ qei Fei cNi ¼ qei B Q  es  cNi ; (9.30) eqi 0 s.t. Eqs. (9.23)e(9.28).   ne ne ne be an NE solution in the market with platform Let qe1 ; qe2 ; /; qeI integration. Therefore, the following first-order conditions must be satisfied:  !  ne  ne ey  N d N i ne 0 e e b  es qei ne ¼ 0; ci; (9.31) þ qei B Q þ B Q  N de qi  ne eqi ¼eqi   ! ! v e ne ne N d N i ne 0 e e es  0; ci; (9.32) B Q þ qei B Q þ b  N de qi   ne eqi ¼eqi where ene ¼ Q

I X

ene qene i ;q i  0:

(9.33)

i¼1

e is Social welfare in the market with platform integration (denoted as S) defined as the sum of the consumer surplus and profits of all platforms and the integrator. In the case of es < es1 , the objective function at the SO can be mathematically written as follows: max Se¼ eqi 0

ZQe

s.t. Eqs. (9.23)e(9.28).

BðxÞdx  0

I X i¼1


v  qei b Te þ W Ne  cN;

(9.34)

204 Supply and Demand Management in Ride-Sourcing Markets

Let

  so so e e qeso ; q ; /; q denote the optimal solution to Eq. (9.34). 1 2 I

Accordingly, the following first-order conditions hold:  1 0  ! v e C so B eso þ Ni bd N  Aqei ¼ 0; ci; @B Q N de qi   so eqi ¼eqi   ! v e so N d N i e  0; ci; B Q þ b  N de qi   so eqi ¼eqi

(9.35)

(9.36)

where I X

eso ¼ Q

eso qeso i ;q i  0:

(9.37)

i¼1

Eq. (9.23) demonstrates that the optimal solution

  so so e e qeso ; q ; /; q is 1 2 I

independent of the commission fee charged by the integrator because the trip fare and commission fee can be packaged as an auxiliary decision variable for use by a social planner. Substituting Eq. (9.28) into Eq. (9.24) and then taking the first-order and second-order derivatives of both sides of Eq. (9.24) with respect to qei yields:

d 2 Ne ¼ de q2i v

v e dNe Te þ W ¼ ; ci; Ni de qi e0 þ qei W N 0 1 ! v e e 0 þ qei W e B e 00 d N C Te þ W @ 2W A de qi

Ni e0 þ qei W N

(9.38)

; ci;

!2

(9.39)

! 0

e and W e represent W Ne where, for simplicity, W respectively,

,

The signs of dNe

v

v

!, and dW Ne

v

d Ne , v

, de qi and

v d 2 Ne

de q2i are undetermined because the sign

e 0 is indeterminate. If Ni =N þ qei W e 0 > 0, then N v decreases of Ni =N þ qei W

Third-party platform integration in ride-sourcing markets Chapter | 9

205

with the passenger demand, which represents a normal non-WGC regime. We can then prove this by using the objective function (Eq. 9.30) and the firstorder condition Eq. (9.36) that the optimal solutions in the market with platform integration are not located in the WGC region, as stated below. Lemma 9-2. In the market with platform integration, the optimal solutions at the NE and the SO are not located in the WGC region. See Appendix 9.C. for the proof. e 0 > 0, implying that Lemma 9-2 clearly , specifies that Ni =N þ qei W , d Ne

v

de qi < 0 and d 2 Ne

v

de q2i < 0, which indicates the existence and/or

uniqueness of the optimisation solutions in the market with platform integration. Theorem 9-2. In the market with platform integration, there is an optimal solution at the NE and a unique optimal solution at the SO. See Appendix 9.D. for the proof. Similar to the market without platform integration, the uniqueness of the Nash game in the market with platform integration is generally not guaranteed. Nevertheless, the optimal solution at the NE in the market with platform integration is unique, as shown in the numerical studies described in Section 9.4. Summing the I equations on both sides of Eq. (9.24) yields: ! v e Te þ W e . (9.40) N ¼ Ne þ Q Then, taking the first-order and second-order derivatives of both sides of e affords: Eq. (9.40) with respect to Q

d 2 Ne ¼ e2 dQ v

v e Te þ W d Ne ; ¼ e eW e0 dQ 1þQ 0 1 ! v e eW e0 þ Q e B e 00 d N C Te þ W @ 2W A e dQ

!2 eW e 1þQ

(9.41)

:

(9.42)

0

The problem represented by Eq. (9.34) can be reformulated with regard to e Q, as follows: ! Z e Q e Te þ W e  cN. BðxÞdx  Qb (9.43) maxS ¼ e>0 0 Q

206 Supply and Demand Management in Ride-Sourcing Markets

The constraint defined by Eq. (9.40) demonstrates that the maximum social welfare in the market with platform integration is also independent of the ewm > 0 (according to Assumption 9-2-(c)), then number of platforms I.3 If Q the first-order condition for Eq. (9.43) with an interior solution should satisfy: ! v so d Ne e so ¼ 0: (9.44) B Q j þb e Qe¼Qe dQ If es lies between es1 and es2 , platform i’s optimal solutions at the NE and the SO can be obtained by solving the following problem: !     e max Pei ¼ qei1 þ qei2 Fei cNi ¼ qei1 þ qei2 B Q eqi1 ;eqi2 0 !!! (9.45) v e e b T þ W N i  cNi ; s.t. Eqs (9.21), (9.22) and (9.29), and max Se¼ eqi1 ;eqi2 0

ZQe BðxÞdx 

I X



v  qei1 b Te þ W Ne

(9.46)

i¼1

0



I X



v   cN; qei2 b Te þ W Nei

i¼1

s.t. Eqs (9.21), (9.22) and (9.29). As the above two optimisation problems are challenging to solve, we facilitate our theoretical analysis by focusing on the case in which all passengers make a request via the integrator (i.e., es < es1 ). Accordingly, unless otherwise specified, in the following theoretical derivations all passengers in the market with platform integration are assumed to make a request via the integrator. We also conduct numerical studies to analyse the general mixedmarket equilibrium, in which some passengers make requests via the integrator and some make direct requests to individual platforms.

3. On this basis, in the case of es < es1 and when the supply capacity N (i.e., vehicle fleet size) is fixed, the solutions in the monopoly market at the SO equal the solutions in the market with platform integration at the SO.

207

Third-party platform integration in ride-sourcing markets Chapter | 9

9.3 Evaluation of the performance of platform integration In this section, we compare the optimal solutions of the markets with and without platform integration at the NE and the SO. This reveals the effects of platform integration and quantifies the extent to which platform integration influences the realised passenger demand, the vehicle utilisation rate, the platform profit, the consumer surplus and the social welfare.

9.3.1 Effect of vehicle fleet size at the Nash equilibrium/social optimum Let Ui and Uei , denote the vehicle utilisation rate of platform i in the markets without and with platform integration, where these rates are measured by the fraction of occupied service time, as follows: qi Ti ; ci Ni   qei1 þ qei2 Te Uei ¼ ; ci: Ni Ui ¼

(9.47)

(9.48)

We present the following lemma to illustrate the influence of vehicle fleet size on the system performance measures and optimal solutions. Lemma 9-3. If es < es1 , then the following relations hold. ne ne ne vne > N vne . This also (a) If Ni > Nj and qne i ; qj > 0, then qi > qj and Ni j applies to the market with platform integration. so so so vso > N vso . This also (b) If Ni > Nj and qso i ; qj > 0, then qi > qj and Ni j applies to the market with platform integration. so (c) If N v $W 0 ðN v Þ is strictly increasing, and Ni > Nj and qso i ; qj > 0, then so so Ui > Uj . (d) In the market with platform integration, all vehicles are utilised to the same extent (i.e., Uei ¼ Uej ; ci; j).

See Appendix 9.E. for the proof. Lemma 9-3 stipulates that compared with a platform with a smaller fleet size, a platform with a larger fleet size has a greater market share at the NE and the SO. In addition, compared with a platform with a smaller fleet size, a platform with a larger fleet size has a higher vehicle utilisation rate at the SO if N v $W 0 ðN v Þ is a strictly increasing function, such as WðN v Þ ¼ AðN v Þk , which has often been used in the literature (Arnott, 1996; Li et al., 2019).4 If k ¼ 12, 4. A is an exogenous parameter that represents the factors associated with the matching technology and depends on the area, vehicle fleet size and vehicle velocity; and k is a parameter representing sensitivity to N v , where 0 < k  1.

208 Supply and Demand Management in Ride-Sourcing Markets

the average sum of waiting and pick-up time of passengers is inversely proportional to the square root of N v . Moreover, the relationship between vehicle fleet size and utilisation rate is not monotonic at the NE in a market without platform integration, which is further discussed in the numerical examples.

9.3.2 Effect of platform integration at the Nash equilibrium We first introduce the following lemma, which deals with the total realised demand at the NE.

 ene > Qne for I  2, where s is given Lemma 9-4. If es < min s; es1 , then Q as follows:

!

ene s¼B Q

1 ene 0 ene B Q þ Q I

!

  1 ne 0 ne ne  BðQ Þ þ Q B ðQ Þ . I

(9.49)

See Appendix 9.F. for the proof. s is the difference between the mean marginal total generalised costs perceived by passengers at the NE in markets with integration and those perceived by passengers at the NE in markets without platform. Lemma 9-4 implies that platform integration can increase realised passenger demand at the NE if the commission charged by the integrator is reasonably low. That is, platform integration leads to more passengers being attracted to platforms, due to its reducing generalised costs. ne Let Sne and Se denote the social welfare at the NE in a market without and with platform integration, respectively, where these variables are defined as follows: Z Qne I X 
ne Sne ¼  cN (9.50) BðxÞdx  qne i b Ti þ Wi 0

i¼1

Z Qe

ne

Se ¼ ne

0

BðxÞdx 

I X

! e e qene i b T þW

ne

 cN.

(9.51)

i¼1

Thus, based on Lemma 9-4,

we present the following theorem. Theorem 9-3. If es < min s; es1 and qi ; qei > 0, then the social welfare at the NE in a market with platform integration is greater than that in a market ne without platform integration (i.e., Se > Sne for I  2). See Appendix 9.G. for the proof. Theorem 9-3 holds that platform integration can improve social welfare at the NE, as it reduces the loss of matching efficiency caused by market fragmentation.

Third-party platform integration in ride-sourcing markets Chapter | 9

209

However, the effect of platform integration on platform profit is uncertain. In a market with platform integration, two opposing market forces affect platform profit: the positive force of increased passenger demand and the negative force of decreased trip fares. Consequently, the relative strengths of these two forces dictate whether profit-maximising platforms benefit from platform integration. In a system with a low supply capacity (i.e., small vehicle fleet size) and/or few platforms, there are long waiting and pick-up times. Thus, upon platform integration, there is a large reduction in the waiting and pick-up times due to the convexity of the waiting time function. This attracts more passengers to the system than before integration, thereby offsetting the negative effect of the reduced trip fare, leading to a greater profit and social welfare. Therefore, a winewin situation is achieved in which the passengers and ride-sourcing platforms are both better off due to platform integration. However, in a system with a certain higher supply capacity and/or more platforms, an increased passenger demand due to integration may not offset the negative effect of a greatly reduced trip fare, leading to a decreased profit for the platforms. Nevertheless, platform integration increases the social welfare in this system because the increase in the consumer surplus is greater than the decrease in the platform profit. These aspects are further explored in numerical experiments presented in Section 9.4.

9.3.3 Effect of platform integration at the Social optimum We present the following lemma to compare the total realised demand at the SO in markets with platform integration with that in markets without platform integration. vso Lemma 9-5. If es < es1 and Ne  Nivso , then given a waiting time function eso > Qso for I  2. WðN v Þ ¼ AðN v Þk , it follows that Q See Appendix 9.H. for the proof. vso The sufficient condition, Ne  Nivso , indicates that the total number of idle vehicles at the SO in a market with platform integration is greater than or equal to N v in each platform in a market without platform integration. This mild assumption generally holds in actual operations. As aforementioned, the waiting time function WðN v Þ ¼ AðN v Þk has been widely used in the literature. Given Lemmas 9-4 and 9-5, we expect that the total realised demand in a market with platform integration is greater than that in a market without platform integration at both the NE and the SO, implying that the former market is more attractive to passengers than the latter market. so Let Sso and Se denote the social welfare at the SO in a market without and with platform integration, respectively, where these variables are defined as follows:

210 Supply and Demand Management in Ride-Sourcing Markets

Z S ¼

Qso

so

BðxÞdx 

0

so Se ¼

Z Qeso

I X


 so qso  cN i b T i þ Wi

(9.52)

i¼1

BðxÞdx 

0

I X

! qeso i b

e Te þ W

so

 cN.

(9.53)

i¼1

Thus, based on Lemma 9-5, we present the following theorem. Theorem 9-4. If es < es1 , then the social welfare at the SO in a market with platform integration is greater than or equal to that in a market without so platform integration (i.e., Se  Sso ). See Appendix 9.I. for the proof. By considering Theorem 9-5 and Theorem 9-3, we find that platform integration helps to improve social welfare at the NE and the SO in a market.

9.4 Numerical studies In this section, we conduct several numerical experiments to illustrate the effects of platform integration on the optimal decision variable (i.e., trip fare), key endogenous variables (e.g., realised demand) and system performance measures (e.g., vehicle utilisation rate, platform profit and social welfare). Specifically, we compare the following two ride-sourcing markets that have multiple platforms: (1) a competitive market without platform integration and (2) a competitive market with platform integration. In addition, we use the experiments to demonstrate the extent to which the supply (in terms of the number of platforms I and the vehicle fleet size Ni ) affects the results of platform integration. We also evaluate the effect of the commission fee charged by the integrator in a general mixed market, in which some passengers make a request via an integrator and others do not. Consider the following negative exponential demand function: Q ¼ Q expðaCÞ;

(9.54)

where a > 0 is a cost sensitivity parameter, C is the generalised travel cost given by F þ bðT þWÞ and Q is the potential demand when C ¼ 0. The waiting time is assumed to be inversely proportional to the square root of N v pffiffiffiffiffiffi (i.e., WðN v Þ ¼ A= N v ). Table 9.2 reports the values of the key model parameters used in the numerical studies. These parameter values are selected with partial reference to previous studies (Zha et al., 2016; Ke et al., 2020) and for illustrative purposes only. In actual operations, real data can be used to calibrate the parameters.

9.4.1 Effect of market fragmentation In this section, we investigate how market fragmentation, which is reflected by the number of platforms in a market, affects market performance measures

Third-party platform integration in ride-sourcing markets Chapter | 9

211

TABLE 9.2 Values of key model parameters in the numerical studies. Symbol

Parameter

Value

Q

Potential demand

1:0  105 (trip/h)

a

Cost sensitivity of demand function

0:013 (1/HKD)

b

Value of time

120 (HKD/h)

c

Unit operating cost per vehicle

50 (HKD/h)

T

Average trip time

0:4 (h)

A

Sensitivity of waiting time function

5 (h)

es

Commission fee charged by the integrator

0 (HKD)

(e.g., platform profit and social welfare). For illustrative purposes, we assume that the vehicle fleet sizes are equal across all platforms (i.e., Ni ¼ Nj ; ci; j). The total supplydi.e., the sum of the number of vehicles of all platformsdis P set to a constant 1:0  104 (veh) (i.e., N ¼ Ni ¼ 1:0  104 (veh)). We ini

crease the number of platforms from 1 to 10 to demonstrate how the extent of market fragmentation affects the total realised demand, trip fare, total profit, and social welfare. The results are shown in Fig. 9.1, wherein the green solid line demonstrates how the values of the abovementioned four market measures for platforms change after the implementation of platform integration with respect to the extent of market fragmentation if the competing platforms do not re-optimise their trip fares after integration. By contrast, the orange solid line indicates the trend in the values of the market measures after the implementation of platform integration with respect to the extent of market fragmentation if the competing platforms do re-optimise their trip fares after integration, such that a new NE is formed. We study the asymptotic properties of some key measures ‘as the number of platforms increases to infinity’ to obtain some practical insights into what occurs when the number of platforms is large but far from infinity. We do not consider congestion effects as these may affect the asymptotic properties and make it infeasible for the number of operators to approach infinity. In the following numerical studies, we demonstrate that these asymptotic properties are observed as the number of platforms increases to 10. Fig. 9.1 demonstrates that the competitive market is fully efficient regardless of the existence of platform integration as the number of platforms increases to infinity. Specifically, the optimal solutions and measures (e.g., the realised demand, profit and social welfare) at the NE of the markets with and without platform integration converge to those at the social optima of the markets. As mentioned, the SO

212 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 9.1 Effect of the number of platforms on the (A) total realised demand, (B) trip fare, (C) total profit and (D) social welfare in a market with/without platform integration.

solution in a market with platform integration is independent of the number of platforms (i.e., if the market fragmentation cost is eliminated), as shown by the constant values of total demand, profit and social welfare represented by the horizontal orange dashed lines in Fig. 9.1. Total realised demand: Fig. 9.1A shows that platform integration increases the total realised demand at the NE and the SO, with these increases being greater in magnitude as the number of platforms increases. This indicates that the larger the number of platforms (i.e., the greater the extent of market fragmentation), the greater the demand that is generated by platform integration. Moreover, the total demand at the NE in the market without platform integration initially increases and then decreases with the number of platforms. This indicates that the increase in the number of platforms in the market without platform integration has two opposing effects. The first effect is its enhancement of competition, which prevents the platforms from

Third-party platform integration in ride-sourcing markets Chapter | 9

213

distorting the trip fares from the efficient level (i.e., shifting the market from the monopoly optimum (MO) towards the SO). The second effect is its increasing the market fragmentation costs. The first effect leads to a decrease in the trip fare and an increase in the number of passengers, while the second effect leads to an increase in the passengers’ waiting time and thus discourages passengers from using ride-sourcing services. Crucially, the implementation of platform integration maintains the first effect but eliminates the second effect. As a result, as the number of platforms increases to infinity, the total passenger demand of all platforms in the market with platform integration approaches the level of efficient passenger demand that exists at the SO. By comparing the green solid line and blue solid line in Fig. 9.1A, it can be seen that even without the re-optimisation of trip fares, platform integration increases competing platforms’ number of passengers when there are few platforms. However, this increase is less than that achieved by re-optimising the trip fares, as represented by the orange solid line. Trip fare: Fig. 9.1B shows that in the markets with and without platform integration, trip fares at the NE initially greatly decrease as the number of platforms increases, which is due to the competition between platforms. As the number of platforms increases further, the platforms in the market with platform integration further reduce their trip fares to increase demand, whereas the platforms in the market without platform integration slightly increase their trip fares to decrease demand. This slight increase serves to offset the increase in the matching frictions (i.e., the waiting time and the marginal change in waiting time with respect to a unit increase of supply increase) that results from the increase in market fragmentation as the number of platforms increases in the absence of platform integration. This counter-intuitive phenomenon was also identified by Zha et al. (2017), who noted that it indicated that the change in the price elasticity of demand and that of matching frictions must be explored to determine whether competition will result in lower or higher trip fares. By contrast, platform integration eliminates the market fragmentation cost, so there is no increase in matching frictions, and thus platforms do not need to suppress demand and therefore do not increase their trip fares. Total profit: Fig. 9.1C shows that at the NE, platform integration does not necessarily improve total profit. This is due to the opposing effects of platform integration on the total platform profit; that is, the fact that platform integration (1) increases total realised demand by eliminating the market fragmentation cost and reducing the waiting time; and (2) increases the competition between the platforms, which results in decreased trip fares. When there are few platforms, effect (1) dominates platform integration greatly increases passenger demand but trip fares are similar to those in the absence of platform integration. As a result, platform integration increases platforms’ profit. By contrast, when there are many platforms, effect (2) dominates platform integration leads to a decrease in trip fares that is greater than the increase in

214 Supply and Demand Management in Ride-Sourcing Markets

passenger demand. The latter situation occurs because when there are many platforms but no platform integration, the average pick-up time increases greatly, leading to substantial market fragmentation costs. As aforementioned, at the MO, the platforms set a trip fare that covers the drivers’ average pick-up and travel-time costs plus a monopoly mark-up. Therefore, compared with the average trip fare in the market with platform integration, that in the market without platform integration is higher, as this eliminates the market fragmentation cost. We thus conclude that platform integration induces platforms to decrease their trip fares, such that they have lower total profits than they do in the absence of platform integration. Moreover, a comparison of the blue solid line and green solid lines in Fig. 9.1C indicates that if the trip fares remain unchanged, the total profit in the market with platform integration ultimately continuously increases with the number of platforms. However, profit enhancement becomes less with re-optimised trip fares under a new NE, when the number of platforms is larger than 3. Social welfare: Fig. 9.1D demonstrates that platform integration leads to increased social welfare at the NE and the SO, which is in accordance with Theorem 9-3 and Theorem 9-4. Moreover, these increases are proportional to the number of platforms, as platform integration not only maintains competition between platforms and prevents their charging excessive trip fares but also decreases the market fragmentation cost caused by competition. By comparing Fig. 9.1D and Fig. 9.1A, it can be seen that the shape of the curve depicting the variation in social welfare vs the number of platforms is similar to that of the curve depicting the variation in total realised demand vs the number of platforms, which directly dictates the consumer surplus. Furthermore, in the market with platform integration, when there is a sufficiently large number of platforms, the increase in the consumer surplus is significantly greater than the decrease in the total profit, which causes social welfare to be continuously increasing. In contrast, in the market without platform integration, when there is a sufficiently large number of platforms, the decreases in both the total realised demand and the total profit cause the social welfare to be continuously decreasing. In addition, in the market with platform integration, there is a greater increase in social welfare when the platforms re-optimise their trip fares than when they do not.

9.4.2 Effect of vehicle fleet size This subsection verifies the theoretical findings in Sections 9.2, 9.3, and 9.4 and investigates the effect of platform integration on ride-sourcing platforms that differ in their vehicle fleet sizes. We assume that the ride-sourcing market comprises platforms 1, 2, and 3, with vehicle fleet sizes of N1 ¼ 500 (veh), N2 ¼ 400 (veh) and N3 ¼ 300 (veh), respectively. We then scale the vehicle fleet size of each platform by d ¼ 10% in every instance and depict the results in Figs. 9.2e9.4.

Third-party platform integration in ride-sourcing markets Chapter | 9

215

FIGURE 9.2 Effect of vehicle fleet size on vehicle utilisation rate in a market with/without platform integration at (A) the Nash equilibrium and (B) the social optimum.

FIGURE 9.3 Effect of vehicle fleet sizes on realised demand in a market with/without platform integration and with the trip fares (A) re-optimised and (B) unchanged after platforms join an integrator.

Vehicle utilisation rate: Fig. 9.2 shows that the vehicle utilisation rates of all three platforms first increase and then decrease with vehicle fleet size in both markets, i.e., with or without platform integration.5 Initially, an increase in vehicle fleet size causes a significant increase in realised passenger demand (see Fig. 9.2A) by reducing the passenger waiting time, which is sufficient to increase vehicle utilisation. Subsequently, the realised passenger demand slowly increases with vehicle fleet size, leading to excess vehicle supply and thus lower vehicle utilisation. In addition, at the NE in the market without platform integration, when the vehicle fleet size is not sufficiently large, the platform with the largest fleet size (i.e., platform 1) has the highest vehicle 5. In a market with platform integration, given these vehicle fleet sizes and es ¼ 0, the curves of the vehicle utilisation rates of all platforms are identical because all passengers use the platform integrator.

216 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 9.4 Effects of vehicle fleet sizes on profit in markets with/without platform integration and the trip fares (A) re-optimised or (B) unchanged after platforms join an integrator.

utilisation rate; however, platform 1 ultimately has a lower vehicle utilisation rate than platform 2 (see Fig. 9.2A). By contrast, platform one always has the highest vehicle utilisation rate in the market without platform integration at the SO (see Fig. 9.2B), which is consistent with our theoretical results summarised in Lemma 9-3. The above trends occur because in the market without platform integration, the increasing return to scale of matching means that compared with a platform with a smaller fleet size, a platform with a larger fleet size has a shorter waiting time and attracts more passengers. The resulting increase in passenger demand is sufficient to generate a higher utilisation rate in the latter platform than in the former platform. However, this marginal effect of increasing supply capacity on passenger demand diminishes more quickly in platforms with a larger fleet size than in platforms with a smaller fleet size. Consequently, the increase in passenger demand does not keep pace with the increase in supply capacity, leading to a lower utilisation rate in platforms with larger fleet size. Furthermore, the vehicle utilisation rate of all platforms in the market with platform integration is larger than the utilisation rates of platforms of all fleet sizes in the market without platform integration, which implies that platform integration increases platforms’ vehicle utilisation rate. Realised demand: We find that the realised demand changes after the integration of non-identical platforms. Fig. 9.3A and B compare each platform’s realised demand in the markets with and without platform integration, with re-optimised and unchanged trip fares, respectively. The figures show that the realised demand initially increases rapidly with vehicle fleet size as a result of the increasing return to scale of matching, and then increases slowly because the scale effect has been fully exploited. Platform integration also results in all three platforms having increased passenger demand due to decreased passenger waiting times. By comparing the dashed lines in Fig. 9.3A with those in Fig. 9.3B, it can be seen that competitive re-

Third-party platform integration in ride-sourcing markets Chapter | 9

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optimisation of trip fares after platform integration further increases the number of passengers. Profit: Fig. 9.4A and B compare each platform’s profit in markets with and without platform integration, with re-optimised and unchanged trip fares, respectively. The figures show that the profits initially increase and then decrease with vehicle fleet size. When the vehicle fleet sizes are small, an increase in supply capacity significantly increases demand by decreasing the passenger waiting time, which increases profits due to the increasing returns to scale of the matching function. By comparing the dashed lines (representing the markets with platform integration and re-optimised trip fares) with the solid lines (representing the markets without platform integration) in Fig. 9.4A, it can be seen that platform integration increases the profits of platforms (especially those of smaller platforms, e.g., platform (3) when the vehicle fleet size is not particularly large. However, platform integration may decrease the profits of platforms (especially those of larger platforms, e.g., platform (1) as the vehicle fleet size increases further because in this situation the matching friction is already small and the effect of platform integration diminishes. As previously discussed, whether a platform obtains increased profits subsequent to platform integration depends on the extent to which the impetus of the increased passenger demand mitigates the resistance of the decreased trip fare. A comparison of the dashed lines in Fig. 9.4B (the market with platform integration and unchanged trip fares) and the dashed lines in Fig. 9.4A (the market with platform integration and re-optimised trip fares) indicates that in the market with platform integration and unchanged trip fares, platforms with larger fleet sizes initially have higher profits than platforms with smaller fleet sizes. However, an iterative decision-making process is needed to achieve a stable market equilibrium to evaluate the long-term performance of platforms of various fleet sizes. This reveals that in the market with platform integration and compared with platforms with smaller fleet sizes, the platform with the largest fleet size has smaller increases in profits and may even have decreased profits.

9.4.3 Effect of commission fee In the numerical studies above, we evaluate the case of es  es1 , in which all passengers make orders via the integrator in the market with platform integration. In this section, we present numerical examples to investigate the effect of the commission fee es charged by the integrator in a general mixed-market equilibrium, which is where some passengers

make  orders via the integrator but others do not (i.e., in the case of es ˛ es1 ; es2 ). Assume that there are platforms 1, 2 and 3 in the ride-sourcing market and that their trip fares are F1 ¼ F2 ¼ F3 ¼ 70 HKD/trip and remain unchanged after platform integration, i.e., Fe1 ¼ Fe2 ¼ Fe3 ¼ 70 HKD/trip. Let ðN1 ; N2 ; N3 Þ denote the supply

218 Supply and Demand Management in Ride-Sourcing Markets

capacity of these three platforms. We conduct numerical experiments for three supply scenarios, namely, ð2; 000; 2; 000; 2; 000Þ, ð3; 000; 3; 000; 3; 000Þ, ð3; 000; 2; 000; 1; 000Þ and ð4; 000; 3; 000; 2; 000Þ, where the units are veh. The other parameters are presented in Table 9.2. Fig. 9.5 shows how the total realised demand changes with the increase in commission fee in this general mixed-market equilibrium with various supply capacities. The cross at the left (right) end of each curve indicates the critical  lower (or upper) threshold of the commission (i.e., es1

or es2 )), below (or

above) which all (or no) passengers make orders via the integrator. That

is, the  general mixed-market equilibrium is reached within the range of es ˛ es1 ; es2 . By comparing the orange solid line (the red solid line) with the blue solid line (the green solid line), it can be seen that the lower and upper bounds of this range generally decrease with the vehicle fleet size. This is because for a given fixed potential demand, the matching friction is smaller when the vehicle fleet size is larger, such that the effect of platform integrationdthat is, the reduction of market fragmentationdbecomes marginal. Accordingly, in this situation, the integrator must decrease its commission charge to retain patronage. In addition, by comparing the orange solid line (the blue solid line) with the red solid line (the green solid line), it can be seen that the lower and upper bounds of this range for platforms with equal vehicle fleet sizes are lower than the lower and upper bounds for platforms with various vehicle fleet sizes. This suggests that platform integration attracts more passengers when the

FIGURE 9.5 Effect of commission fee on realised demand given fixed trip fares and with various supply capacities.

Third-party platform integration in ride-sourcing markets Chapter | 9

219

ride-sourcing market comprises platforms of various vehicle fleet sizes than when it comprises platforms of equal vehicle fleet sizes. Hence, compared with the latter case, in the former case the integrator can increase its commission fee to maintain a higher realised demand and thereby achieve a greater profit.

9.5 Conclusion In this chapter, we investigate third-party platform integration, which is a novel business model for a ride-sourcing market in which multiple ridesourcing platforms are collected together. We devise mathematical models to describe a ride-sourcing market with multiple competing platforms and compare the vehicle utilisation rate, realised demand, profit, and social welfare in two market scenarios: (1) a competitive market without platform integration and (2) a competitive market with platform integration. The major findings of the study are summarised below. First, we prove that platform integration increases the total realised demand and social welfare at both the NE and the SO. Second, we find that platform integration not only maintains competition and restrains platforms’ ability to distort trip fares from the efficient level but also decreases the market fragmentation cost caused by competition, leading to an increase in social welfare. Third, we find that with platform integration and thus in the absence of a market fragmentation cost, the total profit may not increase if the market is greatly fragmented (i.e., when it comprises many platforms), as in such a market the competition for passengers may be so fierce that platforms must set extremely low trip fares. By contrast, without platform integration and thus in the presence of a market fragmentation cost, there is a relatively high platform profit as there is no competition between platforms for passengers and thus platforms increase trip fares to suppress passenger demand. Fourth, we find that when the supply capacity is not excessively high, platform integration results in a winewin situation for passengers and platforms. By contrast, when the supply capacity is relatively high, platform integration results in increased profits for platforms with smaller vehicle fleet sizes but may result in decreased profits for platforms with larger vehicle fleet sizes. In addition, we find that the investigation of platform integration in a general mixed-market equilibrium is analytically intractable; extensive numerical studies are thus executed to evaluate this equilibrium. These reveal that this equilibrium is reached within a certain range of commission fees. That is, compared with when the total supply capacity is high and/or platforms are heterogeneous in terms of vehicle fleet size, when the total supply capacity is low, the integrator can set a higher commission fee to maintain a higher realised demand and achieve a greater profit. Our study reveals several avenues that merit further exploration, such as (1) platform integration with elastic supply and its effect on the platform profit, consumer surplus, provider (driver) surplus and social welfare; (2) partial

220 Supply and Demand Management in Ride-Sourcing Markets

platform integration, in which passengers choose a subset of platforms that offer differentiated ride-sourcing services via an integrator; (3) the platform and service mode choices of heterogeneous passengers with various values of time; (4) the effect of passengers cancelling orders via the integrator; (5) equilibrium analyses of a market in which a subset of ride-sourcing platforms are public operators aiming to maximise social welfare; (6) operating strategy design for an integrator, such as dispatching drivers to passengers who select all of the platforms or select only some of the platforms; (7) a market with passengers who make heterogeneous choices via an integrator, i.e., a market in which some passengers choose a subset of platforms with higher trip fares and better service quality and some passengers who choose a subset of platforms with lower trip fares and lower service quality; and (8) a disaggregate network model for ride-sourcing markets with platform integration and heterogeneous trip attributes (e.g., trip length). This chapter is based on one of our recently published articles (Zhou et al., 2022).

Appendix 9.A. Proof of Lemma 9-1 In the market without platform integration,  fleet size equals the
v the
vehicle v sum of the numbers of vacant vehicles Ni þqi W Ni and occupied vehicles (qi T), and the number of vacant vehicles comprises the number of idle vehicles
 (Niv ) and the number of vehicles dispatched to pick up passengers (qi W Niv ):

 Ni ¼ Niv þ qi T þ W Niv .
 W Niv is a decreasing and convex function with respect to Niv with the following properties: lim WðrÞ ¼ 0, and lim WðrÞ ¼ N. We thus obtain the r/N r/0
v Ni Niv inverted-U shape fi Ni ¼ qi ¼ TþW N v shown in Fig. 9.A1. ð iÞ At the SO, the first-order condition(Eq. 9.12) requires that the following As BðQso Þ  0, relation holds: BðQso Þ þ bdNiv dqi jqi ¼qso  0. i  bdNiv dqi jqi ¼qso  0, which implies that the SO solutions are not located in the i WGC region (i.e., the left-hand side of fi ð $Þ in Fig. 9.A1). At the NE, given the optimal passenger demand qne i (the red dashed line in Fig. 9.A1), the two solutions correspond to the two possible numbers of idle drivers. One solution has a large number of idle drivers (i.e., the right-hand side of fi ð $Þ in Fig. 9.A1), which results in a short pick-up time. The other solution is a ‘bad’ equilibrium in which there are few idle drivers (i.e., the lefthand side of fi ð $Þ in Fig. 9.A1) and the pick-up time is long, and therefore, many drivers spend time picking up passengers who are far away. This is a WGC scenario and is revealed by the derivative of the left-hand side of fi ð $Þ with respect to Niv being positive.

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221

FIGURE 9.A1 Solutions for the number of idle drivers as a function of drivers and ride requests.


 Ni Niv shows that the In addition, the inverted-U shape of fi Niv ¼ qi ¼ TþW ðNiv Þ v N in the normal region is always larger than that in the WGC region. Thus, given the optimal passenger demand qne and the lower waiting time Niv , the

i v  higher trip fare Fi ¼ BðQÞ b T þW Ni . This implies that the platform profit in the normal region is greater than that in the WGC region. Thus, the optimal solutions at the NE are also not located in the WGC region.

Appendix 9.B. Proof of theorem 9-1 In Lemma 9-1, we prove that the optimal solutions at both the NE and the SO in the market without platform integration are not located in the WGC region.   That is, dNiv dqi < 0 and d 2 Niv dq2i < 0. This implies that Niv is a concave function of qi . Assumption 9-2-(b) states that BðQÞ is decreasing and QBðQÞ is concave. As a result, the first-order derivative of  platform profit with respect to demand, i.e., vPi =vqi ¼ Q þ qi BðQÞ þ bdNiv dqi , is decreasing with qi . That is, the model defined by Eq. (9.6) is concave. In addition, it can be proved that the first-order derivative of social welfare with respect to demand, i.e., vS=vpi ¼ BðQÞ þ bdNiv dqi , is decreasing with qi . Thus, the model defined by Eq. (9.10) is also concave. Due to
the
non-negativity of qi and
Fi
 (where  v v Fi ¼ BðQÞ b T þW Ni ), i.e., qi  0 and BðQÞ b T þW Ni  0, ci, the feasible set of qi is compact. As the model defined by Eq. (9.6) is concave and continuous and the feasible set of qi is non-empty, compact and convex, there is a solution at the NE in the market without platform integration (Debreu, 1952). Similarly, as the model defined by Eq. (9.10) is concave and continuous and the feasible set of qi is non-empty, compact and convex, there is a unique solution at the SO in the market without platform integration.

222 Supply and Demand Management in Ride-Sourcing Markets

Appendix 9.C. Proof of Lemma 9-2 In the market with platform integration, the vehicle capacity constraint is given by: !! v v Ni ¼ Nei þ qei Te þ W Ne . ! W Ne

v

v is a decreasing and convex function with respect to Ne with the

following properties: lim WðrÞ ¼ 0, and lim WðrÞ ¼ N. We thus obtain the r/N r/0 ! v v e e v Ni  i =NÞN N i  ¼ Ni ðN  shown in inverted-U shape of fei Ne ¼ qei ¼ v v e e TeþW N TeþW N Fig. 9.C1. At the SO, the first-order condition (Eq. 9.36) requires that the following ! ! v so so e Ni d N e e so  0. As B Q relation should hold: B Q þ N b je  0, qi de qi qi ¼e e Ni d N so jq ¼e N b de qi e i qi v

 0, which implies that the SO solutions are not located in the

WGC region (i.e., the left-hand side of fei ð $Þ in Fig. 9.C1). At the NE, given the optimal passenger demand qene i (the red dashed line in Fig. 9.C1), the two solutions correspond to the two possible numbers of idle ! v drivers. Due to the inverted-U shape of fei Ne

e i =NÞN  , it is ¼ qei ¼ Ni ðN v e TeþW N v

invariably the case that N v in the normal region (i.e., the right-hand side of

FIGURE 9.C1 Solutions for the number of idle drivers as a function of drivers and ride requests.

Third-party platform integration in ride-sourcing markets Chapter | 9

223

feð $Þ in Fig. 9.C1) is higher than N v in the WGC region (i.e., the left-hand side of fei ð $Þ in Fig. 9.C1). Thus, given the optimal passenger demand qene the i and!! ! v e b Te þW Nev lower waiting time Ne , the higher trip fare Fei ¼ B Q . This implies that the platform profit in the normal region is greater than that in the WGC region. Therefore, the optimal solutions at the NE are also not located in the WGC region.

Appendix 9.D. Proof of theorem 9-2 In Lemma 9-2, we prove that the optimal solutions at both the NE and the SO in the market are not located in the WGC region. ,with platform integration , That is, d Ne

v

de qi < 0 and d2 Ne

v

de q2i < 0. This implies that Ne is a concave v

function of qi . In Assumption 9-2-(b), it is stated that BðQÞ is decreasing and QBðQÞ is concave. As a result, the first-order derivative of platform profit with respect to i.e., ! , ,demand, vPei

e e þ qei B Q ve qi ¼ Q

þ ðNi =NÞbdNe

v

de qi  es, is decreasing with qei .

That is, the model defined by Eq. (9.30) is concave. In addition, we can prove that , welfare with respect to demand, i.e., ,the first-order!derivative of social vSe

e vpi ¼ B Q

v þ ðNi =NÞbdNe

de qi , is decreasing with qei . Thus, the

model defined by Eq. (9.34) is also concave. Due ! to the non-negativity of qei !! e b Te þW Nev Fei ¼ B Q ), i.e.,

and

Fei

qei  0

(where and

! !! v e b Te þW Ne B Q  0, ci, the feasible set of qei is compact. As the model defined by Eq. (9.30) is concave and continuous, and the feasible set of qei is non-empty, compact, and convex, there is a solution at the NE in the market with platform integration (Debreu, 1952). As the model defined by Eq. (9.34) is concave and continuous, and the feasible set of qei is non-empty, compact, and convex, there is a unique solution at the SO in the market with platform integration.

224 Supply and Demand Management in Ride-Sourcing Markets

Appendix 9.E. Proof of Lemma 9-3 Let fi ðxÞ ¼

Ni  x ; x > 0; T þ WðxÞ

(9.A1)

where fi ðxÞ and x, respectively, denote the ith platform’s realised passenger demand qi and its corresponding number of idle vehicles Niv in the market
 and without platform integration. That is, fi Niv ¼ qi  
 dNiv dqi jqi ¼qne ¼ 1 fi0 Nivne . i The first-order and second-order derivatives with respect to x are as follows: fi0 ðxÞ ¼  fi00 ðxÞ ¼

1 þ fi ðxÞW 0 ðxÞ ; T þ WðxÞ

2W 0 ðxÞð1 þ fi ðxÞW 0 ðxÞÞ ðNi  xÞW 00 ðxÞ ðT þ WðxÞÞ2

(9.A2) .

In the normal region, fi0 ðxÞ < 0 and fi00 ðxÞ < 0. Clearly, if Ni > Nj , then
 fi ðxÞ > fj ðxÞ and fi0 ðxÞ > fj0 ðxÞ. If fi0 ðxi Þ ¼ fj0 xj ¼ 0, we find that fi0 ðxi Þ ¼

 fj0 xj < fi0 xj . Given that fi00 ðxÞ < 0, we thus have xi > xj , which is illustrated in Fig. 9.E1. Let gi ðuÞ ¼

1u T; u > 0; T þ WðNi uÞ

(9.A3)

where gi ðuÞ and u, respectively, denote platform i’s vehicle utilisation rate  v (qi T=Ni ) and ratio of idleness (Ni Ni ), where the latter is the proportion of idle vehicles on the platform. The first-order and second-order derivatives with respect to u are as follows: g0i ðuÞ ¼  g00i ðuÞ ¼

1 þ fi ðNi uÞW 0 ðNi uÞ T; T þ WðNi uÞ

2Ni W 0 ðNi uÞð1 þ fi ðNi uÞW 0 ðNi uÞÞ Ni2 ð1  uÞW 00 ðNi uÞ ðT þ WðNi uÞÞ2

(9.A4)

T.

If u ¼ x=Ni ; 0 < u < 1, it follows that g0i ðuÞ ¼ fi0 ðxÞT, and thus  
 . Therefore, in the normal region, g0i ðuÞ < 0 and dNiv dqi q ¼qpm ¼ T g0i upm i i

i

g00i ðuÞ < 0. Let Hi ðuÞ ¼ 1 þ fi ðNi uÞw0 ðNi uÞ. Consequently, in the normal region, Hi0 ðuÞ ¼ Ni fi0 ðNi uÞw0 ðNi uÞ þ Ni fi ðNi uÞw00 ðNi uÞ > 0. Thus, Hi ðuÞ is a strictly increasing function.

Third-party platform integration in ride-sourcing markets Chapter | 9

225

FIGURE 9.E1 Effect of vehicle fleet size on (A) qi and Niv at the Nash equilibrium; (B) Ui at the nash equilibrium; (C) qi and Niv at the SO; and (D) Ui at the SO.

If Ni > Nj , then xi > xj . As xw0 ðxÞ is a strictly increasing function and
 and < 0, we find that 0 > uNi w0 ðuNi Þ > uNj w0 uNj
 0 < wðuNi Þ < w uNj , leading to


 Ni w0 ðuNi Þ T þw uNj Nj w0 uNj ðT þwðuNi ÞÞ. Thus, we determine:
 Ni ð1  uÞw0 ðuNi Þ Nj ð1  uÞw0 uNj
 . >
ðT þ wðuNi ÞÞ T þ w uNj w0 ðxÞ

This inequality indicates that if Ni > Nj , then Hi ðuÞ > Hj ðuÞ. If Hi ðui Þ ¼ 0


 and Hj uj ¼ 0, then Hi ðui Þ ¼ Hj uj < Hi uj . Given that Hi ðuÞ is an increasing function, we have ui < uj . Thus, as ui ¼ xi =Ni and xi > xj , we determine that ui Ni > uj Nj . Without platform integration: ne Lemma 9-3-(a). If qne i > 0 and qj > 0, then according to the first-order condition (Eq. 9.8),

226 Supply and Demand Management in Ride-Sourcing Markets

ne

BðQ

0 ne Þ þ qne i B ðQ Þ

 dNiv  ¼ b dqi qi ¼qne

(9.A5)

i

and ne

BðQ

0 ne Þ þ qne j B ðQ Þ

 dNjv  ¼ b  dqj 

.

(9.A6)

qj ¼qne j

  As mentioned, dNiv dqi < 0 and d2 Niv dq2i < 0 for ci in the normal region; thus, the effect of fleet size can be illustrated as shown.in Fig.  9.E1A. If    ne ne v v Ni > Nj , q > 0 and q > 0, let dN dqi  ; thus, dqj  ne ¼ dN i

j

i

qi ¼qi

j

B0 ðQne Þ

qj ¼qne j

> However, as < 0, to satisfy Eqs. (9.A5) and (9.A6), it must ne ne ne be the case that qi > qi > qj . From Fig. 9.E1A, we also obtain Nivne >  ne ne Njvne . By substituting Uine ¼ qne i T=Ni and Uj ¼ qj T Nj into Eqs. (9.A5) and (9.A6), respectively, we obtain:  Uine Ni 0 ne dNiv  ne B ðQ Þ ¼ b (9.A7) BðQ Þ þ T dqi qi ¼qne qne i

qne j .

i

and

 Ujne Nj 0 ne dNjv  B ðQ Þ ¼ b BðQ Þ þ  T dqj  ne

.

(9.A8)

qj ¼qne j

However, the relation between Uine and Ujne is uncertain. In Fig. 9.E1B, une1 denotes the case in which Uine > Ujne , while une2 denotes the case in i i which Uine < Ujne . so Lemma 9-3-(b). If qso i > 0 and qj > 0, the first-order condition (Eq. 9.12) means that  dNiv  so BðQ Þ ¼ b dqi qi ¼qso i

and

 dNjv  BðQ Þ ¼ b  dqj  so

. qj ¼qso j

Similarly, the effect of fleet size at the SO is shown in Fig. 9.E1C and D. so so so vso > N vso and That is, if Ni > Nj , qso i > 0 and qj > 0, then qi > qj , Ni i so so Ui > Uj .

Third-party platform integration in ride-sourcing markets Chapter | 9

227

With platform integration: If all of the passengers ,make orders , via the integrator in the market with v v platform integration, Nei qei ¼ Nej qej . Thus, based on the vehicle conservation

constraint !!!,

v qi Te þW Ne Ni e ,

leads to qei Te

qei ¼ ,

(Eq. !!!, 9.24), v Nj e qj Te þW Ne qej , which , ,

Nj , i.e., Uei ¼ Uej . From qei Te

Ni ¼ qej Te

Ni ¼ qej Te

Nj , we

determine that the passenger demand for each platform on the integrator is e Thus, if Ni > proportional to each platform’s vehicle fleet size, i.e., qei ¼ NNi Q. Nj , then qei > qej .

Appendix 9.F. Proof of Lemma 9-4 ene , it follows ene ¼ Qne if I ¼ 1. Then, if qne  1Qne > 0, qene  1Q Proof. Q j k I I 0 ne that BðQne Þ þ 1I Qne B0 ðQne Þ  BðQne Þ þ qne k B ðQ Þand ! ! ! ! ne ne 1 ene 0 ene ne 0 ene e e B Q þ Q B Q B Q þ qej B Q . I 1 ne 1 ene ne ne ene ene ene If ck; j; qne j  I Q , then qk < q j . Thus, Q < Q . k  I Q > 0, and q ene Otherwise, dk; j;such that qne j , and therefore: k q 0 ne BðQne Þ þ qne k B ðQ Þ



¼ b

  ! ! v e Nj d N  ene þ qene B0 Q ene  es; b  b >   B Q  j dqk qk ¼eqnej N de qj   ne eqj ¼eqj (9.A9) 

dNkv 

dqk qk ¼qne k

dNkv 

where, from top to bottom, these relations follow from first-order conditions   Eqs. (9.7) and (9.8), from dNiv dqi < 0 and d 2 Niv dq2i < 0 for ci in the normal region, from the subsequent proof below, and from the first-order condition given by Eq. (9.32). Combining the above results yields: ! ! ne ne ne 1 1 e e B0 Q e BðQne Þ þ Qne B0 ðQne Þ > B Q þ Q  es: I I

228 Supply and Demand Management in Ride-Sourcing Markets

! e If s ¼ B Q

ne

! þ

1Q ene B0 I

e Q

ne

  ne 1 ne 0 ne  BðQ Þ þ I Q B ðQ Þ , then if es
B Q I

!

ne

! 1 ene 0 ene þ Q B Q . I

As Assumption 9-2-(b) holds that QBðQÞ is concave, we can derive that ene ; I  2, if es < s. BðQÞ þ 1I QB0 ðQÞ is decreasing. Thus, Qne < Q The proof is complete. The following proves the result in Eq. (9.44). Let Ni x N ; x > 0: fei ðxÞ ¼ Te þ WðxÞ Ni 

(9.A10)

Then, the first-order and second-order derivatives with respect to x are as follows: 0 fei ðxÞ ¼

Ni e þ f i ðxÞW 0 ðxÞ N  Te þ WðxÞ

and

! 2W ðxÞ 1 þ fei ðxÞW 0 ðxÞ 0

00 fei ðxÞ ¼

(9.A11)

 Ni  Ni  x W 00 ðxÞ N : !2 

Te þ WðxÞ 0 00 In the normal region, fei ðxÞ < 0 and fei ðxÞ < 0. In this work, x and fei ðxÞ v denote the total number of idle vehicles Ne and platform i’s demand qei in the
 , then market with platform integration, respectively. If fk ðxk Þ ¼ fej xj ¼ qene  j  ne with xj > xk ; I  2. Based on these definitions, comparing dNkv dqk q ¼e k qj ,   0
  Nj ev is equivalent to comparing fk00 ðxk Þ with NNj fej xj , that is, if de qj  N dN  ne e qj ¼e qj ,   
  0 v  N j e ne < fk0 ðxk Þ > NNj fej xj , then dNkv dqk q ¼e d N . de q  j N k qj  ne e qj ¼e qj 0 By employing the definition of fe ðxÞ and f 0 ðxk Þ, we find that i

k

229

Third-party platform integration in ride-sourcing markets Chapter | 9

N e0
 f j xj ¼ 
 0
N j 1 þ fej xj W xj >
 Te þ
W xj 1 þ fej xj W 0 ðxk Þ > Te þ Wðxk Þ 1 þ fk ðxk ÞW 0 ðxk Þ ¼ Te þ Wðxk Þ ¼ fk0 ðxk Þ.

1þ

N e
 0
 f xj W xj Nj j
 Te þ W xj

where from top to bottom, these relations follow from the definition in Eq.  0
 (9.17) for fej xj , the fact that N Nj  1, the fact that xk > xj , the fact that
 fk ðxk Þ ¼ fej xj and the definition in Eq. (9.22) for fk0 ðxk Þ.
   N e0 0 Thus, which leads to dNkv dqk  Nj f j xj < fk ðxk Þ, ,    Nj ev ne < d N . de q  j N qk ¼e qj  ne e qj ¼e qj The proof is complete. -

Appendix 9.G. Proof of theorem 9-3 Based on the social welfare at the NE in the market with platform integration given in Eq. (9.51) and that at the NE in the market without platform integration given in Eq. (9.50), we obtain Q Ze

!

ne

Se S ¼ ne

I X

BðxÞdx 

ne

qene i b

e Te þ W

ne

I X



i¼1

Qne




qne i b

T

þ Wine



! :

i¼1

Substituting the vehicle conservation constraints defined by Eqs. (9.2) and (9.24) into the above equation yields: e Q Z

ne

Se Sne ¼ ne

BðxÞdx þ

I X i¼1

Qne

! Ni evne N  Nivne : b N

Because BðxÞ is convex and strictly decreasing, we have Q Ze

!

ne

BðxÞdx  Qne

!

e  Qne B Q e Q ne

ne

:

230 Supply and Demand Management in Ride-Sourcing Markets

Based on the definition of fei ðxÞ (Eq. 9.24) and fi ðxÞ (Eq. 9.25), qene i ¼ !  
 vne vne . If q0ne ¼ fe N N vne , then q0ne  qne . fei Ne and qne i Ni i i ¼ fi Ni i i i , , v v de qi < 0 and d2 Ne de q2i < 0 in the normal region and Because dNe q0 ne  qne i , it follows that

! Ni evne vne N  Ni b  N i¼1      v   v  I I X X e e Ni Ni 0ne d N  ne d N  b qene b qene  .   i  qi i  qi  N de qi N de qi  i¼1  ne i¼1  ne eqi ¼eqi eqi ¼eqi

I X

Combining these results yields: ne Se Sne 

I  X



0

ne B ene qene @B Q i  qi

!

i¼1

1 v e Ni d N C j ne A. þ b N de qi eqi ¼eqi

Substituting the first-order conditions given by Eqs. (9.A10) and (9.A1) into the above inequation affords: ! !  I  X ne ne ne ne e  Qne . e Se Sne  B0 Q s Q qene qene (9.A12) i  qi i þe i¼1

The minimum value of

I P i¼1

  ne q ene q qene i i is i

! e Q

ne

Qne

, e Q

ne

I, which

is easily determined from the first-order conditions of the following problem:  I  X ne  q minZ ¼ qene qene i i i i¼1

subject to Qne ¼

I X

ne qne i ; qi > 0

i¼1

ene ¼ Q

I X i¼1

ene qene i ;q i > 0

Third-party platform integration in ride-sourcing markets Chapter | 9

Thus, we obtain:

! B

!

e Q

e Q Q

ne

0

ne

Se S   ne

231

ene Q

ne

! e Q þ es Q ne

ne

I

ne

:

! e As Q

ne

e > Qne (from Lemma 9-4) and B0 Q

ne

ne < 0, we find that Se >

Sne ; I  2. With regard to the profit, let:   I X
 ne ne ne ne qi BðQ Þ b T þ Wi  es  cN P ¼ i¼1

and Pe ¼ ne

I X

! e qene B Q i

! e b Te þ W

ne

ne

!  es  cN.

i¼1

Thus, ! ne ene B Q ene Pe Pne ¼ Q

Q BðQ Þ þ ne

ne

I X i¼1

! ¼

!

ene ene  Qne B Q Q

þ

I X i¼1

! 0

 B

ene Q

Ni evne b N  Nivne N

! Ni evne N  Nivne b N I  X

qene i



qne i

!

! ene  Qne es Q

! þQ

ne

ene B Q

!

! BðQ Þ ne

ene  Qne es Q

! !  ne ne ne ne e B Q BðQ Þ . qei þ Q

i¼1

Appendix 9.H. Proof of Lemma 9-5 eso if I ¼ 1: Qso ¼ Q Consider the following waiting-time function: WðN v Þ ¼ AðN v Þk :

(9.A13)

The first-order derivative with respect to Nv is W 0 ðN v Þ ¼  kAðN v Þðkþ1Þ :

(9.A14)

232 Supply and Demand Management in Ride-Sourcing Markets

Let Nx feI ðxÞ ¼ ; x > 0: Te þ WðxÞ

(9.A15)

The first-order and second-order derivatives with respect to x are thus 0 1 þ feI ðxÞW 0 ðxÞ feI ðxÞ ¼  Te þ WðxÞ

and

(9.A16)

! 00 feI ðxÞ ¼

2W 0 ðxÞ 1 þ feI ðxÞW 0 ðxÞ

ðN  xÞW 00 ðxÞ !2

.

Te þ WðxÞ 1 so so If qso j  I Q > 0, then based on the definition in Eq. (9.A15), Q ¼ !  vso vso . If cj; qso  1Qso , then Q eso > Iqso ; thus, Q eso > feI Ne and qso j ¼ fj Nj j j I

Qso . eso  Iqso , then Otherwise, if dj; Q j ! ! !  vso 0 evso vso 0 evso e e 1þfI N W N  1 þ Ifj Nj W N j

¼1



kAIfj Njvso



!ðkþ1Þ vso Nej

  ðkþ1Þ > 1 kAfj Njvso Njvso

  ¼ 1 þ fj Njvso W 0 Njvso ;  where the above relations, from top to bottom, follow from the fact that qso !j 1Qso and N 0 evso , the evso  Nevso j , the definition in Eq. (9.A16) for W N j I

vso assumption that Ne  Njvso , the fact that I  2, and the definition in Eq.  (9.A16) for W 0 Njvso .

By using the first-order conditions given in Eqs. (9.12) and (9.44), and the definitions given in Eq. (9.A16), we have

Third-party platform integration in ride-sourcing markets Chapter | 9

233

!!  vso e  ! b T þW N v so d Ne  1 e !¼ ! ! B Q ¼ b ¼ b  e dQ 0 vso vso vso  e eso 0 e e e e e 1 þ fI N W N fI N Q¼Q and

  b T þ W Njvso ¼   . ¼ b  vso 0 N vso 0 N vso 1 þ f f N W j qj ¼qso j j j j j ! so vso e Finally, we obtain B Q < BðQso Þ, assuming that Ne  Njvso . Because  dNiv  so BðQ Þ ¼ b  dqj 

1

eso > Qso . Bð $Þ is strictly decreasing, Q

Appendix 9.I. Proof of theorem 9-4 By proceeding in the same way as in the proof of Theorem 9-3 in Appendix 9.H., it can be proven that at the SO  0 1  !  v I  X e so C so B eso þ Ni bdN  Se Sso  qeso @B Q A. i  qi  N de q i i¼1 so eqi ¼eqi Based on the first-order conditions given in Eqs. (9.35) and (9.36) and the fact that qeso i > 0; ci, it follows that   ! v so Ni dNe  e ¼ 0: B Q þ b  N de qi   so eqi ¼eqi so Thus, Se  Sso .

Appendix 9.J. General matching function Following Zha et al. (2018), the matching mechanism involves the online matching of waiting passengers and idle drivers and the subsequent pick-up of passengers by their matched drivers. In what follows, we show only the formulas that describe the ride-sourcing market equilibrium without and with platform integration; readers can refer to Zha et al. (2018) for more details about the derivations. l

Market without platform integration

234 Supply and Demand Management in Ride-Sourcing Markets

Let Wim and Wic denote the average pick-up (en route) time and the average waiting time for passengers matched with platform i, respectively. Let Niv and Nic denote the number of idle vehicles and the number of waiting passengers on platform i, respectively. Then, given the length d of each matching interval, the following relations hold in the market without platform integration: X
 qi ¼ Fi þ b T þ Wim þ Wic ; ci; B (9.A17) Wic ¼

d ; ci;  Niv 1 exp  c Ni

Wim ¼


A k ; ci; Niv

Nic ¼ qi Wic ; ci;
 Ni ¼ Niv þ qi T þ Wim ; ci;

(9.A18)

(9.A19) (9.A20) (9.A21)

where A is an exogenous parameter related to the study area and vehicle velocity; and k is a sensitivity parameter and expected to vary in ð0; 1. l

Market with platform integration

In the market with platform integration, the driver supplies of the competing platforms are managed in one matching pool without discrimination. In this case, the average pick-up time and matching time for the pasv v sengers on platform i depend on the aggregate driver supply Ne , i.e., Ne ¼ P v Nei . As aforementioned, if the commission fee charged by the integrator is sufficiently low, all of the passengers seek ride services via the integrator, and thus the following relations hold: ! X  X m c e e e e e i F i þ b T þ W þ W i þ es; ci; qei ¼ B (9.A22) u e ci ¼ W

d 0

1 ; ci; v e Ni C B 1 exp@  c A e Ni em ¼ W

P

A P

v Nei

!k ;

(9.A23)

(9.A24)

Third-party platform integration in ride-sourcing markets Chapter | 9 c e ci ; ci; Nei ¼ qei W

235

(9.A25) !

v e m ; ci; Ni ¼ Nei þ qei Te þ W

(9.A26)

where ‘w’ denotes the counterparts of variables with platform integration. Numerical experiments

l

Consider the following negative exponential demand function: Q ¼ QexpðaCÞ, where a > 0 is a cost sensitivity parameter, C is the generalised travel cost and Q is the potential demand. Table J1 reports the values of the key model parameters used in the numerical studies, which are selected with partial reference to previous studies (Zha et al., 2018; Xu et al., 2017) and the numerical experiments in Section 9.4.1. For illustrative purposes, we assume that the vehicle fleet sizes are equal across all platforms, i.e., Ni ¼ Nj ; ci; j. By increasing the number of platforms from 1 to 10, we demonstrate how the values of the various market measures (e.g., platform profit and social welfare) change with the extent of market fragmentation. Fig. 9.J1 shows the effects of market fragmentation on the total realised demand, trip fare, total profit and social welfare in the context of the more general matching mechanism. By comparing Figs. 9.1 and 9.J1, it can be seen that all of the observations with this more generalised matching are reasonably similar to the observations with FCFS matching, which facilitates our theoretical analysis. This verifies that our analytical results are robust with respect

TABLE 9.J1 Parameters and default values in the numerical studies. Symbol

Parameter

Value

Q

Potential demand

1:0  105 (Trip/h)

a

Cost sensitivity of demand function

0:013 (1/HKD)

b

Value of time

120 (HKD/h)

c

Unit operating cost per vehicle

50 (HKD/h)

T

Average trip time

0:4 (h)

A

Pick-up time function

5 (h)

es

Commission fee charged by the integrator

0 (HKD)

d

Matching step duration

3  103 (h)

k

Supply sensitivity of pick-up time function

0:5

N

Total vehicle fleet size

1:0  104 (Veh)

236 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 9.J1 Effect of the number of platforms on the (A) total realised demand, (B) trip fare, (C) total profit and (D) social welfare in a market with/without platform integration.

to the matching functions. In particular, we find that the online matching process has little effect on the market optima. This is reasonable in most situations of actual operations, wherein the waiting time of passengers is dominated by the pick-up time.

Glossary of notation NE Nash equilibrium S social welfare Q maximum possible demand (i.e., the potential arrival rate of passengers) Q realised passenger demand qi realised demand for platform i b value of time Fi platform i’s trip fare N vi platform i’s number of idle vehicles

Third-party platform integration in ride-sourcing markets Chapter | 9

237


W N vi waiting and pick-up times of passengers opting for platform i N i fleet size in equilibrium for platform i N vi number of idle vehicles in equilibrium for platform i N total fleet size in equilibrium in the market N v total number of idle vehicles in equilibrium in the market T average trip time c average operating cost of a vehicle per time unit Pne I optimal profit of each platform at the NE Qso passenger demand at SO qso i realised demand for platform i at SO uei market share of platform i in the integrator se commission set by the integrator Nev total number of idle vehicles in the integrator N iv number of idle vehicles in equilibrium for platform i in the integrator e Fei platform i’s trip fare in the integrator e q i realised demand for platform i in the integrator Qe realised passenger demand in the integrator Te average trip time in the market with platform integration Ce1 generalised equilibrium trip costs of passengers who make a request via the integrator Ce2 generalised equilibrium trip costs of passengers make a direct request to platform i k index for the platform charging the highest trip fare j ndex for the platform charging the lowest trip fare A Sensitivity of waiting time function

References Arnott, R., 1996. Taxi travel should be subsidized. Journal of Urban Economics 40 (3), 316e333. Castillo, J.C., Knoepfle, D., Weyl, G., 2017. Surge pricing solves the wild goose chase. In: Proceedings of the 2017 ACM Conference on Economics and Computation. ACM, pp. 241e242. Debreu, G., 1952. A social equilibrium existence theorem*. Proceedings of the National Academy of Sciences 38 (10), 886e893. https://doi.org/10.1073/pnas.38.10.886. Ke, J., Yang, H., Li, X., Wang, H., Ye, J., 2020. Pricing and equilibrium in on-demand ride-pooling markets. Transportation Research Part B: Methodological 139, 411e431. Li, S., Tavafoghi, H., Poolla, K., Varaiya, P., 2019. Regulating TNCs: Should Uber and Lyft set their own rules? Transportation Research Part B: Methodological 129, 193e225. Song, J., 2019. Ctrip Integrates Ride-Hailing Service on Baidu Map to Expand Globally. https://krasia.com/ctrips-integrates-its-ride-hailing-service-on-baidu-map-to-expand-globally. Wang, H., Yang, H., 2019. Ride-sourcing systems: a framework and review. Transportation Research Part B: Methodological 129, 122e155. Xu, Z., Yin, Y., Zha, L., 2017. Optimal parking provision for ride-sourcing services. Transportation Research Part B: Methodological 105, 559e578. Zha, L., Yin, Y., Du, Y., 2017. Surge pricing and labor supply in the ride-sourcing market. Transportation Research Procedia 23, 2e21. Zha, L., Yin, Y., Xu, Z., 2018. Geometric matching and spatial pricing in ride-sourcing markets. Transportation Research Part C: Emerging Technologies 92, 58e75. Zha, L., Yin, Y., Yang, H., 2016. Economic analysis of ride-sourcing markets. Transportation Research Part C: Emerging Technologies 71, 249e266. Zhou, Y., Yang, H., Ke, J., Wang, H., Li, X., 2022. Competition and third-party platformintegration in ride-sourcing markets. Transportation Research Part B: Methodological 159, 76e103.

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Chapter 10

Ride-sourcing services and public transit Jintao Ke1, Zhu Zheng2 and Hai Yang3 1

Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China; Department of Civil Engineering, Zhejiang University, Hangzhou, China; 3Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China 2

10.1 Background Urban transportation systems have undergone rapid changes in recent years, due to the increasing popularity of dynamic (or on-demand) ride-sourcing services provided by transportation network companies (TNCs) such as Uber, Lyft and Didi Chuxing. TNCs offer their ride-sourcing services via online platforms or smartphone applications, which are efficient ways to connect passengers and drivers. Consequently, ride-sourcing services are now playing an indispensable role in transportation systems worldwide for serving passengers’ mobility needs, together with other mobility components such as public transit, taxis and private vehicles. It was reported that Uber operated services in 72 countries, and Didi Chuxing provides over 25 million trips every day in 400 cities in China. These emerging ride-sourcing services have transformed the way people move and the patterns of urban mobility. It is therefore important to understand urban transportation systems that include ride-sourcing markets to assist TNCs and governments to design operating strategies that achieve sustainable and efficient urban mobility. Ride-sourcing services have attracted much attention from researchers. Efforts thus far have been primarily focused on theoretical and empirical analyses of the properties of ride-sourcing markets (Zha et al., 2016; Xu et al., 2018; Ke et al., 2019; Yang et al., 2020). However, although the relationship between ride-sourcing and other travel modes plays a key role in the success of urban mobility systems, the complexity of this relationship means that few theoretical frameworks have been devised for characterising the effects of ridesourcing services in a multi-modal transportation system. Some case-specific empirical investigations have been performed (e.g., Li et al., 2016; Hall et al., 2018; Schaller, 2018), but there remains a need for analytical Supply and Demand Management in Ride-Sourcing Markets. https://doi.org/10.1016/B978-0-443-18937-1.00002-4 Copyright © 2023 Elsevier Inc. All rights reserved.

239

240 Supply and Demand Management in Ride-Sourcing Markets

investigations to settle ongoing debates on the nature of the interaction of ridesourcing services with other travel modes. Advocates of ride-sourcing services maintain that such services complement other travel modes. Moreover, efficient matching between drivers and passengers is generally expected to increase vehicle utilisation rates, thereby achieving various socially beneficial objectives such as alleviating traffic congestion and air pollution. However, critics of ride-sourcing services claim that as TNCs provide more convenient and comfortable ride services than other travel modes, TNCs add to vehicle traffic by attracting travellers from space-efficient modes such as walking, public transit and cycling. Accordingly, we analyse the complementary and substitutive relationships between ride-sourcing and public transit services in a multi-modal transportation market. In a multi-modal urban transportation market, ride-sourcing and public transit have distinctive features. On the one hand, ride-sourcing services accommodate on-demand door-to-door travel needs but are generally highcost (i.e., have high fares); on the other hand, public transit services generally have fixed schedules and limited accessibility but are low-cost. These two modes are either substitutive or complementary; passengers can use a ridesourcing service or a combination of ride-sourcing and public transit services to reach their destinations. In the latter case, ride-sourcing provides first-mile and (or last-mile) services, meaning it acts as a feeder to public transit. The demands for different travel modes therefore depend on the competition between and complementarity of the two modes, which in turn depend on factors such as passengers’ total trip distances (from their origin to their destination), the trip fares for the two modes and the first-mile/last-mile distances. Consequently, cooperation between a government, a platform operator and other service providers is essential for achieving a well-balanced multi-modal transportation system. In this chapter, we develop a mathematical model to characterise the intricate relationships between system decision variables (such as ridesourcing fare rates, i.e., the trip fare per unit distance), exogenous variables (such as network topologies and speeds of various modes) and endogenous variables (such as ride-sourcing waiting time) in a multi-modal transportation market that offers ride-sourcing and public transit services. Based on the modelling framework, we analytically and numerically investigate the effects of the decision variables (i.e., ride-sourcing fare rates and vehicle fleet size) on passengers’ mode splits and examine the substitutivity and complementarity between ride-sourcing services and public transit services. Moreover, we examine the optimal strategies for maximising a TNC’s profit in monopoly markets and for maximising social welfare in socially preferable markets. We find that a TNC can identify a Pareto-efficient strategy that satisfies both social welfare and its own profit objectives. The contributions of this work are summarised below

Ride-sourcing services and public transit Chapter | 10

l

l

l

241

We develop a modelling framework to describe a market equilibrium in which passengers choose between direct ride-sourcing services, public transit services and a bundle of first-mile/last-mile ride-sourcing services and public transit services. This differs from most previous studies, e.g., Zha et al. (2016) and Ke et al. (2020), which have only considered one type of ride-sourcing service (i.e., a standard ride-sourcing service or a ridepooling service) and analysed passengers’ choices between a single ridesourcing service and other non-ride-sourcing modes. We investigate the effects of a TNC’s differentiated pricing strategy for direct and first-mile/last-mile ride-sourcing service, and vehicle fleet size, on passengers’ mode choices, the TNC’s profit and social welfare. In particular, we analytically determine the optimal pricing formulas at the monopoly optimum and the social optimum. Our numerical and theoretical analyses offer guidance on how a TNC can design appropriate operating strategies for various circumstances. In addition, we identify a Pareto-efficient frontier comprising the set of second-best solutions that effectively balance the trade-off between a TNC’s profit and social welfare, which can help governments to design efficient regulatory policies.

10.2 Model description Consider a continuous marketplace with several transportation hubs (which offer public transit services) and in which passengers’ origins and destinations are uniformly spatially distributed. The number of transportation hubs is finite, which means that the distances between them and passengers’ origins/destinations are not negligible. Passengers use three travel modes, as follows. l

l

l

Mode b (a bundled mode): a combination of two separate servicesda public transit service and a ride-sourcing servicedwhere the latter solves first-mile/last-mile problems. For example, passengers may take an Uber to their closest transportation hub and then take a bus to their destination. Mode p (a public transit mode): passengers first walk (or cycle) to their closest transportation hub and then use public transit to travel to their destinations. They also use walking or biking to solve first-mile/last-mile problems. Mode r (a direct ride-sourcing mode): passengers use a direct ride-sourcing service to travel from their origins to their destinations.

We use an aggregate model and do not consider the road network, as this model, with its three major travel options, is a reasonable model of the travel behaviour of commuters who do not own private cars. Thus, the model accounts for the first-mile/last-mile problem associated with using public transit, which is not negligible in many cities, as people’s houses are sparsely

242 Supply and Demand Management in Ride-Sourcing Markets

distributed in suburban areas without good accessibility to public transit services. It follows that a key question faced by public transit users is whether to save time but incur a monetary cost by using a ride-sourcing service to travel to/from their closest transportation hub or to spend a long time but avoid a monetary cost by walking/cycling to/from their closest transportation hub. Alternatively, users can use a ride-sourcing service to travel directly from their origin to their destination, which saves much time but is expensive. Clearly, the relationship between ride-sourcing services and public transit services can be substitutive or complementary: a first-mile/last-mile ride-sourcing service complements public transit but a direct ride-sourcing service substitutes for public transit. On the demand side, let q denote the total passenger demand rate (i.e., the number of passengers per unit time); and let qb , qp and qr represent the demand rate for modes b, p, and r, respectively. On the supply side, let N denote the vehicle fleet size of a ride-sourcing platform (i.e., a TNC). Regarding geographical topology, let lom denote the average distance between passengers’ origins/destinations and their nearest transportation hubs, let lmd denote the average distance between the transportation hubs and the nearest origins/ destinations of passengers and let lod denote the average distance from passengers’ origins to their destinations. We use average distances in this work because they generally reflect the average accessibility of various service modes in a multi-modal transportation system. In contrast, heterogeneous service distances would generate complexity in the model formulation without affording additional analytical insights. Let vw , vr and vp indicate the average speeds of walking/biking, ride-sourcing vehicles and public transit, respectively. For simplicity, we do not consider traffic congestion and thus assume that speeds are constant. Let wfr , wdr and wp denote the average waiting times of first-mile/last-mile ride-sourcing services, direct ride-sourcing services and public transit services, respectively. We assume that the waiting time of public transit services is a constant, while the waiting time of first-mile/last-mile (or direct ride-sourcing) services is affected by the availability of ride-sourcing vehicles, i.e., the number of idle vehicles N v in the marketplace. In turn, N v depends on N, qb and qr . Without loss of generality, we assume that passengers have a homogeneous value of time (VOT), denoted by b. Let sb , sr and sp indicate the trip fare rate (i.e., monetary trip fare per unit distance) of firstmile/last-mile ride-sourcing services, direct ride-sourcing services and public transit services, respectively. In summary, lom , lmd , lod , vw , vr , vp , wp , sp , q and b are exogenous variables; qb , qp , qr N v , wfr and wdr are endogenous variables; and sb , sr and N are decision variables determined by the TNC. Given the exogenous variables and decision variables, passengers’ choices between the three modes are given by qb, qp and qr , and affect wfr and wdr . There are several ways to approximate average waiting times. For example, Arnott (1996) and Zha et al. (2017) assumed passengers’ average waiting time

Ride-sourcing services and public transit Chapter | 10

243

to be inversely proportional to the square root of N v , and Bai et al. (2019) approximated passengers’ average waiting time via a queuing model. In this chapter, we assume that the TNC implements a first-come-first-served mechanism, such that a passenger making a request (for either a first-mile/last-mile service or a direct ride-sourcing service) is immediately matched with the closest idle driver. This implies that the TNC does not discriminate between either type of ride-sourcing service in the driverepassenger matching process. In this case, the matching time for both ride-sourcing services (the period beginning when the passenger makes the request and ending when the passenger is matched online with a driver) is negligible, such that the waiting time is dominated by the pick-up time (the period beginning when the passenger is matched online with the driver and ending when the passenger is picked up by the driver) in both first-mile/last-mile services and direct ride-sourcing services. Moreover, due to the indiscrimination in dispatching, the average waiting time of a first-mile/last-mile ride-sourcing service is equal to the average waiting time of a direct ride-sourcing service and given by wfr ¼ wðN v Þ

(10.1)

wdr ¼ wðN v Þ

(10.2) w0

w00

with the properties < 0 and > 0. In a where wð $Þ is a function of stationary equilibrium state, each vehicle will be in one of the following disjoint states: (1) idle and waiting for dispatch; (2) en route to pick up a passenger who has opted for a first-mile/last-mile ride-sourcing service; (3) en route to deliver a passenger who has opted for a first-mile/last-mile ridesourcing service; (4) en route to pick up a passenger who has opted for a direct ride-sourcing service; or (5) en route to deliver a passenger who has opted for direct ride-sourcing service. Therefore, we have the following vehicle conservation equation: Nv

N ¼ N v þ qb wfr þ qb

lom lod þ qr wdr þ qr vr vr

(10.3)

where N v applies to the aforementioned state (1), qb wfr is the number of vehicles in state (2), qb lom =vr is the number of vehicles in state (3), qr wdr is the number of vehicles in state (4), and qr lod =vr is the number of vehicles in state (5). Based on the performance measurements (i.e., waiting times, in-vehicle times, and trip fares), the average fares and trip times for the three modes (i.e., b, p, and r) are given in Table 10.1. As aforementioned, qb , qp and qr are determined by the generalised cost of the three modes. With a homogeneous VOT, the generalised costs of passengers opting for modes b, p, and r, respectively, are given by

244 Supply and Demand Management in Ride-Sourcing Markets

TABLE 10.1 Average trip fare and time cost of three modes. Mode

Average trip fare

Average trip time

B

sb lom þ sp lmd

ðlvomr Þ þ ðlvmdp Þ þ wfr þ wp

P

sp lmd

ðlvomw Þ þ ðlvmdp Þ þ wp

R

sr lod

ðlvodr Þ þ wdr



lom lmd Cb ¼ sb lom þ sp lmd þ b þ þ wfr þ wp vr vp   lom lmd Cp ¼ sp lmd þ b þ þ wp vw vp   lod Cr ¼ sr lod þ b þ wdr vr

 (10.4)

(10.5)

(10.6)

Let Fb ¼ sb lom , which represents the average trip fare spent for the firstmile/last-mile ride-sourcing services; and let Fr ¼ sr lod , which is the average trip fare for the direct ride-sourcing services. Thus, given the arrival rate of all passengers q, the demands for the three modes are given by
(10.7) qb ¼ Lb Cb ; Cr ; Cp
qr ¼ Lr Cb ; Cr ; Cp (10.8)

where Lb Cb ; Cr ; Cp




qp ¼ q  qb  q r

(10.9)
is the demand function for mode b, and Lr Cb ; Cr ; Cp ð1Þ

ð2Þ

ð3Þ

is the demand function for mode r. We assume that Lb < 0, Lb  0, Lb  0, ð1Þ

ð2Þ

ð3Þ

ðiÞ

ðiÞ

Lr  0, Lr < 0, Lb  0, where Lb and Lr are the first-order partial

derivatives of demand function Lb Cb ; Cr ; Cp and Lr Cb ; Cr ; Cp with respect to their ith argument, respectively. This assumption indicates that the demand for a given mode is negatively dependent on the generalised cost for this mode but positively (or not) dependent on the generalised cost for the alternative modes. In summary, the market equilibrium state can be solved by a system of nonlinear simultaneous equations consisting of Eqs. (10.1)e(10.9). The endogenous waiting times affect passengers’ mode choices through the demand functions (Eqs. 10.7e10.9) and the generalised cost functions (Eqs. 10.4e10.6), while the changes in passenger arrival rates affect the waiting times via the waiting time functions and the vehicle conservation equation, (Eqs. 10.1e10.3). The interactions between the waiting times and the demand rates for the three modes

Ride-sourcing services and public transit Chapter | 10

245

constitute the market equilibrium. Given fixed exogenous settings, the TNC can influence this market equilibrium by varying sb and sr (or Fb and Fr ) and N.

10.3 Optimal strategy design In this section, we investigate the optimal strategies (i.e., decision variables) under the equilibrium condition in the aforementioned multi-modal market. The following three market scenarios are considered and examined: l

l

l

The monopoly optimum (MO): this involves the TNC freely choosing a strategy (i.e., a group of decision variables) to maximise its own profit, which is equal to the total trip fares charged for both first-mile/last-mile and direct ride-sourcing services less the total operating cost of its vehicles. The social optimum (SO): this strategy is chosen to maximise the total social welfare, which is defined as the sum of consumer surplus and the TNC’s profit. The SO is also called the first-best (FB) solution; this may not be achievable, because at the SO the TNC’s profit is generally negative. The second-best (SB) solution: this strategy is chosen to maximise the total social welfare, while guaranteeing that the TNC’s profit is higher than a certain level. The SB solution is a trade-off between the MO and SO, as it balances their two distinct objectives to maximise social welfare and the TNC’s profit.

10.3.1 Monopoly optimum Consider a monopoly market in which the TNC aims to maximise its profit by varying the three major decision variables: sr , sb and N. The ride-sourcing platforms (i.e., the TNCs) do not own vehicles and thus offer wages to attract drivers to offer on-demand ride services. However, Zha et al. (2017) showed that a TNC behaves like a taxi platform that directly controls N if the drivers’ reservation cost is homogeneous and the potential driver supply is sufficient. In this case, potential drivers continue entering the ride-sourcing market until their average net earnings decrease to zero. This means that the TNC can control N via the wages it pays to drivers, such that indirectly controlling N is equivalent to directly controlling wages. In this section, for simplicity, we adopt the assumption of Zha et al. (2017), which means that the profit-maximisation problem can be formulated as follows: ðP1Þ max P ¼ qb Fb þ qr Fr  cN Fb ;Fr ;N

(10.10)

s:t: Eqs:ð10:1Þ ð10:3Þ where P is the profit per hour of the TNC, c is the operating cost per unit time of a ride-sourcing vehicle (or driver) and qb and qr are endogenously

246 Supply and Demand Management in Ride-Sourcing Markets

determined by the market equilibrium represented by Eqs. (10.1)e(10.3). In Appendix 10.A., we show that under the driver supply assumption made by Zha et al. (2017), P1 is equivalent to a profit-maximising problem with trip fares and wages as decision variables. Fb (i.e., sb lom ), Fr (i.e., sr lod ), and N are treated as the three decision variables. The first-order conditions of P1 are given by   ð2Þ lom qr Lð1Þ r  qr Lb Fb ¼ c þ wfr þ (10.11) ð1Þ ð2Þ ð1Þ vr Lð2Þ r Lb  Lb Lr .

  ð2Þ lom qr Lð1Þ r  qr Lb þ wfr þ Fb ¼ c ð1Þ ð2Þ vr Lð2Þ L  L Lð1Þ

(10.12)

c ¼ ðqb þ qr Þðb þ cÞw0

(10.13)

r

b

b

r

where Eq. (10.11) represents the pricing formula for the first-mile/last-mile ride-sourcing service and takes the form of the Lerner formula (Lerner, 1934). The right-hand side of Eq. (10.11) consists of two terms: the first term represents the marginal cost of using a vehicle to serve a passenger in the trip phase (clom =vr ) and pick-up phase (cwfr ), while the second term is a monopoly mark-up that represents the market power of the monopoly platform to distort the trip fare from its efficient level. This monopoly mark-up not only depends ð2Þ

on the direct elasticities Lr ð1Þ

ð2Þ

ð1Þ

and Lb but is also affected by the cross elasð1Þ

ð2Þ

ticities Lr and Lb . Specifically, if Lr and Lb are equal to zerodmeaning that the demand for one mode only depends on the generalised cost of this mode and not on the generalised costs of other modesdthe monopoly mark-up  ð1Þ reduces to qb Lb , which is consistent with the MO pricing formula in Ke et al. (2020) for a market with one type of ride-sourcing service. In addition, Eq. (10.12) describes the pricing formula for the direct ridesourcing service and also follows the Lerner formula. The right-hand side of Eq. (10.12) consists of two terms: the first term indicates the marginal cost of using a vehicle to serve a passenger in both the trip phase and pick-up phase, while the second term is a monopoly mark-up. Although this mark-up also ð2Þ

contains Lr

ð1Þ

ð1Þ

and Lb and Lr

ð2Þ

and Lb , it has a different form to the moð1Þ

ð2Þ

nopoly mark-up in Eq. (10.11). In particular, as Lr and Lb are equal to zero,  ð2Þ the monopoly mark-up reduces to qr Lr , which is related only to the demand and elasticity of the demand function of the direct ride-sourcing service.

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10.3.2 Social optimum We now consider an idealised scenario in which a strategy is chosen to maximise social welfare, which is defined as the sum of the consumer surplus and the TNC’s profit. Let VðCb ; Cr Þ be the customers’ surplus from completing a ride-sourcing trip (Zha et al., 2016). Then, under certain regularity conditions, the following properties hold by construction: vV ¼  qr vCr

(10.14)

vV ¼  qb vCb

(10.15)

The social-welfare maximisation problem can then be formulated as ðP2Þ max S ¼ VðCb ; Cr Þ þ qb Fb þ qr Fr  cN Fb ;Fr ;N

(10.16)

s:t: Eqs. ð10:1Þ ð10:9Þ where S is the social welfare, which is equal to the sum of consumer surplus VðCb ; Cr Þ and the TNC’s profit qb Fb þ qr Fr  cN; and Cb , Cr , qb , and qr are endogenously determined by the market equilibrium represented by Eqs. (10.1)e(10.9). The first-order conditions of P2 are thus given by   lom Fb ¼ c þ wfr (10.17) vr   lod þ wdr Fr ¼ c (10.18) vr c ¼ ðqb þ qr Þðb þ cÞw0

(10.19)

It can be shown that Fb , which here represents the SO trip fare for the firstmile/last-mile ride-sourcing service, is equal to the corresponding MO trip fare less the monopoly mark-up. Namely, Fb at the SO is equal to the marginal cost of using one vehicle to serve a passenger in both the pick-up and delivery phases. Similarly, Fr , which here represents the trip fare for a direct ridesourcing service at the SO, is also given by the marginal cost of deploying one vehicle to serve a passenger in pick-up (cwdr ) and delivery (clod =vr ). In addition, Eq. (10.19) takes exactly the same form as Eq. (10.13), which indicates that this property holds at both the MO and SO. At the SO, the TNC’s profit is given by

248 Supply and Demand Management in Ride-Sourcing Markets

 Pso ¼ qb c

lom þ wfr vr

 þ qr c

  lod þ wdr cN vr

¼ cN v < 0ðP2Þ max S ¼ VðCb ; Cr Þ þ qb Fb þ qr Fr  cN

(10.20)

Fb ;Fr ;N

which shows that the TNC’s profit at the SO is always negative. This implies that the SO is unattainable unless the government subsidises the TNC.

10.3.3 Second-best solution As the TNC’s profit may be in deficit at the SO (i.e., the FB), we consider an SB scenario in which a strategy is chosen to maximise S, while ensuring a certain level of profit: ðP3Þ max S ¼ VðCb ; Cr Þ þ qb Fb þ qr Fr  cN

(10.21)

s:t: qb Fb þ qr Fr  cN  P

(10.22)

Fb ;Fr ;N

and Eqs. ð10:1Þ ð10:9Þ where Objective (10.21) maximises S, Constraint (10.22) guarantees that the TNC’s profit is higher than a targeted level P , while Eqs. (10.1)e(10.9) ensure that all of the system’s endogenous variables meet the market equilibrium conditions. To solve this problem, we formulate the following Lagrangian: ðP3:1Þ max L ¼ S þ xðqb Fb þ qr Fr  cN  P Þ Fb ;Fr ;N

(10.23)

s:t: Eqs.ð10:1Þ ð10:9Þ where x is the Lagrange multiplier. The first-order conditions of P3.1 are given by   ð2Þ lom qr Lð1Þ r  q b Lr Fb ¼ c þ wfr þ x (10.24) ð1Þ ð2Þ ð1Þ vr Lð2Þ r Lb  Lb Lr   ð2Þ ð1Þ lod qb Lb  qr Lb þ wdr þ x Fr ¼ c ð1Þ ð2Þ vr Lð2Þ L  L Lð1Þ

(10.25)

c ¼ ðqb þ qr Þðb þ cÞw0

(10.26)

r

b

b

r

Clearly, the pricing formulas for the SB solution, i.e., Eqs. (10.24) and (10.25), are linear combinations of the pricing formulas at the MO, i.e., Eqs. (10.11) and (10.12), and the pricing formulas at the SO, i.e., Eqs. (10.17) and (10.18). In addition, Eq. (10.26) has the same form as Eqs. (10.13) and (10.19). This indicates that Eq. (10.26) holds for all SB solutions at the Pareto-efficient frontier that connects the MO with the SO (Yang and Yang, 2011). As no one

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SB solution can simultaneously increase the TNC’s profit and S at any point along the Pareto-efficient frontier, the Pareto-efficient frontier is generally regarded as an important measure of the performance of a given regulation or policy. A regulation/policy is said to be Pareto-efficient if it can induce a TNCdwhich aims to maximise its own profit in the feasible space constrained by the policy/regulationdto choose a targeted Pareto-efficient strategy (or SB solution).

10.4 Numerical case study The equilibrium-based model developed in Sections 10.2 and 10.3 characterises the behaviour of a transportation market with ride-sourcing and public transit services. Optimal strategies can be implemented by the TNC to affect the equilibrium mode demands and thus reach the MO (P1 in Section 10.3.1), the FB situation (P2 in Section 10.3.2), or the SB situation (P3 or P3.1 in Section 10.3.3). Due to the complexity of the model, the complementary and substitutive relationship between ride-sourcing and public transit cannot be fully delineated via analytical derivations. Accordingly, in this section, numerical case studies are conducted to provide an intuitive sense of the equilibrium states, and profit and/or S maximisation, in the multi-modal market. Without loss of generality, we use the logit model as the demand function (Fisk, 1980), such that expð kCb Þ
expðkCb Þ þ expðkCr Þ þ exp  kCp
exp  kCp
qp ¼ q expðkCb Þ þ expðkCr Þ þ exp  kCp

qb ¼ q

qr ¼ q

expð kCr Þ
expðkCb Þ þ expðkCr Þ þ exp  kCp

(10.27)

(10.28)

(10.29)

where k denotes the sensitivity coefficient of passengers’ generalised costs. Based on this logit model, the consumers’ surplus is given by the following logarithm-sum formula:
 p  VðCb ; Cr Þ ¼ log expðkCb Þ þ expðkCr Þ þ exp  kCp þ V0 k

(10.30)

which satisfies Eqs. (10.14) and (10.15). Moreover, the waiting time for ride-sourcing services is estimated by the following formula, which we used in a previous study (Ke et al., 2020): wfr ¼ wdr ¼

A N va

(10.31)

250 Supply and Demand Management in Ride-Sourcing Markets

where A is a coefficient depending on the area and network structures of an examined city, vehicular speed and other aspects, while a is a dimensionless parameter representing the returns to scale. Normally, a is set to 0.5, indicating that the average waiting time is inversely proportional to the square root of N v . We consider multi-modal travel scenarios in Hong Kong with the following exogenous parameters: vw ¼ 5:6 km/h, vr ¼ 50:0 km/h, vp ¼ 45:0 km/h, sp ¼ 1:0 HKD/km, b ¼ 80 HKD/h, c ¼ 50 HKD/vehicle$h, q ¼ 3; 000 passengers/ h, A ¼ 2:0 h, a ¼ 0:5, wp ¼ 0:10 h, k ¼ 0:1 and V0 ¼ 200 k HKD/h. We let lmd ¼ 10:0 km and lod ¼ 10:0 km and then examine the following three scenarios, which have various first-mile/last-mile distances (1) lom ¼ 1:0 km, which represents a short (i.e., walking/cycling-accessible): first-mile/last-mile scenario; (2) lom ¼ 1:5 km, which represents a scenario with a medium-length first-mile/last-mile distance; and (3) lom ¼ 2:0 km, which represents a long first-mile/last-mile distance scenario. Although numerical experiments in a single study could be biased with respect to all possible real-world scenarios, the aforementioned numerical setting is representative of many large urban areas. That is, the first-mile/last-mile problem in reality is the ‘first-severalmile’ or ‘last-several-mile’ problem (Chen and Wang, 2018). First, we discuss the changes in the equilibrium conditions with the TNC’s decision variables (i.e., sb , sr , and N). Second, we analyse the TNC’s strategies for profit and/or S maximisation (i.e., P1, P2, and P3).

10.4.1 Analysis of equilibrium states The numerical results below illustrate the equilibrium states for scenario (2). We allow the decision variables to vary within a large range to comprehensively examine the effect on passengers’ choices. Fig. 10.1 depicts qb under equilibrium states with various sb . In the subfigures, the horizontal and vertical axes denote the variation in sr and N, respectively; the solid black contour lines represent the variation in qb ; and the dashed red contour lines represent N v according to the equilibrium state. Under each sb , N v monotonously increases with N. In addition, if sr is too low or too high, N v is insensitive to sr ; while if sr is within some medium interval, N v monotonously increases with sr (see Fig. 10.1A where sr is between w1:0 HKD/km and 4:0 HKD/km). We denote such a direct ride-sourcing fare rate interval as the idle vehicle shifting interval (IVSI). The variation in qb is significantly affected by the IVSI. First, when sr is lower than the IVSI, there is only a low qb ; this is because the fare for direct ride-sourcing services is low and mode b becomes unattractive to passengers. Second, if sr is within the IVSI, qb rapidly increases with sr and becomes less sensitive to N. That is, within the IVSI, the fare rates of ride-sourcing services play more important roles than N in affecting passengers’ choices. Third, when sr is above the IVSI, qb monotonously increases with N but is insensitive to sr . This variation in qb with N at a high sr (i.e., above the IVSI) reflects the competition between

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FIGURE 10.1 Equilibrium demand rate for bundled mode qb ; (A) sb ¼ 0:0 HKD/km; (B) sb ¼ 4:0 HKD/km; (C) sb ¼ 8:0 HKD/km; (D)sb ¼ 12:0 HKD=km.

mode b and mode p: as N increases, the waiting time of ride-sourcing services decreases and passengers is more inclined to use ride-sourcing services than other services for solving their first-mile/last-mile problems. In addition, the lower/upper boundaries of the IVSI move higher as sb increase (see Fig. 10.1AeD), which is intuitive. Similar to Fig. 10.1, Fig. 10.2 shows the variation in qp with various sb . The solid black contour lines denote qp , and the dashed red contour lines represent N v . When both N and sr are small, qp decreases with sr . This is because passengers’ preferences between ride-sourcing services and public transit are dominated by N v, as it determines the ride-sourcing waiting time. Moreover, and according to the equilibrium condition, an increase in sr will lead to a larger N v, qb (see Fig. 10.1) and qr (see Fig. 10.3). With a low sr, qp decreases markedly as N increases from w1100 to 1200 (in Fig. 10.2AeD), which is due to the competitive demand pattern. That is, if the generalised cost of mode r is substantially greater than that of the other two modes, a small change in passengers’ waiting time (due to a change in N) significantly affects the demand rates of the existing modes (i.e., modes b and p in this case),

252 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 10.2 Equilibrium demand rate for public transit mode qp ; (A) sb ¼ 0:0 HKD/km; (B) sb ¼ 4:0 HKD/km; (C) sb ¼ 8:0 HKD/km; (D) sb ¼ 12:0 HKD/km.

leading to a marked demand-shift. Fig. 10.2D shows that when both sb and sr are high, many passengers use mode p, due to its low trip fare. Finally, we use Fig. 10.3 to illustrate the variation in qr with sb . Similar to Figs. 10.1 and 10.2, the solid black and dashed red contour lines represent the variation in qr and N v , respectively. Corresponding to the findings in Fig. 10.2, when sr is within the IVSI, qr is significantly sensitive to (i.e., monotonously decreases with) sr ; when sr is below the IVSI, qr is determined by (i.e., monotonously increases with) N; and when sr is above the IVSI, qr is low. Analogous to the decrease in qp depicted in Figs. 10.1 and 10.2, the rapid demand shift between modes b and r with respect to sr within the IVSI is caused by the competitive demand pattern. Based on the results of the medium first-mile/last-mile distance scenario in Figs. 10.1, 10.2, and 10.3, we arrive at the following conclusions. First, if there are sufficient ride-sourcing vehicles (i.e., N is high) in the market, the demand for mode p is low. Thus, the fare rates of ride-sourcing services play a decisive

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FIGURE 10.3 Equilibrium demand rate for direct ride-sourcing mode qb ; (A) sb ¼ 0:0 HKD/km; (B) sb ¼ 4:0 HKD/km; (C) sb ¼ 8:0 HKD/km; (D) sb ¼ 12:0 HKD/km.

role in the competition between mode r and mode b. With a fixed first-mile/ last-mile ride-sourcing fare rate, there exists a range of direct service fare rates (i.e., the IVSI) that cause passengers’ choice between mode b and mode r to be significantly sensitive to the direct service fare rate but insensitive to N. Second, mode p absorbs many passengers when N is too low. In this manner, if the direct ride-sourcing fare rate is low, the demand for mode p is determined by both fare rates and N (i.e., it decreases with direct ride-sourcing fare rate and N); conversely, it is determined by N if the direct ride-sourcing fare rate is high. These findings indicate that N has a critical effect on the substitutive and complementary relationship between public transit and ride-sourcing services, and that fare rates affect the market share between first-mile/last-mile and direct ride-sourcing services. Accordingly, governments or decision-makers could develop regulatory schemes to promote any of these three modes according to their potential social benefits.

254 Supply and Demand Management in Ride-Sourcing Markets

10.4.2 Analysis of profit- and/or social welfare-maximising strategies To better understand the optimal strategies of the TNC under various optimisation scenarios, we numerically solve the aforementioned problems, P1, P2 and P3 (P3.1), for scenarios (1), (2) and (3). First, we illustrate the effect of the decision variables on the objective values of P1 and P2. Fig. 10.4AeC depicts the numerical results for scenarios (1), (2) and (3), respectively. The subfigures are drawn in the domain of sr  N. For each pair of ðsr ; NÞ, we let sb vary within a large range and numerically obtain the corresponding maximum S and maximum P. The solid blue contour lines and dashed red contour lines represent the obtained S and P (measured in k HKD/h), respectively. As the obtained S and P for a fixed ðsr ; NÞ pair can be regarded as the local optimal objective values of P1 and P2, the global optimal (i.e., maximum) S and P are found in the two-dimensional figure and are denoted by SO and MO, respectively. For each pair of ðsr ; NÞ, the

FIGURE 10.4 Profit and social welfare of the transportation network company (TNC); (A) MO and SO with lom ¼ 1:0 km; (B) MO and SO with lom ¼ 1:5 km; (C) MO and SO with lom ¼ 2:0 km; (D) SB solution with variouslom .

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corresponding S and P could occur under different values of sb , due to the distinctive first-order conditions of P1 and P2 (see Sections 10.3.1 and 10.3.2). In scenario (1), the optimal solution of P1 (i.e., the MO) is at sb ¼ 20:6 HKD/km, sr ¼ 3:2, and N ¼ 496, which gives a P of 5:1 k HKD/h; and the optimal solution of P2 (i.e., the SO) is at sb ¼ 7:1 HKD/km, sr ¼ 1:6 and N ¼ 942, which gives an S of 77:9 k HKD/h (see Fig. 10.4A). When the first-mile/ last-mile distance is short, passengers are inclined to choose mode p, making ride-sourcing services uncompetitive. Thus, to maintain the market share of ride-sourcing services, the TNC may have to increase N or decrease sr , where the former strategy increases operational costs and the latter strategy decreases net revenue. As a result, we note that the TNC’s profit is negative for most of the ðsr ; NÞ pairs (see the contour lines in Fig. 10.4A). In scenario (2), the MO is at sb ¼ 17:5 HKD/km, sr ¼ 3:4, and N ¼ 659, where P ¼ 13:2 k HKD/h; and the SO is at sb ¼ 4:6 HKD/km, sr ¼ 1:6 and N ¼ 1; 066, where S ¼ 73:6 k HKD/h (see Fig. 10.4B). In scenario (3), the MO is at sb ¼ 14:8 HKD/km, sr ¼ 3:8 and N ¼ 768, where P ¼ 23:7 k HKD/h; and the SO is at sb ¼ 3:9 HKD/km, sr ¼ 1:6 and N ¼ 1; 142, where S ¼ 71:0 k HKD/h (see Fig. 10.4C). These results indicate that as the first-mile/last-mile distance increases, the maximum P of the TNC increases, as mode p loses passengers due to the notable increase in the walking/biking time from origins to the nearest transportation hubs (or from the hubs to destinations). Fig. 10.4AeC shows that for a fixed lom , the ride-sourcing fare rates at the MO are always higher than those at the SO, while N at the MO is lower than at the SO. These results are consistent with the analytical derivations in Sections 10.3.1 and 10.3.2. This is intuitive, as when pursuing P, the TNC needs higher fare rates to ensure high revenue and a small N to reduce its operational costs. Another interesting finding is that when the first-mile/last-mile distance shifts from short to long, the S with a low direct ride-sourcing fare rate and a small N (i.e., the left-bottom corner in Fig. 10.4AeC) substantially decreases, from a high positive level to a negative level. Second, as the decision variables and the objective values (S and P) at the SO are notably different from those at the MO, we illustrate the SB solution for problem P3 (P3.1) and examine whether there is a Pareto-efficient condition for both S and P. Pareto-efficiency is an economic state in which resources cannot be reallocated to make one stakeholder better off without making the other stakeholder worse off (Yang and Yang, 2011; Ke et al., 2020). In Fig. 10.4D, we illustrate the SB conditions by plotting the curves of P vs its corresponding S (i.e., solution to P3). As the curves with different first-mile/last-mile distances have a similar pattern, we take scenario (3) as an example. Compared with the MO, at which the maximum P is 23:7 k HKD/h and S is 55:7 k HKD/h, a slight reduction of P in the SB condition can significantly improve S. That is, we can regulate the target P to be w20:0 k HKD/h and maintain a high S (w65:0 k HKD). Such a target P is near the MO and the corresponding S is near the SO condition (i.e., 71:0 k HKD),

256 Supply and Demand Management in Ride-Sourcing Markets

which indicates that the TNC can find a Pareto-efficient strategy that satisfies both S and its need for a good P.

10.5 Conclusion This study investigates a multi-modal transportation market in which passengers can opt to take one of three modes of transport: mode r, mode p, or mode b, where the ride-sourcing services in these modes both complement (by providing first-mile/last-mile service) and compete with (by providing direct service) public transit services in the modes. Under this framework, we examine the effects of operating strategies, i.e., the fare rates for direct and first-mile/last-mile ride-sourcing services, and N, on passengers’ mode choices. We also investigate the optimisation problems for maximising the TNC’s P at the MO of a market and for maximising S at the SO (i.e., finding the FB) of an SB market. The findings of our theoretical and numerical studies offer valuable insights for TNCs and governments. They show that governments can set an optimal trip fare equal to the marginal cost of using a vehicle to serve a passenger in both the in-trip and pick-up phases at the SO. In contrast, a TNC can set an optimal trip fare equal to the SO trip fare plus a monopoly mark-up, ð2Þ

ð1Þ

ð1Þ

ð2Þ

which depends on both Lr and Lb , and Lr and Lb . Moreover, we find that N critically affects the complementary and substitutive relationship between ride-sourcing services and public transit, and the fare rates of direct and first-mile/last-mile ride-sourcing services affect the market share between the two types of ride-sourcing services. First, if N is high, passengers are inclined to use modes with ride-sourcing services. As a result, public transit ridership is affected by the competition between mode r and mode b. Passengers’ choice between these two modes is determined by the differentiated ride-sourcing fare rates. Second, mode p absorbs a large number of passengers if N is too low. The sensitivity of such a public transit mode is notably affected by the direct ride-sourcing fare rate: if this fare rate is low, the mode demand will decrease with this rate and N; while if this rate is high, the mode demand is affected by N. For optimisation problems, we find that the maximum P increases with the first-mile/last-mile distance; with a fixed first-mile/last-mile distance, the ridesourcing fare rates at the MO are always higher than those at the SO, while N at the MO is lower than that at the SO. Based on the results of the SB markets, we note that the TNC can find a Pareto-efficient operating strategy that generates an acceptable S and P. This study opens up some new avenues for future work: (1) modelling multi-modal transportation markets with heterogeneous values of time for passengers; (2) incorporating the effect of traffic congestion on passengers’ mode choices in a multi-modal transportation system; (3) studying the effects of ride-splitting services (a special ride-sourcing service that allows one

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vehicle to service two or more riders at once) on transit usage and traffic congestion; (4) investigating the equilibrium games between ride-sourcing platforms and public transit operators, each of which aims to maximise its own objectives; (5) ascertaining cooperative operational strategies between a platform and public TNCs, such as integrated trip fares and coordinated public transit schedules, to maximise the total P or S; and (6) calibrating and validating the modelling results with actual multi-modal mobility data, and understanding passengers’ multi-modal choice behaviour via statistical or deep learning approaches. This chapter is based on one of our recent articles (Ke et al., 2021).

Glossary of notation S social welfare q total passenger demand rate (i.e., the number of passengers per unit time) qb demand rate for modes b qp demand rate for modes p qr demand rate for modes r N vehicle fleet size of a ride-sourcing platform (i.e., a TNC) lom average distance between passengers’ origins/destinations and their nearest transportation hubs lmd average distance between the transportation hubs and the nearest origins/destinations of passengers lod average distance from passengers’ origins to their destinations vw average speeds of walking/biking vr average speeds of ride-sourcing vehicles vp average speeds of public transit wfr average waiting times of first-mile/last-mile ride-sourcing services wdr average waiting times of direct ride-sourcing services N v number of idle vehicles wp average waiting times of public transit services b passengers have a homogeneous value of time Lb Cb ; Cr ; Cp demand function for mode b Lr Cb ; Cr ; Cp demand function for mode r P profit per hour of the TNC c operating cost per unit time of a ride-sourcing vehicle (or driver) A a coefficient depending on the area and network structures of an examined city, vehicular speed, and other aspects k sensitivity coefficient of passengers’ generalised costs

References Arnott, R., 1996. Taxi travel should be subsidized. Journal of Urban Economics 40 (3), 316e333. Bai, J., So, K.C., Tang, C.S., Chen, X., Wang, H., 2019. Coordinating supply and demand on an ondemand service platform with impatient customers. Manufacturing & Service Operations Management 21 (3), 556e570.

258 Supply and Demand Management in Ride-Sourcing Markets Chen, Y., Wang, H., 2018. Pricing for a last-mile transportation system. Transportation Research Part B: Methodological 107, 57e69. Fisk, C., 1980. Some developments in equilibrium traffic assignment. Transportation Research Part B: Methodological 14 (3), 243e255. Hall, J.D., Palsson, C., Price, J., 2018. Is Uber a substitute or complement for public transit? Journal of Urban Economics 108, 36e50. Ke, J., Cen, X., Yang, H., Chen, X., Ye, J., 2019. Modelling drivers’ working and recharging schedules in a ride-sourcing market with electric vehicles and gasoline vehicles. Transportation Research Part E: Logistics and Transportation Review 125, 160e180. Ke, J., Yang, H., Li, X., Wang, H., Ye, J., 2020. Pricing and equilibrium in on-demand ride-pooling markets. Transportation Research Part B: Methodological 139, 411e431. Ke, J., Zhu, Z., Yang, H., He, Q., 2021. Equilibrium analyses and operational designs of a coupled market with substitutive and complementary ride-sourcing services to public transits. Transportation Research Part E: Logistics and Transportation Review 148, 102236. Lerner, A.P., 1934. The concept of monopoly and the measurement of monopoly power. The Review of Economic Studies 1, 157e175. Li, Z., Hong, Y., Zhang, Z., 2016. Do ridesharing services affect traffic congestion? An empirical study of Uber entry. Social Science Research Network 2002, 1e29. Schaller Consulting, 2018. The New Automobility: Lyft, Uber and the Future of American Cities. http://www.schallerconsult.com/rideservices/automobility.pdf. Xu, Z., Li, Z., Guan, Q., Zhang, D., Li, Q., Nan, J., Ye, J., 2018. Large-scale order dispatch in ondemand ride-hailing platforms: a learning and planning approach. In: Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. ACM, pp. 905e913. Yang, H., Yang, T., 2011. Equilibrium properties of taxi markets with search frictions. Transportation Research Part B: Methodological 45 (4), 696e713. Yang, H., Qin, X., Ke, J., Ye, J., 2020. Optimizing matching time interval and matching radius in on-demand ride-sourcing markets. Transportation Research Part B: Methodological 131, 84e105. Zha, L., Yin, Y., Du, Y., 2017. Surge pricing and labor supply in the ride-sourcing market. Transportation Research Procedia 23, 2e21. Zha, L., Yin, Y., Yang, H., 2016. Economic analysis of ride-sourcing markets. Transportation Research Part C: Emerging Technologies 71, 249e266.

Appendix 10.A In this appendix, under the assumption made by Zha et al. (2017), we demonstrate that a ride-sourcing platform that controls trip fares and wages behaves like a taxi platform that controls trip fares and N. Suppose the drivers’ wages per km for first-mile and direct ride-sourcing services are sb and sr , respectively. Then, the P-maximisation problem of the ride-sourcing platform can be formulated as ðPA1Þ max P ¼ qb ðsb  sb Þlom þ qr ðsr  sr Þlod sb ;sr ;sb ;sr

(A.1)

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s:t: Eqs. ð10:1Þ ð10:9Þ qb sb lom þ qr sr lod c ¼ 0 N

(A.2)

where qb sb lom þ qr sr lod represents the total earnings of N drivers per unit time, and c is the operating cost per unit time of one ride-sourcing vehicle (or driver). Eq. (A.2) indicates that the net earnings of drivers reach zero at equilibrium. In addition, another driver supply model can be adopted; for example, by assuming that drivers’ reservation costs are distributed over a range, N is given by an increasing function of drivers’ average net earnings ðqb sb lom þqr sr lod Þ=N  c. In addition, the elasticity of drivers’ willingness to provide ride services with respect to their net earnings can also be examined and incorporated into the supply function. For analytical tractability, we adopt the assumption of Zha et al. (2017) and use P1 to obtain the MO solutions. By substituting Eq. (10.29) into Eq. (10.28), we can easily prove that problem PA1 is equivalent to problem P1. Similarly, under the assumption made by Zha et al. (2017), we can prove that problems P2 and P3 are equivalent to an S-maximisation problem and an SBmaximisation problem that regard trip fares and drivers’ wages as decision variables.

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Chapter 11

Optimization of matching-time interval and matching radius in ride-sourcing markets Jintao Ke1 and Hai Yang2 1

Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China; Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China 2

11.1 Research problem Passengers and drivers are generally matched using one of two methods: instant matching or batch matching. The former immediately matches a passenger with the closest idle driver upon submission of the passenger’s order, whereas the latter waits for a certain time interval to accumulate driverepassenger pairs for superior matching, and then executes either greedy matching with the shortest pick-up time or optimal bipartite-graph matching with one or more objectives. An appropriate average matching time interval (Dt) is essential for optimal matching: if Dt is too long, passengers may become impatient, whereas if Dt is too short, a platform cannot collect sufficient driverepassenger pairs in the matching pool; consequently, the expected pick-up distance is lengthened. As shown in Fig. 11.1, as Dt increases (from Dt1 to Dt2 ), the numbers of idle drivers (N v ) and waiting passengers (N c ) increase. This situation allows for the matching of waiting passengers and idle drivers over short mean pick-up distances. However, the average passengers’ waiting time (w) and drivers’ idle time (wv) may correspondingly increase, and thus a matching radius is usually imposed in most matching algorithms to prevent undesirably long-distance matching between drivers and passengers. As Fig. 11.2 illustrates, a passenger can be matched with a nearby driver if the driver is within a circle centred at the passenger that has a radius equal to R. As R increases (from R1 to R2 ), a passenger is increasingly likely to be matched with a driver. However, the average matching or pick-up distance also increases. The abovementioned trade-offs associated with Dt and R also depend on these variables’ joint effects. For example, a long average pickup distance Lp due to a large R can be offset by a prolonged Dt with increased N v and N c . Supply and Demand Management in Ride-Sourcing Markets. https://doi.org/10.1016/B978-0-443-18937-1.00009-7 Copyright © 2023 Elsevier Inc. All rights reserved.

261

262 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 11.1

Effect of matching-time interval ðDtÞ.

FIGURE 11.2 Effect of matching radius (R).

Moreover, the trade-offs between system performance and the selection of Dt and R are critically dependent upon real-time supplyedemand conditions, such as the arrival rates of passengers and idle drivers, and N v and N c remaining from a previous interval. For example, when a large N v remains from a previous interval, any new waiting passenger can be easily matched with a nearby idle vehicle. Thus, a platform should minimise Dt to reduce w. Therefore, examining the effects of Dt and R and optimising these two key decision variables is particularly important for maximising system efficiency under various traffic conditions.

11.2 Modelling and optimising the matching process This section establishes a spatial probability model for describing the matching of idle drivers with waiting passengers in a ride-sourcing system.

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Subsequently, the effects of matching strategies on system performance are discussed in terms of the effects of Dt and R.

11.2.1 Online matching process We analyse the effects of two key decision variables (Dt and R) on system performance in an online matching process, rather than design an efficient matching algorithm. For analytical tractability, certain simplified assumptions are made regarding passengers’ and drivers’ arrival processes. The online matching process of a ride-sourcing platform is illustrated below. Passengers and idle drivers arrive at the matching pool continuously over time at different rates, and the platform makes sequential decisions regarding Dt and R. Each waiting passenger is matched with only one idle driver in the matching pool; RS scenarios (two or more passengers sharing a ride) are not considered. At the beginning t of a time interval, the platform first observes the numbers of waiting passengers (Noc ) and idle drivers (Nov ) remaining from the previous interval. Then, the platform checks the predicted arrival rates of new q v ); these rates can be determined using passengers (b q c ) and idle drivers (b various machine-learning approaches and real-time Q data (Ke et al., 2017, 2019b; Tong et al., 2017). On the basis of Noc , Nov , qbc , and qbv , the platform sets Dt and R to maximise the overall system performance, which is measured by the weighted sum of multiple objectives of interest, such as the matching rate and w. At the end of Dt, the platform matches waiting passengers and idle drivers within R. The ex-post arrival rates of passengers and idle drivers during Dt may differ from the ex-ante predictions. Furthermore, as certain passengers may become impatient and leave the matching pool as Dt increases, we incorporate passengers’ abandonment behaviours into our model by assuming that the proportion of passengers who exit the matching queue increases with w. The proportion remaining in the matching pool at the end of Dt is denoted by SðDtÞ and decreases with Dt, where SðDtÞε½0; 1Þ and decreases with Dt. SðDtÞ can be inferred from the distribution of passengers’ willingness to wait, and the actual w of individual passengers who join the queue at different times or time intervals are recorded by the platform. c v Given these considerations, we can estimate Nb and Nb using the following equations:   c (11.1) Nb ¼ qbc $ Dt þ Noc $SðDtÞ v Nb ¼ qbv $Dt þ Nov

(11.2)

where qbc $Dt and qbv $Dt are the predicted numbers of new passengers (demand) and idle drivers (supply) during Dt, respectively.

264 Supply and Demand Management in Ride-Sourcing Markets

11.2.2 Matched passengeredriver pairs For simplicity and analytical tractability, we consider a simplified matching c v procedure. Given Nb and Nb , the platform loops through each waiting passenger and finds the closest idle driver. If the distance between a passenger and the nearest driver is within R, the passenger and the driver are matched. The model for delineating the matching process should satisfy the conditions outlined below. Sequence dependence does not affect the matching outcomes. As illustrated in Fig. 2.1A, consider Dt with two passengers (A and B) and one vehicle (C) in the matching pool. If the platform first matches A and then B, then A is matched with C, and B is unmatched. However, the pick-up distance between B and C is less than that between A and C. Thus, the A-to-B matching sequence results in less efficient system performance than the opposite matching sequence (B-to-A). To avoid these conflicts of sequence dependence, Xu et al. (2017) proposed the use of a dominant zone and a few criteria for matching every mutually closest driverepassenger pair at each time step. The dominant zone is the neighbouring area of each passenger, within which the distance from any point to the passenger is shorter than that from the point to any other waiting passenger (Fig. 2.1B). During each time interval, a waiting passenger is matched with an idle driver if and only if the following conditions are satisfied. First, the idle driver must lie within the dominant zone of the waiting passenger. Second, the idle driver must be the closest idle driver to the waiting passenger. Third, the distance from the idle driver to the waiting passenger must be within R. We assume that waiting passengers are uniformly distributed over the examined space, whereas the spatial distribution of idle drivers follows a spatial Poisson point process (Chiu et al., 2013). This assumption has been used by others (Arnott, 1996; Xu et al., 2017). We denote the area of the studied space as A, such that the estimated densities of demand and supply (b rc v and b r , respectively) can be expressed as follows: b rc ¼

c Nb A

(11.3)

b rv ¼

v Nb A

(11.4)

 c The area of each passenger’s dominant zone equals ðb r c Þ1 ¼ A Nb . Thus, as each waiting passenger can be matched only with an idle driver who is within R, we further assume that the shape of the dominant zone of each qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  passenger can be approximated by a circle of radius ðb r c Þ1 p. It follows that the matching area (AM ), which represents the area surrounding a waiting

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265

passenger within which the passenger can be matched with an idle driver, is given by  c 1  AM ¼ min ðb r Þ ; pR2 (11.5) c where pR2 is the area of the circle centred on each waiting passenger. Nb c should be a positive integer, as the extreme case of Nb ¼ 0 is not considered.  c Thus, AM  ðb r c Þ1 ¼ A Nb  A, which implies that AM does not exceed A. Clearly, the mean of the number of idle drivers within each v r v ¼ Nb $AM =A. Therefore, the probability that n idle AM is equal to AM $b drivers are within AM of each waiting passenger can be written as follows:

Pfng ¼

1 expðAM b r v ÞðAM b r v Þn n!

(11.6)

The probability of each passenger being matched equals the probability of having at least one idle driver within each passenger’s AM , which is given as follows: rvÞ PM ¼ 1 Pf0g ¼ 1 expðAM b

(11.7)

b Therefore, the expected number of matched passengeredriver pairs ( M) equals    v  c 1 b ¼ Nbc $PM ¼ Nbc $ 1 exp b (11.8) r min ðb r Þ ; pR2 M

11.2.3 Expected pick-up distance The expected pick-up distance is an important metric for evaluating system performance. A short expected pick-up distance reduces time wastage for passengers and drivers and may prevent certain passengers from abandoning their requests (which they do if the pick-up time is long, due a long pick-up distance). Let Tb denote the expected pick-up time, which can be derived on the basis of the matching process presented in Section 3.2. The distance from each waiting passenger to the closest driver, which is denoted x, is described by a cumulative distribution function FðxÞ and a probability density function f ðxÞ. If the distribution of idle drivers follows a spatial Poisson distribution, FðxÞ and f ðxÞ are given by the following equations:   FðxÞ ¼ 1 Pf0g ¼ 1 exp  px2 b rv (11.9)   rv f ðxÞ ¼ 2pxb r v exp  px2 b (11.10) As AM is approximated as the minimum of two circles, it is always a circle pffiffiffiffiffiffiffiffiffiffiffiffi of radius r ¼ AM =p. The expected pick-up distance, which is denoted RbE ,

266 Supply and Demand Management in Ride-Sourcing Markets

is thus approximated by

RbE ymFðrÞ1

Z

r

x$f ðxÞdx

0

pffiffiffiffiffiffiffiffiv   m 1  pffiffiffiffiffiv erf ð pb rv r $rÞ r$exp  pr 2 b 2 v r 1 expð pr b r 2 b 

¼

pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi   pffiffiffiffiffiffiffiffiffiffiffiffi m 1 AM ffiffiffiffiffi p pb rv $ erf  AM =p $expðb ¼ r v AM Þ v v 1 expðb r AM Þ 2 b p r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  c 1 ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  c 1  erf  b r v min ðb r Þ ; pR2  c 1   v min ðb r Þ ; pR2 pffiffiffiffiffiv  m r Þ ; pR2 $exp b r min ðb p r 2 b ¼  v  c 1  1 exp b r min ðb r Þ ; pR2 (11.11)

where erf ðxÞ ¼ p2ffiffipffi

Rx 0

et dt is a Gaussian error function; and m is a detour 2

ratio, which is the ratio of the actual road distance to the Euclidean distance. Yang et al. (2018) found that the distribution of detour ratios can be characterised by a universal horn-shaped distribution law. The mean of the detour ratios should be inversely proportional to the Euclidean distance with an intercept, and this mean approaches a constant of approximately 1.27 if the Euclidean distance is sufficiently long. This constant has been empirically and theoretically identified (e.g., Arnott, 1996; Boscoe et al., 2012; Fairthorne et al., 1963). Under the assumption that the average speed of vehicles (v) is a constant, Tb is given as follows: RbE Tb ¼ v

(11.12)

11.2.4 System performance measure In general, ride-sourcing platforms consider multiple objectives when adjusting Dt and R. Their first objective is to minimise the total pick-up and w costs of passengers, as this objective aligns with enhancing passengers’ satisfaction and thus reduces order abandonment. Their second objective is to b (the number of successfully matched driverepassenger pairs per maximise M unit time), which is in accordance with maximising the platform’s revenue and

Optimization of matching-time interval and matching radius Chapter | 11

267

reducing the order abandonment rate. P to evaluate the overall system performance in a given Dt.    1 b 2 Tb$ M b Dt$ Noc þ Nov  1Dtðb q c $Dt þ qbv $DtÞ P¼ a$ M Dt 2 ¼

b    1 c M q þ qbv Þ$Dt a  2 Tb  Noc þ Nov  ðb Dt 2

(11.13)

where a is a positive weighting factor that measures the benefit of successfully b represents the total pick-up time cost of the matching one passenger; 2 Tb$ M   matched passengers and drivers; a 2 Tb represents the net benefit generated   by one matched pair; Dt$ Noc þNov represents the total w cost of all waiting passengers and idle drivers who remain from the previous time interval; 1Dtðb q c $Dt þb q v $DtÞ represents the w of all new passengers and idle drivers, 2 assuming their arrival is uniform during Dt; and P is the net benefit (benefit minus cost) in a unit time interval (total net benefit divided by Dt). b and Tb are expressed as functions of Dt and R. For As derived above, M b ¼ MðDt; b simplicity, let Tb ¼ TbðDt; RÞ and M RÞ. Then, the platform optimisation problem can be written as follows: maxPðDt; RÞ ¼

b    1 c M a  2 Tb  Noc þ Nov  ðb q þ qbv Þ$Dt Dt 2

(11.14)

subject to: Tb ¼ TbðDt; RÞ

(11.15)

b ¼ MðDt; b M RÞ

(11.16)

The challenge is therefore to strike a balance between the two objectives by choosing an appropriate matching strategy. Extending Dt increases N v and N c in the matching pool, thereby further reducing the mean pick-up time. However, this directly increases w. Similarly, enlarging R can increase the number of successfully matched driverepassenger pairs but may increase Tb.

11.2.5 General model properties In this section, we explore the general properties of the formulated model. These properties are applicable to a balanced scenario (where demand and supply are minimally different) and an imbalanced scenario (where demand and supply are substantially different).

268 Supply and Demand Management in Ride-Sourcing Markets

11.2.5.1 Effect of matching radius

 b Dt and Tb. Evidently, if First, we analyse the effects of R on M qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  r c Þ1. Thus, in this case R  ðb r c Þ1 p, AM is determined by AM ¼ ðb  b Dt and Tb are independent of R, such that a sufficiently large R does not M qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  affect system performance. However, if R < ðb r c Þ1 p and AM ¼ pR2 , the  b Dt with respect to R is partial derivative of M  b MðDt; RÞ v   2 d v Dt ¼ Nb b r v pR2 > 0 r pRexp  b (11.17) vR Dt  b Dt monotonically increases with R. In addition, the which indicates that M partial derivative of Tb with respect to R is    v 2  pffiffiffiffiffiv ffi pffiffiffiffiffiffiffiffiffiffiffiffi v TbðDt; RÞ m$exp b r pR $ b r $pR  pffiffiffiffiffiv b ¼   v r v pR2 r $ R erf  2 $ 2 b vR v exp b r pR2  1

(11.18) pffiffiffiffiffiv pffiffiffiffiffiffiffiffiffiffiffiffiffi  b r v pR2 . Thus, FðRÞ  0 because We define FðRÞ ¼ 2 b r $R erf pffiffiffiffiffiv    r v pR2 0. Therefore, Fð0Þ ¼ 0; and dFðRÞ=dR ¼ 2 b r 1 exp  b   v TbðDt; RÞ vR  0, which indicates that Tb monotonically increases with R. In summary, we have the following proposition. Proposition 11-1. If R is less than the radius of the dominant area of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   b Dt and Tb monotonically increase passengers, that is, ðb r c Þ1 p, then M  b Dt and Tb are independent of R. with R. Otherwise, M We then study the following partial derivative with respect to R of the overall system performance in Eq. (11.13), as follows:  b b  vM vP 1  b vT ¼ a  2 Tb $  M$ vR Dt vR vR  ¼

    b vM b R v a  2 Tb a  2 Tb M R þ   b vR vR M RDt a  2 Tb 

  b a  2 Tb M ^ T^ εRM þ εa2 ¼ R RDt

(11.19)

^ T^ are the elasticity of M b and the net benefit per matching a where εRM and εa2 R   b vR  0 and v Tb vR  0, 2 Tb with respect to R, respectively. Given that v M ^ T^ < 0 if a > 2 T T^  0 otherwise. Hence, if b, and εa2 εRM > 0. Furthermore, εa2 R R a < 2 Tb, then vP=vR < 0; thus, P decreases with R if the net benefit per ^ matching is negative. If a > 2 Tb, the sign of vP=vR depends on the sign of εRM þ

Optimization of matching-time interval and matching radius Chapter | 11 ^

^

^

^

269 ^

T : vP=vR > 0 if ε M > εa2T , and vP=vR  0 if ε M  εa2T . In εa2 R R R R R ^ ^ M a2 view of εR > 0 and εR T < 0 for a > 2 Tb, P increases with R if the elasticity of M in R is larger than the absolute value of the elasticity of a  2 Tb in R, and vice versa.

11.2.5.2 Effect of matching-time interval

 b Dt We next study the effects of Dt on system performance (including on M and the expected pick-up distance). For simplicity, we first consider a situation in which the demand and supply that are carried over from the previous time interval are approximately zero, that is, Noc zNov z0; we also simplify the discussion by ignoring passengers’ abandonment behaviours and letting SðDtÞ ¼ 1. These conditions are realistic if few waiting passengers are carried over from the previous time interval(s) and Dt is restrained by a threshold value that is tolerable to most passengers. However, the model also attempts to  b Dt, which implicitly minimises the abandonment rate over a maximise M certain period or multiple matching intervals.   rc r c Þ1 > pR2 and AM ¼ pR2 . Therefore, if b If Dt < A qbc pR2 , then ðb c is low (due to a short Dt for a given qb ), then AM is governed by R, rather than by the dominant area of each passenger. In this case, the partial derivatives of  b b MðDt; RÞ and MðDt; RÞ Dt with respect to Dt are: 

 b DtpR2 q^v DtpR2 q^v vM 1 ¼ e A $ A  1 þ e A (11.20) qbc þ DtpR2 qbv qbc $ > 0 vDt A   DtpR2 q^v b v M=Dt e A $pR2 $b q v qbc >0 ¼ A vDt

(11.21)

 b Dt increase indicating that if Dt is short, the number of matched pairs and M with Dt.   If Dt  A qbc pR2 , then ðb r c Þ1 . Thus, AM is r c Þ1  pR2 and AM ¼ ðb governed by the dominant area of each passenger, rather than by R. In this  b b case, the partial derivatives of MðDt; RÞ and MðDt; RÞ Dt with respect to Dt are 

 b vM qbv ¼ qbc $ 1 exp  c >0 (11.22) vDt qb   b v M=Dt ¼0 (11.23) vDt  b Dt is independent of Dt. The latter result indicates that if Dt is long, M Summarising the above results leads to the following proposition.

270 Supply and Demand Management in Ride-Sourcing Markets

Proposition 11-2. If the carryover demand and supply from the previous time interval are approximately zero, i.e., Noc zNov z0, passengers’ abandon b Dt first inment behaviours are ignored, and Dt is sufficiently long, then M creases and then remains unchanged with Dt. Subsequently, we investigate the influence of Dt on Tb. As discussed above,   if Dt < A qbc pR2 , AM ¼ pR2 . Thus, the partial derivative of Tb with respect to Dt is sffiffiffiffiffiffi  v Tb m A 1 2 pffiffiffiffiffi pffiffiffi c1 ð2ec1 c1  1 þ ec1 Þ ¼ $ vDt 4v qbv ð1 þ ec1 Þ2 p (11.24) pffiffiffiffiffi c1 c1 32 ð2c1  1 þ e Þerf ð c1 Þe Dt  where c1 ¼ Dt qbv pR2 A, which has a positive dimensionless value. Evidently, the pffiffiffiffiffiffiffiffiffiffi sign of vT=vDt depends on that of the term gðc1 Þ ¼ 2 c1 =pð2ec1 c1 1 þec1 Þ pffiffiffiffiffi ð2c1 1 þec1 Þerf ð c1 Þec1 , where gð0Þ ¼ 0. It is therefore difficult to analytically determine the sign of gðc1 Þ; however, gðc1 Þ decreases with c1 , implying that gðc1 Þ < 0 for c1 > 0. Thus, vT=vDt < 0.   r c Þ1 . Then, the first- and second-order parIf Dt  A qbc pR2 , AM ¼ ðb tial derivatives of T with respect to Dt are 11 2 sffiffiffiffiffi sffiffiffiffiffi3 sffiffiffiffiffi 0

v v v Tb m A B qbv 7 3 6 2 qq^^vc qb qq^^c C p ffiffiffi ¼ erf $ 1  e $ e 5$Dt 2 (11.25) @ A 4 vDt 4v qbv qbc qbc p v Tb 3m ¼ vDt2 8v 2

3 0 11 2 sffiffiffiffiffi sffiffiffiffiffi

sffiffiffiffiffiv  v v v ^ ^ q q qb qb 7 5 A B C 6 2 erf $@1  eq^c A $4pffiffiffieq^c 5$Dt 2 qbv qbc qbc p 

(11.26)

Given that 1  e^q =^q > 0, the signs of vT=vDt and v2 T vDt2 depend on pffiffi pffiffiffi pffiffi that of the term 2ez z= p erf ð zÞ, where z ¼ qbv =b q c . Let hðzÞ ¼ p ffiffiffi p ffiffi p ffiffi pffiffi pffiffiffi z z= p erf ð zÞ. Thus, we have vhðzÞ=vz ¼ 2ez z= p < 0, 2e implying that hðzÞ monotonically decreases with z. Considering that z ¼ q c > 0 and hð0Þ ¼ 0, we can conclude that hðzÞ < 0 for any value of z > qbv =b   0. Therefore, v Tb vDt < 0 and v2 Tb vDt2 > 0. These findings are summarised in the following proposition. Proposition 11-3. If the carryover demand and supply from the previous time interval are approximately zero, i.e., Noc zNov z0, and passengers’ abandonment behaviours are ignored, then the average pick-up time T  monotonically decreases with Dt (that is, v Tb vDt < 0) and at a decreasing  A . rate (that is, v2 Tb vDt2 > 0) if Dt  ð^qc pR 2Þ v

c

Optimization of matching-time interval and matching radius Chapter | 11

271

The propositions above are analytically derived based on appropriate assumptions about qbv and qbc , and their distributions over space. These propositions provide a good understanding of the effect of the two key decision variables (Dt and R) on system performance. Thus, they can aid in the development of an efficient on-demand ride-sourcing platform.

11.3 Model properties in imbalanced scenarios Imbalanced scenarios, in which supply is considerably larger or smaller than demand, are commonly observed in actual operations. In these scenarios, the optimisation of Dt and R depends on the magnitudes of demand and supply and their relative imbalance.

11.3.1 Effects of matching-time interval We first consider a scenario in which supply is substantially larger than demand (a supply-dominated scenario). We restrict our discussion to a case in which R is larger than the radius of the dominant zone of each waiting passenger or AM is governed by b rc. r c. Thus, AM ¼ 1=b r c , where b r s AM ¼ b r v =b Moreover, we let SðDtÞ ¼ 1 by ignoring passengers’ abandonment behaviours.  d r v AM Þ If b rs b r is greater than a certain value, then the exponential term expðb pffiffiffiffiffiffiffiffiffiffiffi ffi v z 0, and the Gaussian error-function term erf ð b r AM Þz1. For example, if pffiffiffiffiffiffiffiffiffiffiffi ffi v c v b r =b r ¼ 3, then expðb r AM Þ  0:049z0, and erf ð b r v AM Þ  0:99998z1. In addition, the number of matched passengeredriver pairs and Tb can be approximated as follows: c b ¼ Nbc $½1 expðb M r v AM Þz Nb ¼ Noc þ qbc Dt

(11.27)

m m RbE Tb ¼ z pffiffiffiffiffiv ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi v 2v b r 2v N v þ qbv Dt =A

(11.28)

o

As the supply is considerably larger than the demand, it is reasonable to b q c Dt. The optimal design for matching also assume that Noc z0 and thus Mzb strategies in this case can be formulated as the following mathematical problem: maxPðDtÞ ¼

b    1 c M a  2 Tb  Noc þ Nov  ðb q þ qbv Þ$Dt Dt 2

(11.29)

subject to: Tb ¼

m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  v  ffi 2v N o þ qbv Dt =A

(11.30)

272 Supply and Demand Management in Ride-Sourcing Markets

b ¼ qbc Dt M

(11.31)

The properties of (P2) are summarised in the following proposition: Proposition 11-4. If the supply is considerably larger than the demand, then  3 (1) if qv > pv ffiffiffi Nov 2 ð1 þqv =qc Þ, P first increases and then decreases with m A

Dt, with

 2

2 3 3 1 Nv 13  v v 1 3 Dt ¼ A m ðq Þ  qvo ; or 1þqv =qc

with



v ffiffiffi p m A Dt ¼

(2) if qv 

3 Nov 2 ð1 þqv =qc Þ, then P monotonically decreases with Dt,

0.

Thus, qv , Nov and qv =qc jointly determine the trend of P (which first increases and then decreases or monotonically decreases) with Dt. For a given qv =qc , if qv is greater than Nov , P first increases and then decreases with Dt (Case (1) in Proposition 11-4.). This may be because T first decreases rapidly and then gradually with Dt, as indicated in Eq. (11.27). In contrast, w linearly increases with Dt. Therefore, the marginally favourable (i.e., T-reducing) effect of increasing Dt becomes less than its marginally unfavourable (i.e., wincreasing) effect when Dt exceeds a certain value. However, in extreme cases where Nov is extremely large and qv is small (Case (2) in Proposition 11-4), each newly arriving passenger can be immediately matched with a very close driver. In such a scenario, increasing Dt generates little or no reduction in T but directly increases w. Thus, the optimal Dt z 0. For example, outside an airport many idle drivers wait in queues, and thus each newly arriving passenger can and should be matched immediately with an idle driver with a near-zero pick-up distance. Thus, a platform should set Dt ¼ 0. A rigorous proof of Proposition 11-4 is given below. Proof. The first-order condition of Eq. (P2) is as follows: pffiffiffi  3 1 c vP m A qbv qbc  v ¼ q þ qbv Þ (11.32) No þ qbv Dt 2  ðb vDt 2 2v The limits of the first-order derivative are pffiffiffi vP m A qbv qbc  v 32 1 c ¼ q þ qbv Þ No lim  ðb Dt/0vDt 2 2v

(11.33)

Optimization of matching-time interval and matching radius Chapter | 11

lim

vP

Dt/NvDt

1 c ¼  ðb q þ qbv Þ 2

273

(11.34)

Furthermore, the second-order derivative is non-positive: pffiffiffi  5 v2 P 3m A qbc qbv  v ¼ (11.35) No þ qbv Dt 2  0 2 vDt 4v  3 vP > 0. Eqs. (11.34) and (11.35) q c Þ, then lim vDt q v =b If qbv > pv ffiffiffi Nov 2 ð1 þb m A

Dt/0

m A

Dt/0

verify the existence of one global Dt for maximising P. This condition also indicates that P first increases and then decreases with Dt. To calculate Dt , we let vP=vDt ¼ 0, and thereby obtain

23

 1 v 1 Nov 1 2  3  Dt ¼ A3 ðb q v Þ3 (11.36) m 1 þ qbv =b qc qbv  3 vP  0. Thus, vP is always nonq c Þ, then lim vDt q v =b If qbs  pv ffiffiffi Nov 2 ð1 þb vDt positive. Therefore, P monotonically decreases with Dt, and Dt ¼ 0. This completes the proof. -

11.3.2 Properties of optimal matching-time interval In Case (1) of Proposition 11-4, Dt can be identified. Of particular interest is the relationship between Dt and the real-time supplyedemand conditions, such as qbv and Nov . An interesting finding is given in the following proposition. Proposition 11-5. If supply is considerably larger than demand, and  3 v q c Þ, then Dt decreases with Nov . In addition, q v =b qb > pv ffiffiffi Nov 2 ð1 þb m A " #1=3 if Nov  16

2Am2 ð^ qv Þ 2 v ð1þ^ q =^ qc Þ 2 v 2

, Dt increases with qbv ; otherwise, Dt decreases

with qbv . It can be intuitively understood that Dt should decrease with Nov . If many idle vehicles are carried over (Nov is large), each newly arriving passenger has a good opportunity to be matched with a nearby driver, as Nv is already high. Thus, the marginal benefit of increasing Dt to reducing the expected pick-up distance decreases. In addition, increasing Dt itself increases w and wv. Therefore, the optimal Dt is relatively short, given that a long Dt affords only a small benefit or may be harmful. In contrast, Dt may not be monotonic with respect to qbv , as the optimal Dt is short if qbv is excessively high or low. If qbv is too high, setting a short Dt reduces T to a certain low level. If qbv is too small, the marginally favourable (T-reducing) effect of increasing Dt may become less than its marginally unfavourable (w-increasing) effect. Therefore, the optimal Dt is short. A formal proof of Proposition 11-5 is given below.

274 Supply and Demand Management in Ride-Sourcing Markets

Proof. With reference to Eq. (11.36), the partial derivatives of Dt with respect to Nov and qbv are vDt 1 ¼  v0 qb vNov 8 > > >
> > v 2 7 = 6 Am2 ðb vDt 1 q Þ 6 7 v 2 v N ¼ ðb q Þ  6 7

 o > 34 vb qv qbv 2 5 > > > > > : ; 1 þ c v2 qb

(11.37)

2

(11.38)

We then obtain " #1=3 8 2 v 2 > 1 Am ðb q Þ > >  0; if Nov  > > 3 ð1 þ qbv =b <  q c Þ2 v 2 vDt ¼ " #1=3 vb qv > > 2 v 2 > 1 Am ðb q Þ > v > < 0; if N < : o 3 ð1 þ qbv =b q c Þ2 v 2

(11.39)

This completes the proof. -

11.3.3 Further discussion Dt is theoretically determined under a supply dominated imbalanced scenario. It can be analytically obtained if qbv is relatively large and Nov is relatively small. However, if qbv is relatively small and Nov is relatively large, P monotonically decreases with Dt. Therefore, Dt approaches zero. In a demand-dominated imbalanced scenario in which the supply is substantially less than the demand, the number of matched passengeredriver pairs and Tb can be approximated by v b ¼ Nbc $½1 expðb M r v AM Þz Nb ¼ Nov þ qbv Dt

(11.40)

RbE 2m 2m Tb ¼ z pffiffiffiffiffic ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi v 3v b r 3v N c þ qbc Dt =A

(11.41)

o

Eqs. (11.40) and (11.41) for the demand-dominated scenario are analogous to Eqs. (11.27) and (11.28) for the supply dominated scenario. This condition suggests that if the sizes of the supply and demand pools are sufficiently different, the number of matched passengeredriver pairs approximately equals the number in the smaller-sized pool, whereas Tb is inversely proportional to the square root of number in the larger-sized pool. The difference between the coefficients (2/3 in Eq. (11.41) and 1/2 in Eq. (11.27)) is due to the different assumptions used (a spatially uniform distribution for waiting passengers and a

Optimization of matching-time interval and matching radius Chapter | 11

275

spatial Poisson distribution for idle drivers). These observations suggest that conclusions similar to those in Proposition 11-4 and Proposition 11-5 can be drawn regarding the demand-dominated scenario. Moreover, similar derivations can be performed by assuming that there is a spatially uniform distribution of idle drivers and a spatial Poisson distribution of waiting passengers.

11.4 Numerical studies In this section, we provide numerical examples to investigate how the two key  b Dt, and other system perdecision variables (Dt and R) jointly affect Tb, M formance measures in balanced and imbalanced scenarios. Furthermore, we compare the performance of our model on a simulation platform with that of a benchmark algorithm with constant Dt and R.

11.4.1 Balanced scenario First, we examine the effects of the matching strategies on the system performance in a balanced scenario in which there is only a small difference between supply and demand, and in which passengers’ abandonment behaviours are ignored. In this example, we study an area of 500 km2 with v ¼ 20 km/h, and consider a simple case in which Nov ¼ Noc ¼ 0, qbv ¼ qbc ¼ 1  104 pass=h and a ¼ 500 s=pair. We perform sensitivity analyses with various values of Dt (from 2 to 240 s, with a step size of 1 s) and R (from 0.5 to 2 km, with a step size of 0.05 km). r c (i.e., AM ¼ ðb r c Þ1 ) Fig. 11.3A demonstrates whether AM is bounded by b or by R (i.e., AM ¼ pR2 ) in the two-dimensional space of Dt and R. It reveals r c if Dt is long (which leads to a high b r c ) or if R is that AM is bounded by b large.  b Dt and P in the twoFig. 11.3BeD depicts the contours of Tb, M dimensional space of Dt and R. In Fig. 11.3B, Tb decreases with Dt and increases with R if AM is governed by R. In contrast, Tb decreases with Dt but is r c. These numerical observations are independent of R if AM is governed by b consistent with the theoretical findings in Proposition 11-1 and Proposition 11-3.  b Dt first increases with Dt (if Dt is short and AM is In Fig. 11.3C, M governed by R) and then remains unchanged with Dt (if Dt is long, such that  b Dt increases with R if AM is governed by r c ). In addition, M AM is governed b r c. These numerical results R, but is independent of R if AM is governed by b support Proposition 11-1 and Proposition 11-2.  b Dt via a, which reveals that P is Fig. 11.3D jointly considers Tb and M convex with respect to the decision variables (proving that it is theoretically difficult to determine). We can also identify the combination of Dt and R that

276 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 11.3 Effects of matching-time interval and matching radius without passenger abandonment; (A) matching area; (B) expected pick-up time; (C) matching rate; (D) System performance.

achieves the optimal system performance. Specifically, at each instant in actual operations with a known Nov and Noc , and a predicted qbv and qbc , a ride-sourcing platform can determine the optimal combination of Dt and R. Thus, a platform can improve system efficiency by dynamically adjusting its matching strategies in accordance with the latest supply and demand conditions. Second, we investigate a balanced scenario and consider passengers’ abandonment behaviours. In the simulation, SðDtÞ is numerically generated by assuming that passengers’ willingness to wait follows a normal distribution with a mean of 5 min and a standard deviation of 3 min. Fig. 11.4AeC depicts  b Dt, and P in the two-dimensional space of Dt and R. the contours of Tb, M Fig. 11.4A and B reveals that Proposition 11-1 continues to hold because the  b Dt and Tb is independent of passengers’ abandonment beeffect of R on M haviours. However, Proposition 11-2 no longer holds if passengers’ aban b Dt first donment behaviours are considered: as illustrated in Fig. 11.4B, M increases and then decreases with Dt. This is because an increase in Dt

Optimization of matching-time interval and matching radius Chapter | 11

277

FIGURE 11.4 Effects of matching-time interval and matching radius with passenger abandonment; (A) expected pick-up time; (B) matching rate; (C) system performance.

produces two opposite effects in this scenario. On the one hand, a long Dt accumulates additional waiting passengers and idle drivers in the matching  b Dt. On the other hand, it pool, thereby improving matching and increasing M makes certain passengers impatient and thus increases the abandonment rate. Finally, Fig. 11.4C shows that P is approximately convex with respect to the two decision variables (Dt and R).

11.4.2 Imbalanced scenarios We examine the effects of matching strategies on system performance under imbalanced scenarios. We consider an area of 500 km2 with v equal to 20 km/ h, and an imbalanced scenario in which Noc ¼ zero and qbv =b q c ¼ 3. Fig. 11.5 depicts P for this scenario, with Dt varying from 0 to 300 s with a step size of 2 s. 4 Fig. 11.5A shows the trend of P vs Dt, with qbv ¼ 1:0   10 veh=h and  3 v v ffiffiffi N v 2 1 þq^ p ¼ 10:06 > 1. In Nov ¼ 50 veh. This implies that qbv o q^c m A

FIGURE 11.5 System performance vs matching-time interval.

278 Supply and Demand Management in Ride-Sourcing Markets

this scenario, P first increases and then decreases with Dt, which verifies Case (1) in Proposition 11-4. Fig. 11.5B displays the change in P with respect Dt, with qbv ¼ 1:0  104 veh=h and Nov ¼ 500 veh. This reveals that 

 3   v v ffiffiffi N v 2 1 þq^ p qbv ¼ 0:32 < 1. In this scenario, P monotonically o q^c m A

decreases with Dt, and Dt ¼ 0. This observation is consistent with Case (2) in Proposition 11-4. This section examines how Dt is affected by the real-time supplye demand conditions. First, we set qbv as 1:0  104 veh=h and calculate Dt for various Nov (from 50 veh to 300 veh, with a step size of 10 veh). As displayed in Fig. 11.6A, Dt monotonically decreases with Nov . Second, we set Nov to  100 veh and calculate Dt for various qbv (from 0:5  104 to 3  104 veh h, with a step size of 0:1  104 veh=h). As illustrated in Fig. 11.6B, Dt first increases and then decreases with qbv . This situation is consistent with that in Proposition 11-5. If qbv is low, the magnitude of the favourable effect (reduced T) caused by increasing Dt is less than that of the unfavourable effect caused by increasing Dt. Therefore, Dt must be short. If qbv is high, the absolute magnitude of the favourable effect of increasing Dt decreases as Dt increases, due to the convexity of Tb with respect to Dt. Thus, Dt should not be excessively long. These two opposite forces drive Dt first to increase and then decrease with qbv .

11.4.3 Model performance in a dynamic simulation environment The abovementioned theoretical and numerical studies show how the two key decision variables (Dt and R) affect system performance. Such research also demonstrates how the optimal values of decision variables can be determined

FIGURE 11.6 Optimal matching-time interval vs (A) number of carryover idle vehicles (Nov ) and (B) arrival rate of idle vehicles (b q v ).

Optimization of matching-time interval and matching radius Chapter | 11

279

or the matching strategies can be dynamically adjusted for each time interval. To further verify the effectiveness of the proposed method, we simulate the stochastic online matching process using Python. We also apply our optimisation strategy, which dynamically (interval-by-interval) solves the optimisation problem (with two decision variables) using the CVXOPT solver. The experiments are performed on a 3.40 GHz i7-6700 CPU Windows PC with 16 GB of RAM. The simulator is established on a space with 10 km  10 km area and a minimum simulation-updating time-step of 5 s for an entire day. The following steps are executed. (1) New passenger requests are generated with random origins and destinations in accordance with the qbv shown in Fig. 11.7A. (2) Certain drivers enter the platform (go online) for order dispatching, whereas other drivers leave the platform (go offline), as drivers have different working schedules. (3) After dropping off a passenger, a vehicle becomes available for matching. (4) Certain passengers become impatient and abandon their orders if their accumulated w exceeds the maximum time they are willing to wait. In the simulation, it is assumed that passengers exhibit a normal distribution of maximum tolerated w (with a mean of 20 min and a standard deviation of 5 min). (5) If the platform executes a matching with a given R, then a bipartite matching is conducted between waiting passengers and idle drivers. Subsequently, the matched passengers leave the waiting list and the matched idle drivers become occupied. The daily pattern of qbv is calibrated by a taxi trip dataset from New York (https://www1.nyc.gov/site/tlc/about/tlc-trip-recorddata.page). The number of drivers in service at each minute of a day is shown in Fig. 11.7B. The dynamic optimisation strategy starts from the initial time and solves (P1) sequentially to determine the optimal matching decision variables for execution. Specifically, at an instant or time slot t, the platform determines Dt and R by solving P1, where qbv and qbc are predicted from historical averages. From t to t þ Dt (which may include multiple simulation

FIGURE 11.7 Simulation settings: (A) Arrival rate of passenger requests over time; (B) number of drivers over time.

280 Supply and Demand Management in Ride-Sourcing Markets

time-steps), the simulator generates new requests and then updates the status of in-trip/idle/offline/online drivers and waiting passengers on the basis of the abovementioned stochastic procedure ([1]e[4]). Then, at time t þ Dt, the platform executes bipartite matching between idle drivers and waiting passengers. After the statuses of idle drivers and waiting passengers are updated in accordance with the matching outcomes, the platform determines a new pair of Dt and R on the basis of the newly observed and predicted supplyedemand conditions. We now compare the performance of the dynamic optimisation strategy and benchmark algorithms for a given and fixed pair of Dt and R in the matching platform. Two benchmark scenarios are considered. The first scenario has a fixed short Dt of 10 s and a fixed R ranging from 1 to 9 km, with a step size of 2 km. The second scenario has a fixed long Dt of 60 s and a fixed R ranging from 1 to 9 km, with a step size of 2 km. The evaluation metrics are the mean pick-up time, mean w, and mean total w (sum of pick-up time and w) of passengers and the completion rate (the proportion of passenger requests served). The results in Table 11.1 show that the dynamic optimisation strategy achieves a higher completion rate (0.895) than all of the benchmark scenarios. The strategy also achieves a reasonably low mean total w; it is only large in the case of Dt ¼ 10 s and R ¼ 1 km, which has a low completion rate of only 0.805 (the lowest of all of the cases). Overall, our dynamic optimisation strategy achieves the best system performance of all of the simulated cases.

11.5 Conclusion This chapter examines optimal matching strategies in an on-demand ridesourcing platform. A spatial probability model is established to characterise the real-time online matching process. Two key decision variables (Dt and R)  b Dt, Tb , and are determined to optimise system performance in terms of M expected w. We investigate the effects of these decision variables on P under various supplyedemand conditions (Nov , Noc qbv , and qbc ). Theoretical deriva b Dt and Tb first increase with R but tions and numerical analyses reveal that M then become independent of R when R exceeds a certain threshold. This independence stems from the fact that when R is sufficiently large, the bilateral matching process is dominated by rv and rc . Moreover, Dt can maximise system performance if the supply is considerably larger than the demand (and vice versa), as Dt (in a supply-dominated scenario) is jointly determined by Nov and qbv . This interval can also be zero if Nov is sufficiently large. As demonstrated in the simulation, the proposed model can be easily implemented in a dynamic environment by iteratively determining and executing the two key decision variables. The simulation results indicate that this dynamic optimisation strategy outperforms strategies with fixed decision variables. This chapter is based on one of our recent articles (Yang et al., 2020).

Matching-time interval Dt (s)

Matching radius R (m)

Mean pick-up time (s)

Mean waiting time (s)

Mean total waiting time (s)

Completion rate

Dynamic optimisation strategy

e

e

144.47

280.71

425.18

0.895

Fixed strategy (short Dt)

10

1000

84.23

307.16

391.39

0.805

10

3000

135.22

296.09

431.31

0.881

10

5000

148.21

289.74

437.95

0.892

10

7000

145.59

299.52

445.10

0.884

10

9000

145.31

290.01

435.31

0.892

60

1000

81.48

350.99

432.46

0.814

60

3000

128.96

323.65

452.61

0.875

60

5000

127.16

348.16

475.33

0.873

60

7000

136.02

330.64

466.66

0.881

60

9000

130.82

367.13

497.95

0.849

Model

Fixed strategy (long Dt)

Optimization of matching-time interval and matching radius Chapter | 11

TABLE 11.1 Comparison of simulation results.

281

282 Supply and Demand Management in Ride-Sourcing Markets

Glossary of notation R matching radius RS ride-pooling (ride-splitting) wv average drivers’ idle time Dt average matching time interval N v numbers of idle drivers N c numbers of waiting passengers w average passengers’ waiting time Lp average pickup distance ^qc predicted arrival rates of new passengers ^qv predicted arrival rates of idle drivers N co numbers of waiting passengers remaining from the previous interval N vo numbers of idle drivers remaining from the previous interval S(Dt) proportion remaining in the matching pool at the end of Dt A area of the studied space ^pc estimated densities of demand ^pv estimated densities of supply AM area surrounding a waiting passenger within which the passenger can be matched with an idle driver ^ M ^ expected number of matched passengeredriver pairs T expected pick-up time x distance from each waiting passenger to the closest driver ^ E expected pick-up distance R Rx 2 erf ð xÞ [ p2ffiffipffi 0 eLt dt a Gaussian error function m a detour ratio

References Arnott, R., 1996. Taxi travel should be subsidized. Journal of Urban Economics 40 (3), 316e333. Boscoe, F.P., Henry, K.A., Zdeb, M.S., 2012. A nationwide comparison of driving distance versus straight-line distance to hospitals. The Professional Geographer 64 (2), 188e196. Chiu, S.N., Stoyan, D., Kendall, W.S., Mecke, J., 2013. Stochastic Geometry and its Applications. John Wiley & Sons. Fairthorne, D.B., 1964. The distance between pairs of points in towns of simple geometric shape. In: Proceedings of the 2nd International Symposium on the Theory of Traffic Flow. OECD. Ke, J., Xiao, F., Yang, H., Ye, J., 2019. Optimizing Online Matching for Ride-Sourcing Services with Multi-Agent Deep Reinforcement Learning arXiv preprint arXiv:1902.06228. Ke, J., Zheng, H., Yang, H., Chen, X.M., 2017. Short-term forecasting of passenger demand under on-demand ride services: a spatio-temporal deep learning approach. Transportation Research Part C: Emerging Technologies 85, 591e608. Tong, Y., Chen, Y., Zhou, Z., Chen, L., Wang, J., Yang, Q., Lv, W., 2017. The simpler the better: a unified approach to predicting original taxi demands based on large-scale online platforms. In: Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. SIGKDD, pp. 1653e1662. Xu, Z., Yin, Y., Zha, L., 2017. Optimal parking provision for ride-sourcing services. Transportation Research Part B: Methodological 105, 559e578. Yang, H., Ke, J., Ye, J., 2018. A universal distribution law of network detour ratios. Transportation Research Part C: Emerging Technologies 96, 22e37. Yang, H., Qin, X., Ke, J., Ye, J., 2020. Optimizing matching time interval and matching radius in ondemand ride-sourcing markets. Transportation Research Part B: Methodological 131, 84e105.

Chapter 12

Labour supply analysis of ride-sourcing services Sun Hao1, Hai Wang2 and Zhixi Wan1 1 Faculty of Business, The University of Hong Kong, Pokfulam, Hong Kong, China; 2School of Computing and Information Systems, Singapore Management University, Bras Basah, Singapore

12.1 Background The rapid development and popularisation of mobile and wireless communication technologies has enabled ride-sourcing platforms such as Didi Chuxing and Uber, which use mobile wireless technology to connect passengers and drivers, to disrupt the transportation industry worldwide. In the traditional taxi industry, drivers are typically required to obtain an occupational licence, or ‘medallion’, to provide transport services to passengers. Moreover, the number of drivers is limited by the number of medallions issued, the regions in which drivers can pick up passengers are dictated by the jurisdiction that issued their medallion, and fares are often set by regulatory bodies (Cramer and Krueger, 2016). In contrast, in ride-sourcing markets, drivers work as freelancers and thus can have flexible schedules; that is, they can use their own or leased cars to offer transport services whenever and wherever they choose. Ride-sourcing platforms match incoming requests from passengers with nearby drivers and adjust fares dynamically when demand is high relative to the supply of drivers in a local region. Drivers’ incomes depend on their working time and location, the supply of drivers and the passenger demand (Chen et al., 2019). Crucially, because drivers are autonomous, they are able to adjust their supply of labour in response to income fluctuations. Due to the flexible labour supply of ride-sourcing platforms, such platforms offer drivers large payments to incentivise them to offer their services. For example, Uber offers a driver US$500 for completing 120 trips in a week or a guaranteed income of US$1800 for the first 200 trips. Moreover, a new Uber incentive, ‘Boost’, multiplies drivers’ trip fares by a certain amount for

Supply and Demand Management in Ride-Sourcing Markets. https://doi.org/10.1016/B978-0-443-18937-1.00004-8 Copyright © 2023 Elsevier Inc. All rights reserved.

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284 Supply and Demand Management in Ride-Sourcing Markets

all trips within designated busy areas during specified periods.1 Such incentive payments may represent 20%e40% of the income of typical drivers, according to an analysis conducted by Los Angeles-based Uber driver Harry Campbell, who runs the popular blog The Rideshare Guy, and these incentive payments were one of Uber’s biggest operational expenses for the six quarters that ended in the first half of 2017 (Bensinger, 2017). Accordingly, the effects of hourly income rates on ride-sourcing platforms’ labour supply must be examined.

12.1.1 Motivation It is challenging to investigate the effect of hourly income rates on the labour supply of ride-sourcing platforms. This is because standard labour economics theory holds that income rates exert two distinct effects on labour supply: they affect both an intensive margin, which relates to the number of hours worked, and an extensive margin, which relates to the decision on whether to participate in a labour market (Cahuc et al., 2014). For nearly 20 years, there has been debate amongst labour economists on how taxi drivers decide how many hours to work (Scheiber, 2016). Colin Camerer, a behavioural economist, found that there was a negative relationship between the hours worked by taxi drivers in New York City and transitory changes in their income and argued that drivers entered the market with an income target and stopped working when they reached this target (Camerer et al., 1997). In contrast, Henry Farber found that taxi drivers tended to respond positively to increases in earnings opportunities and that their estimated income elasticities were generally positive (Farber, 2015). In observational studies, bias in the estimations of effects can arise due to unobservable factors and the endogeneity problem. First, drivers who are able to participate in ride-sourcing platforms face a two-step decision: whether to participate, and if so, how many hours to work. However, only historical data are available for participating drivers, which indicates that data on drivers’ current working hours are censored. Given that unobservable factors affect both the participation decision and the working-hour decision of drivers, performing fitting on non-randomly selected samples leads to sample selection bias (Heckman, 1979).1 Second, drivers do not determine their working hours immediately after obtaining fixed income rates because the hours they work affect their income rates. This simultaneity bias creates an endogeneity problem in model identification. Furthermore, many empirical studies have sought to analyse labour supply using observed hourly income rates, which are calculated by dividing total daily income by total hours worked. This approach introduces measurement 1. In this book, we use ‘sample selection bias’ and ‘sample self-selection bias’ interchangeably [It may be clearer to use only one of these terms, as ‘selection’ does not mean the same as ‘self-selection’.].

Labour supply analysis of ride-sourcing services Chapter | 12

285

error, which generates a negative correlation between calculated hourly income rates and hours worked.

12.1.2 Research questions Motivated by these challenges associated with evaluating the effects of income rate on the labour supply of ride-sourcing platforms, we seek to answer the following three research questions. (1) What method can we use to analyse the influence of income rates on overall labour supply, while also controlling for sample self-selection, endogeneity of the income rate and errors in the measurement of working hours? (2) Does income-targeting behaviour influence overall labour supply as a function of daily participation and daily hours worked? (3) How is this influence affected by driver heterogeneity?

12.1.3 Methodology To answer the above research questions, we first discuss a labour supply model that decides the number of daily hours worked on the basis of the referencedependent preference theory, which predicts the sign of the intensive margin elasticity of labour supply. Next, we devise an econometric approach to estimate the effect of income rates on the daily labour supply response on both the extensive and intensive margins, which can accommodate the sample selection of labour force participation and endogeneity in the income rate. In particular, we use an instrumental variable (IV) approach to address potential omitted variable bias and simultaneity bias. We apply our empirical framework to analyse data from Didi Chuxing, the leading ride-sourcing platform in China. Crucially, we exploit the driver income multiplier as the exogenous shock in a large-scale natural experiment, which enables us to avoid the endogeneity problem in model identification. We first classify drivers by their extensive and intensive margins of labour supply; then, we estimate the participation elasticity and working-hour elasticity in the presence of driver heterogeneity.

12.1.4 Results We derive two key findings from our empirical analysis of labour. First, by analysing a dataset from a ride-sourcing platform, we estimate that participation elasticity and working-hour elasticity are positive and significantly different from zero: 0.107e0.524 and 0.023e1.037, respectively. These results do not exclude the possibility that drivers exhibit income-targeting behaviour. However, these results do imply that when an average driver makes a daily labour supply decision based on the practical income levels in our dataset, the behavioural forces that favour neoclassical intertemporal substitution dominate the behavioural forces that favour income targets. Second, in the presence of driver heterogeneity, we find that participation elasticity generally decreases

286 Supply and Demand Management in Ride-Sourcing Markets

along both the extensive and intensive margins, whereas working-hour elasticity decreases only along the intensive margin. Our results also reveal that labour supply elasticity is affected by driver gender and age.

12.1.5 Main contributions The main contributions of this work are as follows. l

l

l

We devise an econometric framework with closed-form measures to estimate the participation elasticity (i.e., extensive margin elasticity) and the working-hour elasticity (i.e., intensive margin elasticity) of labour supply. We estimate the effects of income rates on the total labour supply of ridesourcing platforms using a framework that models the sample selfselection bias of labour force participation and the simultaneity bias of working hours. We suggest that this framework can also be applied to analyse the labour supply of other on-demand platforms with independent agents. We construct a natural experimental environment in which a platform implements adjustment of driver incentive, which serves as an exogenous shock. The incentive, which is denoted an income multiplier in this work, multiplies drivers’ trip fares by a certain amount for all trips during specific periods in a day. In our identification approach, the exogenous income multiplier is adopted as an IV to solve the simultaneity bias problem. We obtain empirical evidence on the labour supply elasticity of a ridesourcing platform, finding that it is positive in all groups of drivers in the data of the ride-sourcing platform we examine. We also highlight the important influence of driver heterogeneity on the effects of income rates on the labour supply of the ride-sourcing platform. In particular, we find that participation elasticity decreases along both the extensive and intensive margins, whereas working-hour elasticity decreases along only the intensive margin.

12.2 Related literature Our work connects two strands of research in the literature. The first strand is the empirical analysis of the daily labour supply of workers who decide whether to participate in a platform, and if so, decide the number of hours they will work flexibly. Table 12.1 summarises prior numerical studies on the following six aspects of this strand of research: labour markets in which both income levels and the quantity of labour supplied are varied each day; labour supply responses along both the extensive and the intensive margins; the theoretical background; data sources; identification approaches to address endogeneity problems; and estimates of labour supply elasticities.

TABLE 12.1 Values of key model parameters in prior numerical studies. Labour supply

Theory

Data

Identification

Elasticity

Taxi drivers (Camerer et al.,. 1997)

Hours worked

Referencedependent preference

Trip sheets of NYC cabdrivers (1988, 1990, 1994)

Average income as IV

Negative hours elas. (
0.355 to
0.618)

Taxi drivers (Chou, 2002)

Hours worked

Referencedependent preference

Survey of Singapore taxi drivers

Average income as IV

Negative hours elas. (
0.3 to
0.9)

Bicycle messengers (Fehr and Goette, 2007 )

Hours worked, effort per hour

Referencedependent preference

Number of shifts and revenue per shift

RCT

Negative effort elas. (0.24); positive hours elas. (1.34e1.50)

Workers on the Trans-Alaska pipeline (Carrington, 1996)

Participation, hours worked

Neoclassical intertemporal labour supply

Unemployment insurance reports

Temporary demand shocks as an IV

Positive participation elas. (0.738); positive hours elas. (0.583)

Baseball stadium vendors (Oettinger, 1999)

Participation

Marginal analysis

Participation and income

Demand shift as an IV

Positive participation elas. (0.6)

Taxi drivers (Farber 2015)

Hours worked

Neoclassical intertemporal labour supply

Trip sheets of NYC cabdrivers (2009e13)

Average income as IV

Positive hours elas. (0.36 e0.62)

Commercial trap fishermen (Stafford, 2015)

Participation, hours worked

Neoclassical intertemporal labour supply

Marine fisheries trip tickets

Moon phase as an IV, selectivity correction

Positive participation elas. (0.062e0.066), positive hours elas. (1.05e1.26)

287

Continued

Labour supply analysis of ride-sourcing services Chapter | 12

Labour market

Labour market

Labour supply

Theory

Data

Identification

Elasticity

South Indian boat owners (Gine´ et al., 2017)

Participation

Neoclassical intertemporal labour supply

Sales and loan transactions

Demand-shifter as an IV, selectivity correction

Positive participation elas. (0.8e1.3), negative participation elas. On accum. Income. (-0.05 to
0.007)

Drivers in a ridesourcing firm (Sheldon, 2016)

Hours worked

Neoclassical intertemporal labour supply

Trips and hourly activity

Average income as an IV

Positive hours elas. (0.13 e0.25)

Uber drivers (Angrist et al., 2017)

Participation, weekly hours worked

Neoclassical intertemporal labour supply

RCT

Incentive offer as an IV

Almost zero participation elas., positive hours elas. (1.2)

Note. IV ¼ instrumental variable; RCT ¼ randomised controlled trial; elas. ¼ elasticity; NYC ¼ New York City; and accum. ¼ (weekly) accumulated. Farber (2005) analysed the stopping behaviour of New York City taxi drivers and showed that their probability of stopping driving on a day was positively related to the number of hours they had already worked. Crawford and Meng (2011) also made this finding.

288 Supply and Demand Management in Ride-Sourcing Markets

TABLE 12.1 Values of key model parameters in prior numerical studies.dcont’d

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Three important differences distinguish our research from these prior studies. First, we use a large dataset from a ride-sourcing platform to conduct a comprehensive analysis of drivers’ labour-supply responses to changes in income opportunities, namely responses in terms of the number of working hours per day and the number of days worked (i.e., the daily participation margin). Second, we devise a causal inference method to control for sample self-selection bias due to participation decisions and endogeneity of hours worked per day. Our research is most similar to the work of Stafford (2015), which estimated the income elasticities of participation and the number of working hours per day using a framework that deals with sample self-selection bias and the endogeneity of income rates. Moreover, Stafford (2015) used the phase of the Moon as an instrument for hourly income rates of commercial trap fishermen in a two-stage least squares (2SLS) regression, which removed bias due to endogeneity in these income rates. In our work, we take advantage of a natural experiment that contains exogenous shocks and instrument the endogenous hourly income rates of drivers with the driver incentive income multiplier, which is uncorrelated with measurement error. Third, by taking drivers’ heterogeneity into consideration, we obtain labour supply elasticities for diverse groups of drivers. Then, by comparing the estimated elasticities of these groups, we obtain additional information on labour supply behaviour on the ride-sourcing platform. The second strand is staffing and pricing for shared transportation platforms. Some studies (Cachon et al., 2017; Taylor, 2018; Bai et al., 2019) have focused on agents’ rational decisions without considering reference-dependent preferences. Zha et al. (2017) developed equilibrium models to characterise the labour supply (i.e., working hours) of drivers. These models were based on neoclassical or income-targeting hypotheses, which are competing hypotheses on how drivers determine their working hours. Our empirical analysis of data from the largest ridesharing platform in China shows that average drivers’ daily labour-supply decisions are affected more by behavioural forces favouring neoclassical intertemporal substitution than by behavioural forces favouring income targets. Similar conclusions have been reached by Fehr and Goette (2007), Stafford (2015) and Gine´ et al. (2017).

12.3 Labour supply model 12.3.1 Optimal decisions on hours worked based on income targets This section introduces a theoretical model of drivers’ labour supply based on income targets. We treat each day separately and focus on drivers’ decisions on the number of hours to work on a given day (also see Crawford and Meng, 2011; Farber, 2015; Zha et al., 2017). According to the reference-dependent preference model developed by KTszegi and Rabin (2006) and Farber (2015),

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on a given day on a platform with a hourly income rate of W, a driver’s utility of working hours H (also denoted ‘hours worked’) is the sum of the driver’s positive utility from income earned (I, where I ¼ WH) and disutility from hours worked. The utility function of a driver with a daily income target T can be specified as follows: 8 q > 1þhh > ; if I < T; > < ð1 þ aÞðI
TÞ
1 þ h H h UðI; HÞ ¼ (12.1) > q > 1þhh > H ; if I  T; : ð1
aÞðI
TÞ
1 þ hh where a  0 and represents the change in marginal utility at the income target T, q  0 and controls the disutility from hours worked, and hh ˛ R is related to the income rate elasticity of labour supply along the intensive margin. The utility with  reference-dependent  preferences is based on a neoclassical utility function

q H 1þhh ðI
TÞ
1þh h

augmented with a gaineloss

utility (aðI
TÞ) around the reference point T. This is demonstrated by the fact that when a ¼ 0, UðI; HÞ is the neoclassical utility function and the optimal number of working hours is given by  h1 W h H0 ¼ ; (12.2) q which implies that the intensive margin elasticity of labour supply is h1 . If h reference-dependent preferences are considered (a > 0), it can be found that drivers decide their optimal number of working hours by maximising the utility function represented by Eq. (12.1), which gives the following optimal solution:  1þh1 h 8 ð1 þ aÞW h1h h h q > ; if W < T 1þhh ; > > > q 1þa > > > > 1    1þh1 h < 1þhh hh h h T q q 1þhh H ¼ ; if T

W 1 þ a 1
a > > > > >  1 1þh1  > hh > h q : ð1
aÞW hh ; if W > T 1þhh q 1
a (12.3) Eq. (12.3) shows that when income rates are very low, the number of working hours required to achieve the income target is so high that the disutility from hours worked is greater than the positive utility from income T and satearned. Hence, the optimal number of working hours is less than W isfies the condition that the marginal utility from consumption is equal to the

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marginal utility from working. When income rates are intermediate in magnitude, drivers choose their number of working hours such that their total income is equal to their target income, and the range of income rates in which   1 h   1 h  1þhh 1þhh h h q q 1þhh a driver is a target earner is T ; T 1þhh . If a is close to 1þa 1
a zero, which implies that there is little gaineloss utility, the range of income rates becomes very small and the reference-dependent preference (or income target) has little effect on the labour supply. When income rates are sufficiently high, the number of hours drivers must work to achieve their income targets is so  it is optimal for them to work for more than this number of hours  low that T . H > W

The intensive margin elasticity of labour supply for the optimal number of working hours given by Eq. (12.3) is 8   1þh1 1þh1 hh hh h h > q q > 1þhh > T

> 1 > : ; otherwise: hh The reference-dependent model predicts that when income levels are intermediate in magnitude, it is optimal for drivers to stop working when they reach their income target. In this case, the income rate elasticity of labour supply at the intensive margin is close to
1, which means that the labour supply decreases if the platform offers higher income rates to drivers.

12.3.2 Importance of the extensive margin in the labour supply model The above-described labour supply model with income targeting predicts the sign or magnitude of income rate elasticity along the intensive margin. However, we also wish to understand the effects of income rates on labour supply along the extensive margin, as these influence drivers’ daily participation decisions. This is because the hourly income rate in ride-sourcing markets fluctuates continually, which means that both the probability of drivers’ participation and the number of hours they work per day can vary. Specifically, drivers increase their amount of leisure time and non-market work when hourly income rates are relatively low, whereas they decrease their amount of leisure time and non-market work when hourly income rates are relatively high (Lucas and Rapping, 1969). It is crucial to understand how the labour supply along the extensive margin (or the number of active drivers) affects the operational practices of ridesharing platforms, as this is relevant to their development. As platforms offer a flexible schedule with low barriers to entry, drivers who have qualified

292 Supply and Demand Management in Ride-Sourcing Markets

to work on platforms are free to offer their preferred number of days of transport services to passengers. Moreover, although many drivers try working on platforms, some stop working after a certain period, while others continue working indefinitely. For example, UberX was launched in July 2012, and from then until late 2014, there was exponential growth in the number of active Uber drivers in the United States, demonstrating that this Uber service had provided new opportunities to a large and growing segment of the labour force.2 In addition, predictors of the growth in the number of Uber drivers across cities, such as city population, provide insights into the forces underlying Uber’s success (Hall and Krueger, 2018). A two-sided matching market must be designed to provide sufficient thickness (Roth, 2018). This can be achieved in a ride-sourcing market by having a sufficiently large number of active drivers (i.e., a high extensive margin), as this ensures that a satisfactory service is provided to passengers, such as a high matching efficiency and a short waiting time before pick-up, resulting in a low proportion of unsatisfied passenger requests. The distinction between the extensive and intensive margins of labour supply has long been recognised in micro-econometric studies (e.g., Blundell et al., 2011). As mentioned, standard labour economics theory holds that income rates exert two distinct effects on labour supply; that is, income rates influence intensive margins and extensive margins (Cahuc et al., 2014). It follows that by splitting the labour supply behaviour of drivers into participation in work and the intensity of work they supply, we can obtain a comprehensive understanding of the effect of hourly income rates on total labour supply. We recognise that micro-data-based estimates of the elasticity of labour supply can be too low, due to a failure to account for the extensive margin (i.e., drivers’ participation decisions) (Rogerson, 1988). Therefore, in the next section, we devise a method to model drivers’ labour-force participation decisions.

12.4 Modelling endogeneity of income rates and selfselected participation in the labour force 12.4.1 Methodological implications of self-selection and endogeneity As drivers self-select a participation strategy that is suitable for them, an empirical model that does not account for sample self-selection bias could be misspecified, and thus elasticity estimates and normative conclusions drawn from an analysis using this model could be misleading. Accordingly, motivated by the income-selectivity bias problem introduced by Gronau (1974), we 2. Active drivers were defined as those who provided at least four trips to passengers in a given month (Hall and Krueger, 2018).

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model drivers’ self-selection using the sample selection model presented by Heckman (1976), describe drivers’ two-step labour supply decisions along the extensive and intensive margins and adopt IVs for the endogenous variables.

12.4.1.1 Modelling self-selected participation in the labour force Drivers considering working on ride-sourcing platforms face a two-step decision: (1) whether to participate in the platform, and if so, (2) how many hours to work. Consider a random sample of I drivers. Let Wit denote driver i’s reservation income rate on a given day t, with this rate being unobservable by the platform. Let Wit denote an index of labour force attachment, which in the absence of fixed costs of work may be interpreted as the difference between income rates Wit and Wit0 , i.e., Wit ¼ Wit
Wit0 (Gronau, 1974; Heckman, 1976). According to the analysis of labour-force participation by Gronau (1974), a job seeker uses an income rate Wit to decide which income offers to accept and which to reject. Wit0 denotes a job seeker’s cost for time spent at home, and an income offer less than Wit0 is rejected by the job seeker. In our problem, a driver does not participate in the platform on days when the market hourly income rate Wit is less than Wit0 , i.e., Wit  Wit0 . Wit is unobservable by the platform, but both driver i’s participation decision Yit ˛ f0; 1g and the number of hours worked Hit by the driver on day t are observable by the platform. We specify driver i’s participation decision using an indicator function, where 1 ¼ ‘participation’ and 0 ¼ ‘no participation’, and thus the labour supply of driver i on day t is represented by the following equations:   Yit ¼ II Wit > 0 ; (12.5) Wit ¼ X1i b1 þ єit ;

(12.6)

Hit ¼ X2i b2 þ εit ;

(12.7)

where X1i and X2i are vectors of independent variables, including hourly income rate Wit ; b1 and b2 are vectors of parameters; and єit and εit are error terms that capture the effects that cannot be identified or measured. We assume that E½єit  ¼ 0 and E½εit  ¼ 0. The covariance of random terms for driver i is E½єit εit  ¼ s12 . As a consequence of a random sampling scheme, we have E½єit εi0 t  ¼ s12 for different drivers isi0 (Heckman, 1979). As data for the number of working hours Hit are available only if the corresponding participation decision Yit ¼ 1, the population regression function of the number of hours worked for the subsample of available data is E½Hit jX2i ; participation decision strategy ¼ X2i b2 þ E½εit jparticipation decision strategy ¼ X2i b2 þ E½εit jX2i єit >
X1i b1 .

(12.8)

294 Supply and Demand Management in Ride-Sourcing Markets

If εit is independent of єit , this implies that the driver’s participation decision is independent of the number of hours worked by the driver. Thus, the conditional mean of εit is zero and Eq. (12.8) reduces to E½Hit jX2i ; participation decision strategy ¼ X2i b2 þ E½εit j participation decision strategy ¼ X2i b2 .

(12.9)

For example, due to their contracting requirements, New York City taxi drivers work almost every day (Farber, 2015). This means that the participation decision strategy is irrelevant to their decisions, as these are only evidenced by the number of hours they work each day, which depends on the income rate fluctuation. In contrast, drivers working as freelancers on ridesourcing platforms can have more flexible schedules, and certain factors affect both their participation decision and the number of hours they work. For example, an exogenous shock that reduces a driver’s cost of time at home, e.g., a negative shock to household employment (Stafford, 2015), can render a driver more likely to participate in the platform and more likely to work longer hours for any given hourly income rate. Therefore, the conditional mean of εit is non-zero and the regression estimates for the parameters in Eq. (12.7) fit onto the non-randomly selected samples. Consequently, omitting the conditional mean as a regressor generates sample selection bias. To solve this sample self-selection problem, we construct a Heckman sample-selection model to correct selection bias (Heckman, 1976; Cameron and Trivedi, 2005). Assume that єit and εit follow a bivariate normal distri  bution N 0; 0; s2є ; s2ε ; r . In practice, although labour force attachment Wit is unobservable, the parameters of the probability that a driver chooses to participate in a platform can be estimated by performing a probit analysis of the full sample. The specification of a probit model to describe the participation decision is P½Yit ¼ 1jXi  ¼ FðX1i ; bi Þ;

(12.10)

where the standard deviation sє of єit is normalised to 1; and F and f are the cumulative density function and probability density function of the standard normal distribution, respectively. Let Hitj represent the number of working hours of driver i on day t corresponding to the participation decision j, where j ¼ 1 represents the driver’s decision to participate in the platform and j ¼ 0 represents the driver’s decision to not participate in the platform. Then, by accounting for the non-zero covariances due to the unobservable factors that affect the participation (i.e., selection) and hours worked (i.e., outcome) equations above, the expected

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number of working hours of drivers that participate in the platform can be respecified as E½Hit1 j j ¼ 1 ¼ X2i b21 þ rsε

fðX1i b1 Þ FðX1i b1 Þ

(12.11)

Similarly, the expected number of working hours of drivers that choose not to participate in the platform can be respecified as E½Hit0 jj ¼ 0 ¼ X2i b20
rsε

fðXi aÞ . 1
FðXi aÞ

(12.12)

Eqs. (12.11) and (12.12) define the expected number of working hours as the sum of two components: common features and heterogenous features. The common features component represents the effects of independent variables that influence the number of hours worked, so its coefficients for drivers who participate in the platform are different from its coefficients for drivers who do not participate in the platform. The heterogeneous features component, which is different in each equation, represents the influence of the expected value of the error terms εit1 and εit0 , given the truncation in єit . The truncated density of the error term єi owing to drivers’ selection is known as the inverse Mills ratio and is calculated from the selection equation as fðX1i b1 Þ=FðX1i b1 Þ for drivers who participate and fðX1i b1 Þ=ð1
FðX1i b1 ÞÞ for drivers who do not participate. These inverse Mills ratio terms capture the role of unobservable contextual factors that affect the participation decision and the working-hour decision, given the self-selected nature of the participation decision. rs 0 indicates the presence of endogeneity, whereas r ¼ 0 indicates that the participation decision is exogenous and independent of the working-hour decision. The outcome from Eq. (12.11) for drivers who choose to participate in the platform implies that when drivers are randomly assigned to participate, their average number of working hours is X2i b2l . However, when drivers’ participation is self-selected, their average number of working hours is given by X2i b2l þ rsε fðX1i b1 Þ =FðX1i b1 Þ. If rsε fðX1i b1 Þ =FðX1i b1 Þi0, we have E½Hit jj ¼ 1iX2i b2l , which means that the number of working hours of drivers who choose to participate in the platform is greater than the average number of working hours of drivers. This indicates that unobservable characteristics are not only influencing drivers’ decisions to participate in the platform but are also causing their number of working hours to be greater than the average number of working hours. The outcome from Eq. (12.12) for drivers who choose to not participate in the platform suggests that when drivers are randomly assigned to participate, their average number of working hours is X2i b20 . However, when drivers’

296 Supply and Demand Management in Ride-Sourcing Markets

non-participation is self-selected, their average number of working hours is given by X2i b20
rsε fðXi aÞ =ð1
FðXi aÞÞ. When rsε fðXi aÞ =ð1
FðXi aÞÞi 0, we have E½Hit0 jj ¼ 0hX2i b20 , which suggests that the presence of unobservable characteristics enables the number of working hours of nonparticipants to be less than what their average number of working hours would have been if they had been participants.

12.4.1.2 Modelling the endogeneity of the hourly income rate The total service time supplied by each driver on each day depends on each driver’s participation decision and working-hour decision. The hourly income rate is determined by the aggregate service time supplied by all participating drivers, meaning that there is reverse causality between the number of working hours and the hourly income rate. As both supply and demand curves shift over time, a simple regression of the number of working hours on the hourly income rate will not yield a generally consistent estimate of the labour supply pattern. In addition, although we can use our historical dataset to adjust for heterogeneous driver characteristics in our model, there may be unobservable factors that influence both the income rates and labour supply. Ignoring this potential source of endogeneity could result in biased inferences being obtained. Consequently, IV-type estimation must be performed to obtain consistent estimates, as this ‘purifies’ the independent variable by removing its nonstochastic element. In principle, an IV should predict the independent variable of interest and be uncorrelated with the dependent variable. However, although this can be an effective way to remove endogeneity bias, problems can arise if the IV is not strongly correlated with the endogenous variable. Specifically, if an instrument is weak, misleading confidence intervals may be obtained by using the asymptotic distribution in a 2SLS regression analysis, such that the IV-based estimates will be biased in the same way as ordinary least squares (OLS)-based estimates (Bound et al., 1995). Additionally, IV estimates based on weak instruments are highly sensitive to small violations of the exclusion restriction (Small and Rosenbaum, 2008). Valid IVs are typically derived from natural experiments or exogenous shocks. In our problem, we instrument the endogenous variable hourly income rate by exploiting the fact that some income adjustments could be suitable IVs because they are exogenously related to hourly income rate and unrelated to the number of hours worked. In addition, factors that alter exogenous demand, such as the number of potential passenger requests, could be used as instruments for observed hourly income rate. In Section 12.6, we discuss our choice of IVs for our setting. Given valid IVs, a 2SLS regression analysis can be used to jointly estimate the endogenous hourly income rate with its instruments and all of the other

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control variables (stage 1), and the main dependent variable hours worked with the purified independent variable and all of the other control variables (stage 2) (Durbin, 1954). Consider the following linear regression model with the endogenous regressor hourly income rate: Hit ¼ X2i b2 þ Wit b2u þ εit ;

(12.13)

where Wit is correlated with εit and this correlation generates the endogeneity problem. Let Zt be an IV (or IVs as a vector) and gt be its parameter. Zt is uncorrelated with εit , i.e., CoyðZt ; εit Þ ¼ 0. In the first stage of the estimation procedure, the instrument equation is Wit ¼ X2i b2 þ Zt gt þ vt ;

(12.14)

b it where vt is a random error term. In the second stage, the predicted values W of Wit are used in Eq. (12.13), which generates the following causal regression model: Hit ¼ X2i b2 þ HatWit b2w þ εit .

(12.15)

Eq. (12.15) implies that the only basis of the relationship between hourly income rate Wit and IV Zt is the first-stage instrument equation; that is, the instrument has no effect on hours worked other than via this first-stage channel.

12.4.2 Model of labour supply elasticity on a ride-sourcing platform As mentioned, drivers make decisions on whether to participate in a ridesourcing platform, and if so, how many hours to work on a given day. With reference to Eqs. (12.5)e(12.7), let random variables Y and H be a driver’s participation decision and the number of hours worked by a driver, respectively. The total labour supply S of a driver is therefore S ¼ E½Y  H ¼ P½Y ¼ 1E½Y  HjY ¼ 1 þ P½Y ¼ 0E½Y  HjY ¼ 0;

(12.16)

where P½Y ¼ 0E½Y HjY ¼ 0. Eq. (12.16) shows that the total number of hours worked by a driver can be decomposed into an extensive component P½Y ¼ 1 and an intensive component E½Y HjY ¼ 1, where the extensive margin of labour supply is defined as the number of days that the driver chooses to participate in the platform. This decomposition was also used by Blundell et al. (2013).

298 Supply and Demand Management in Ride-Sourcing Markets

Let W denote the income rate. The aggregate income elasticity of total labour supply h can be defined as h¼ ¼

vS=S vW=W vP½Y ¼ 1=P½Y ¼ 1 vE½HjY ¼ 1=E½HjY ¼ 1 þ . vW=W vW=W

(12.17)

Eq. (12.17) shows that h is the sum of extensive margin (participation) elasticity and intensive margin (the number of hours worked) elasticity. To consider the two-step decisions of drivers and measure the two parts of hourly income elasticity, we follow the tradition of measuring labour supply elasticity from coefficients in logarithmic versions of equations for estimating income rate. Therefore, we focus on the logarithmic version of Eqs. (12.10)e(12.12) by using log Hit instead of Hit and log Wit instead of Wit . The corresponding regression models are specified as follows: P½Yit ¼ 1jXi  ¼ FðX1i b1 þ b1u log Wit Þ; E½log Hit1 jj ¼ 1 ¼ X2i b21 þ b2u1 log Wit þ rsε E½log Hit0 jj ¼ 0 ¼ X2i b20 þ b2u0 log Wit
rsε

fðX1i b1 þ b1u log Wit Þ . FðX1i b1 þ b1u log Wit Þ

fðX1i b1 þ b1u log Wit Þ . 1
FðX1i b1 þ b1u log Wit Þ (12.18)

Based on the above equations, the elasticity at the extensive margin hp is measured as follows3: hp : ¼

vP½Y ¼ 1=P½Y ¼ 1 fðX1i b1 þ b1u log Wit Þ ¼ b1u . vW=W FðX1i b1 þ b1u log Wit Þ

(12.19)

When drivers choose to participate in the platform, the elasticity at the intensive margin hh1 is measured as follows: hh1 ¼ b2u1
b1u rsε

ðX1i b1 þ b1u log Wit ÞfðX1i b1 þ b1u log Wit ÞFðX1i b1 þ b1u log Wit Þ F2 ðX1i b1 þ b1u log Wit Þ


b1u rsε

f2 ðX1i b1 þ b1u log Wit Þ . F2 ðX1i b1 þ b1u log Wit Þ

(12.20)

3. Chen and Sheldon (2016) studied the labour supply on Uber’s platform and measured labour supply elasticity using an IV model without considering extensive margin (participation) elasticity. Consequently, due to sample self-selection bias, when the two random terms were correlated, their estimated labour supply elasticity on the intensive margin was biased.

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When drivers choose to not participate in the platform, the elasticity at the intensive margin hh0 is measured as follows: hh0 ¼ b2u0 þ b1u rsε


b1u rsε

ðX1i b1 þ b1u log Wit ÞfðX1i b1 þ b1u log Wit ÞFðX1i b1 þ b1u log Wit Þ 2

F ðX1i b1 þ b1u log Wit Þ

f2 ðX1i b1 þ b1u log Wit Þ 2

F ðX1i b1 þ b1u log Wit Þ

.

(12.21)

Based on the above formulations, h is measured as follows: h ¼ hp þ hh1 ¼ b1u

fðX1i b1 þ b1u log Wit Þ þ b2u1
b1u rsε FðX1i b1 þ b1u log Wit Þ

ðX1i b1 þ b1u log Wit ÞfðX1i b1 þ b1u log Wit ÞFðX1i b1 þ b1u log Wit Þ F2 ðX1i b1 þ b1u log Wit Þ
b1u rsε

f2 ðX1i b1 þ b1u log Wit Þ . F2 ðX1i b1 þ b1u log Wit Þ

(12.22)

In the presence of sample self-selection and endogeneity, estimates of labour supply elasticity on ride-sourcing platforms may be biased. For example, as drivers make participation decisions by comparing hourly income rates with their cost of time at home, any shock that affects these factors will induce a correlation between the error terms in Eqs. (12.6) and (12.7). As a result, if the correlation is non-zero and the income rate fluctuation significantly affects drivers’ participation decision (i.e., rs0 and b1u s0), Eqs. (12.20) and (12.22) dictate that the elasticity at the intensive margin for participating drivers will be biased if the sample self-selection problem is not dealt with. In addition, hourly income rates Wit are calculated by dividing driver i’s total daily income on day t by driver i’s total hours worked on that day Hit, and the number of hours worked appears in reciprocal form on the right-hand side of Eq. (12.7). This means that any error in the measurement of the number of hours worked will generate a negative correlation between the number of hours worked and the hourly income rate, such that the elasticity at the intensive margin hh1 will be downward biased. Finally, hourly income rates and the number of hours worked conditional on hourly income rates may be jointly determined by the same factors. If these factors are unobservable, the coefficient b2u will be biased due to simultaneity, which will lead to biased estimates of elasticity along the intensive margin.

300 Supply and Demand Management in Ride-Sourcing Markets

In the next section, we apply an IV approach based on a large-scale natural experiment to address the endogeneity of hourly income rates and reduce the influence of measurement error on the number of hours worked.

12.5 Research design 12.5.1 Research context The data we use are obtained from Didi Chuxing, the largest mobile transportation platform in China. This platform receives approximately 30 million ride requests each day and hosted 21 million registered drivers across 400 cities in China by 2018. It offers a diverse range of transportation services via one mobile application (app), such as Premier, Express, Hitch, Taxi and Chauffeur. In our study, we consider the Express service, in which drivers act as independent transportation service providers who determine their own work schedules. We focus on the Express service for several reasons, as detailed below. First, the Express service is the largest service on the platform, which enables us to collect cumulative data to study labour supply behaviour on the platform. Second, the Express service became the dominant ride-sourcing service in China during the sample period, which distinguishes Didi Chuxing from other domestic platforms. Specifically, the Express service was introduced in May 2015 as an economical version of a private car-hailing service, and in August 2016 Didi Chuxing acquired its biggest competitor in China. Thus, platform competition would have had little impact on the variation in labour supply during the sample period. Third, to control the nationwide budgets for supply costs, the central headquarters team of Didi Chuxing (i.e., rather than the local team in the city from which our dataset is obtained) adjusted the driver incentive income multiplier during the sample period.4 This adjustment was thus independent of and exogenous to the demand and supply status in the local market.

12.5.2 Large-scale natural experiment 12.5.2.1 Income multiplier Ride-sourcing platforms offer various programmes to drivers to encourage them to participate in platforms and thus meet passenger demand. One of the most popular and effective programmes uses an income multiplier (also called an income accelerator), which increases a driver’s trip fare by a certain multiplier for all trips within a specified region during a specified period. For example, let the base fare of an order be P, the dynamic pricing factor a  1, 4. In contrast, before the sample period and the natural experiment, the income multipliers were determined by the local team.

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the commission rate b ˛ ð0; 1Þ and the income multiplier d  1. In this case, the total income for a driver from an order is given by Pð1
bÞ þ P maxða
1; d
1Þ;

(12.23)

where the first component is the income minus a commission fee and the second component is the maximum of the dynamic price and the income multiplier. In practice, a platform charges its drivers a commission fee of approximately 20%, i.e., b ¼ 20%, and the fare P is normally calculated from the base fare, cost per minute  time in ride, cost per kilometer  ride distance and booking fee (if any). All of these parameters are fixed during the sample period. In this context, the income multiplier is reminiscent of dynamic pricing, a well-known tool; however, the former differs from the latter in the following two ways. First, the fluctuation in dynamic pricing factor a changes the price on the passenger’s side and the income on the driver’s side, whereas the income multiplier only changes the income on the driver’s side and thus has little direct influence on the number of passenger requests. Second, the dynamic pricing factor a is contingent on the market supplyedemand condition and dynamically changes in real time, whereas the income multiplier d is set in advance and subsequently communicated to all drivers. Fig. 12.1 shows an

FIGURE 12.1 Example of an income multiplier.

302 Supply and Demand Management in Ride-Sourcing Markets

example of the income multiplier on a single day across five time windows, including the peak periods of 6:30e8:30 a.m. and 5:00e6:30 p.m.

12.5.2.2 Exogenous shocks in natural experiments During our sample period, which was from 13 March to 21 May 2017, the central headquarters team of the platform controlled their nationwide budget for supply costs by adjusting drivers’ income rate using income multipliers. They did this by increasing or decreasing d, which is independent of and exogenous to the demand and supply status in the local market, and no local information was considered to differentiate the income multiplier between cities. In our dataset, the maximum value of the variable income multiplier is 1.55, the minimum value is 1.1 and the standard deviation is 0.10, which represents good inter-day variation in income multiplier. This policy change created a natural experiment that allows us to assess how changes in income rates affected labour supply along both the extensive and intensive margins of drivers. As mentioned, the income multipliers were offered only to drivers and no similar incentive was offered to passengers, such that there was little direct impact on passenger demand. In addition, the income multipliers were subject to nationwide budget controls and independent of local market conditions, and all new daily income multipliers were shared in advance with drivers on the morning of each day. The income multipliers were thus an exogenous shock that affected drivers’ labour supply only by changing their income rates, which partly reduces bias due to the endogeneity problem in our analysis of the labour supply on the platform. 12.5.3 Data description 12.5.3.1 Variable description Our research design allows us to study the effects of income rate changes on the labour supply in the presence of driver heterogeneity.5 The temporal unit in our analysis is a day, but the conceptualisation of a driver’s workday is challenging, as ridesharing platforms lack organisational constraints on time. Consequently, drivers can design flexible work schedules for themselves and thus their labour supply decisions need not result in their having a typical workday from 12:00 a.m. to 11:59 p.m. In informal conversations with drivers, we learned that some began working at 7:00 a.m. and stopped working at 8:00 p.m., whereas others began working at 3:00 p.m. and stopped working at 4:00 a.m. the next day (to avoid fierce competition in the morning and earn more in the evening). Therefore, an approach that analyses drivers’ working hours per calendar day requires a ‘workday’ that spans midnight and consists of two

5. The results in this study are obtained by analysing proprietary data that cannot be released. However, where appropriate, we indicate levels of significance.

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shifts. In our analysis, we adopt the approach of Chen and Sheldon (2016): we define a shift as the cluster of all trip and online service activities that occur without a break of more than 4 h. Hence, a period of driver inactivity greater than 4 h marks the beginning of a new shift in our data.6 Next, we count the number of shifts at each time (in min) and find that 4:00 a.m. was the time with the fewest shifts, i.e., when the fewest drivers were on the road. Finally, by considering the daily adjustment of income multipliers, we define a ‘day’ as the collection of all time during the 24-h period from 4:00 a.m. 1 day to 3:59 a.m. the next day. A driver’s hourly income rate is defined as the ratio of the driver’s total daily income to the driver’s daily number of hours worked. The underlying assumption is that there is an hourly income rate characterising a working day that a driver can use to make a decision on the number of hours to work. Camerer et al. (1997) and Sheldon (2016) suggested conducting an autocorrelation analysis to test this assumption. If the autocorrelation is negative, drivers may stop working early when the hourly income rate is high, because high hourly income rates are likely to be followed by low-wage hours. This labour supply behaviour could lead to the incorrect inference that drivers were making decisions based on income targets (Camerer et al., 1997). Therefore, it is important to understand the time-series properties of the hourly income rates within a day to understand what drivers may infer from income rates in the current hour about possible income rates during the remainder of a day (Farber, 2005). We therefore conduct an autocorrelation analysis on intra-day income rates for each clock hour. The results show that the first-order autocorrelation is positive (0.602) and significantly different from zero, suggesting that when the hourly income rates are high they will probably continue to be high in the subsequent period, such that intra-shift hourly earnings are relatively stable (Sheldon, 2016). To capture observed driver heterogeneity, the vector X of control variables includes two time-invariant driver-specific variables: age (agei ) and gender (genderi ). To account for the learning-by-doing effect and capture driverspecific heterogeneity in experience, the vector X includes work experience (experienceit ), measured as the number of days between the day a driver qualifies to work on a platform and current day t. To capture temporal variation in preference for driving, X includes six dummies for days of the week (i.e., Mondayt ¼ 1 if day t is a Monday; otherwise Mondayt ¼ 0). Finally, the vector X includes a number of weather-related variables, such as the highest temperature (temperaturet ), the highest precipitation within 24 h (intensityt ) and the 2.5-mm particulate matter (PM2.5) concentration (pm25t ), with the latter serving as an air quality index.

6. Farber (2015) defined any gap of more than 6 h (more than 360 min) between trips as marking the end of one shift and the beginning of the next shift.

304 Supply and Demand Management in Ride-Sourcing Markets

12.5.4 Driver classification along the extensive and intensive margins Based on the evidence regarding driver heterogeneity, we present a twodimensional scheme to classify drivers (Table 12.2). As mentioned, standard labour economics theory holds that income rates influence two margins and thus exert two distinct effects on labour supply: they influence the intensive margin and thereby affect the number of hours worked, and they influence the extensive margin and thereby affect the decision to participate in the labour market (Cahuc et al., 2014). Therefore, we classify drivers based on the total number of days they worked (the extensive margin) and the average daily hours they worked (the intensive margin) in the previous 30 working days.7

12.5.5 Empirical analysis 12.5.5.1 Basic model We first use an OLS regression model, as given in Eq. (12.24), to examine the working hours of drivers at different hourly income levels. Camerer et al. (1997) estimated the regression and interpreted b2;1 as the elasticity of labour supply. They showed that estimates of working-hour elasticity are strongly negative and explained that drivers set a daily income target and stopped working once they reached this target.

TABLE 12.2 Driver classification along the extensive and intensive margins. Percentage of observations in each group Working days within 30 days

Working hours per day High

Intermediate

Low

High

(I) 15.919%

(IV) 20.147%

(VII) 10.649%

Intermediate

(II) 1.592%

(V) 7.142%

(VIII) 9.855%

Low

(III) 1.095%

(VI) 4.612%

(IX) 28.989%

7. Cut-off values for classifying drivers are anonymised.

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OLS: logðWorking hoursit Þ ¼ b2;0 þ b2;1 logðincome rateit Þ þ b2;2 agei þ b2;3 genderi þ b2;4 experienceit þ b2;5 temperaturet þ b2;6 intensityt þ b2;7 pm25t þ b2;8 Mondayt þ b2;9 Tuesdayt þ b2;10 Wednesdayt þ

(12.24)

b2;11 Thursdayt þ b2;12 Fridayt þ b2;13 Saturdayt þ εit

However, it has been noted that there are two major problems with this OLS model: a conceptual problem and an econometric problem. Regarding the first type of problem, Farber (2005) noted that Eq. (12.24) can only estimate labour supply elasticity b2;1 if there is a significant exogenous inter-day variation in average income. However, in an empirical analysis, Farber (2005) found no significant exogenous inter-day variation in average income and no autocorrelation in hourly income rate on a particular day. Second, Camerer et al. (1997) highlighted that any measurement error in Working hoursit biased the estimate of b2;1 downward. Accordingly, they instrumented the variable income rateit with the average income rate for other drivers on the same calendar date. Farber (2005) also suggested that as there may be day-specific factors that correlate with drivers’ hourly income rates and drivers’ working hours conditional on their hourly income rate, other drivers’ average hourly income rates could not be employed as IVs to estimate the causal effect of income rates on the number of working hours. In the natural experiment during our sample period, the platform publicly announced early every morning the daily adjustments it had made to the income multiplier to control its budget. This created exogenous inter-day variation in drivers’ average hourly income rates and thus helped us to design our models, which we discuss in section 12.5.5.3.

12.5.5.2 Consideration of potential sample selection bias As discussed in Section 12.4.1.1, if some factors affect both of the decisions made by drivers regarding a platformethat is, the decision on participation and the decision on the number of hours to workethen the OLS estimates of parameters in Eq. (12.24) fit onto the non-randomly selected samples, which leads to sample self-selection bias. The appropriate econometric technique to deal with such a scenario is to model self-selection of labour force participation as presented in Section 12.4.1.1. Accordingly, we add another selection equation (Eq. 12.25) to Eq. (12.24) to describe drivers’ participation decision, which affords the outcome equation below (Eq. 12.26). Our Heckman two-step model specification is therefore also shown below.

306 Supply and Demand Management in Ride-Sourcing Markets

Selection Equation:  P½participateit ¼ 1 ¼ F b1;0 þ b1;1 logðavg last7 income rateit Þþ b1;2 agei þ b1;3 genderi þ b1;4 experienceit þ b1;5 temperaturet þ b1;6 intensityt þ b1;7 pm25t þ b1;8 Mondayt þ b1;9 Tuesdayt þ b1;10 Wednesdayt þ  b2;11 Thursdayt þ b2;12 Fridayt þ b2;13 Saturdayt

(12.25)

Outcome Equation: logðWorking hoursit Þ ¼ b2;0 þ b2;1 logðincome rateit Þþ b2;2 agei þ b2;3 genderi þ b2;4 experienceit þ b2;5 temperaturet þ b2;6 intensityt þ b2;7 pm25t þ b2;8 Mondayt þ b2;9 Tuesdayt þ b2;10 Wednesdayt þ b2;11 Thursdayt þ b2;12 Fridayt þ b2;13 Saturdayt þ εit (12.26)

Thus, in this study, we estimate the income rate elasticities of daily participation probabilities and daily number of hours worked using the Heckman sample selection two-step model described in Section 12.4.2 that controls for self-selection bias. In Eq. (12.26), the variable income rateit is defined as the ratio of the observed total daily income that a driver i receives on a working day t to the daily number of hours worked by driver i. Ideally, Eq. (12.25) should also estimate the probit model of participation using income rateit ; however, driver i does not receive income on the day of making the participation decision, i.e., day t, and it is impossible to observe income rateit if driver i does not participate on day t. Therefore, we estimate the probit model of participation by replacing the observed hourly income rate with avg last7 income rateit , which is defined as the average hourly income rate of driver i over the past seven working days prior to day t that driver i participated in the platform. These days may be within a single week (for fulltime drivers) or may fall within multiple weeks (for part-time drivers). The mean of the relative difference between avg last7 income rateit and income rateit ranges from
5% to 3% across participating drivers in the nine driver groups and across days during the sample period, which indicates that avg last7 income rateit is a good approximation of drivers’ perceptions of the hourly income rate.

12.5.5.3 Identification of the outcome equation Eq. (12.25) represents a situation in which drivers make their participation decision after becoming aware of average income rates during the past seven working days. However, Eq. (12.26) represents a situation in which drivers do not make a decision on their number of working hours immediately after obtaining fixed income rates, which accounts for the fact that working hours also have effects on income rates. This simultaneity bias creates an

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endogeneity problem in model identification. Moreover, we analyse labour supply using observed hourly income rates that are calculated by dividing total daily income by the total number of hours drivers work. This means that any error in the measurement of the number of hours worked would generate a negative correlation between the number of hours worked and hourly income rate and thus result in a downward-biased elasticity at the intensive margin. To address the two problems described above, we adopt the IV 2SLS approach introduced in Section 12.4.1.2 as our identification strategy. A valid instrument for this approach should be (1) correlated with the endogenous regressor (a relevance condition) and (2) uncorrelated with the error term (an exclusion-restriction condition) and thus influence the outcome (i.e., the number of hours worked) only via the endogenous variable (i.e., the hourly income rate). We employ two types of instrument. First, given the exogenous shock created by the natural experiment in our study period, we employ the variable income multipliert . As the income multiplier is time-varying, we use the daily income multiplier that affects the greatest number of drivers as an IV, which is the multiplier during the evening peak period (5:00e6:29 p.m.) on day t. We do so for the following two reasons. (1) The multipliers during the morning peak period (7:00e8:29 a.m.) and the evening peak period (5:00e6:29 p.m.) are equal on most days and their difference is always less than 0.1 in the dataset. (2) We find that the proportions of the number of hours worked by drivers at 5:00e6:29 p.m. are very similar for all nine driver groups (i.e., 13.74%e16.59%). In addition, the standard deviation of the number of hours worked by drivers during peak periods is low for each driver group (i.e., 0.86e1.60), which indicates that there is relative intra-group homogeneity in the driver groups. Therefore, although the treatment effect of the income multiplier on drivers’ labour supply is heterogeneous for each driver, we focus on the average effect for each group. With respect to the total income for a driver from an order, Eq. (12.23) indicates that income multipliert reflects the effect of exogenous driver income on the number of hours worked by drivers, thereby satisfying the relevance condition. Admittedly, this IV estimates the average causal effect of treatment on the subpopulation of drivers who can be induced to change the number of hours they work by a change in the instrument (these drivers are denoted ‘compliers’). According to the local average treatment effect theorem (Imbens and Angrist, 1994), the identified treatment effect is an average for these compliers. It follows that without any further assumptions, the local average treatment effect does not apply to those who never work during peak hours when the income multiplier incentive is offered (Angrist and Pischke, 2008). Second, we follow Oettinger (1999) by supplementing our analysis with another IV: the logarithm of the number of passenger requests on day t (i.e. requestt ), which serves as a shifter of demand for drivers. Traditional taxi data cannot track total potential demand as they can only record satisfied demand;

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however, the total potential demand can be determined from ride-sourcing platform data. In this case, the use of the number of passenger requests as an IV does not represent the number of passengers who request a ride. Rather, it represents the real potential demand, which is defined as the number of passengers who use the platform’s mobile app to indicate their origins and destinations, and then receive an estimated waiting time, based on which they decide whether to request a ride. This IV was also used by Castillo et al. (2018) as demand shifted: the endogenous pick-up time determined in equilibrium was instrumented using the number of passengers who used the Uber app, as market outcomes do not influence this app-usage decision. This is the same logic that underpins our model. Although drivers may be aware of the demand peaks and surge prices during certain periods of the day, it is difficult for them to predict the total passenger requests and income levels on a new day. In addition, the temporal patterns of intra-day demand may be similar across days, which means that an awareness of the intra-day patterns of peak/non-peak periods cannot help drivers to gain much valuable information on the income level in a new day. Thus, in this work, we do not study the intra-day shifts in the number of hours worked by drivers. Instead, we recognise that the number of passenger requests affects the working-hour decisions of drivers by affecting their hourly income rate. Thus, as the IVs income multipliert and requestt are correlated with the hourly income rate but are not directly correlated with the measurement of the number of hours worked, this helps us to reduce the influence of error in the measurement of the number of hours worked by drivers. In Section 12.6.1, we use the two IVs and the 2SLS estimation procedure introduced in Section 12.4.1.2. We also provide results of statistical tests that confirm these IVs are valid.

12.6 Results and discussion In this section, we discuss our results and summarise our important findings. We show that both sample selection bias and endogeneity bias exist, which confirms that some unobservable factors affect drivers’ decisions regarding participation and the number of hours to work. Given that the effect of hourly income rate on labour supply varies between different groups of drivers, we also include driver-specific fixed effects to further control for driver heterogeneity. In Section 12.6.1.1, we compare the estimates obtained from the OLS model, the Heckman two-step sample selection model without IVs, and the Heckman two-step sample selection model with IVs, and show that these results shed light on sample selection bias and endogeneity bias. In Section 12.6.1.2, we evaluate the validity of the two IVs by conducting statistical tests. In Section 12.6.2, based on the framework proposed

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in Section 12.4, we present and discuss the estimates for labour supply elasticity along both the extensive and intensive margins. In section 12.6.3, we present estimates grouped by age and gender.

12.6.1 Model estimation 12.6.1.1 Evidence of sample selection and endogeneity bias As discussed in Section 12.4.2, we jointly estimate driver participation and the number of hours worked, while accounting for sample self-selection bias and endogeneity. Tables 12.3 and 12.4 summarise the results of different model specifications for Group I and Group IX, respectively. First, we obtain OLS estimates via Eq. (12.24) without accounting for sample self-selection and endogeneity; this is similar to the OLS specification employed by Camerer et al. (1997), which directly used observed hourly income rate. The results show that the coefficients of logðincome rateÞ are positive and significantly different from zero. Second, we estimate the Heckman two-step model without IVs and with consideration of sample self-selection bias using the selection equation (Eq. 12.25) and the outcome equation (Eq. 12.26). The estimates we obtain show that the coefficients of logðavg last7 income rateÞ in Eq. (12.25) are positive, which implies that the probability of participation increases as the income rate increases. The coefficients of selectivity-corrected logðincome rateÞ in Eq. (12.26) are also positive, which is consistent with the neoclassical labour supply model. Moreover, the coefficients of invmillsRatio are significantly different from zero, which implies the existence of sample selection bias. Third, we take the endogeneity of the hourly income rates into consideration by building on the Heckman model without IVs and control the bias by instrumenting income rate with income multiplier and request. The estimates we obtain show that the coefficients of the hourly income rates logðavg last7 income rateÞ in the selection equation (Eq. 12.25) and the coefficients of the selectivity-corrected logðincome rateÞ in the outcome equation (Eq. 12.26) are positive and significant. Furthermore, we find evidence of endogeneity in the hourly income rates, i.e., in income rateit ; specifically, the results of the WueHausman test are significantly different from zero at the significance level of a ¼ 0:05. Thus, we reject the null hypothesis that the treatment variable is exogenous with respect to the number of hours worked by drivers. The coefficients of the invMillsRatio are also significantly different from zero, which confirms the existence of sample selection bias. In addition, as these coefficients are also significantly negative, this indicates that (according to Eq. 12.11) the number of working hours of drivers who decide to participate in the platform is less than the average number of working hours. Consequently, if sample selection bias is not taken into consideration when

(2) Heckman model without IVs

Variablesa

(1) OLS

Selection equation

(Intercept)

1.467 ***


0.785 ***

log(income_rate)

0.204 ***

log(avg_last7_income_rate)

Outcome equation 1.276 ***

(3) Heckman model with IVs

Selection equation 2.854 ***

0.217 *** 0.326 ***

Outcome equation

(4) Outcome equation with fixed effects

1.696 *** 0.152 ***

0.111 ***

0.310 ***

Log (request)


0.699 ***

income_multiplier


0.514 ***

Age

0.002 ***

0.012 ***

0.003 ***

0.012 ***

0.002 ***

0.001 ***

Gender

0.008 *

0.078 ***

0.018 ***

0.078 ***

0.007.

0.002

Experience


0.000 ***


0.000 **


0.000 ***


0.000 **

Temperature


0.001 ***


0.002 ***


0.001 ***

Intensity


0.015 ***

0.060 ***


0.009 *

pm25


0.000 ***


0.001 ***

Monday

0.001

Tuesday


0.007 **

0.007 0.033 ***


0.000 *


0.000.

0.000


0.001 ***


0.001 ***

0.190 ***


0.017 ***


0.012 ***


0.001 ***


0.000 ***


0.000 ***

0.002


0.119 ***


0.002


0.003


0.003


0.093 ***


0.011 ***


0.012 ***


0.000 ***

310 Supply and Demand Management in Ride-Sourcing Markets

TABLE 12.3 Estimates for group I (drivers with a high extensive margin and a high intensive margin).

0.011 ***

0.164 ***

0.030 ***

Thursday

0.013 ***

0.198 ***

0.035 ***

Friday

0.023 ***

0.210 ***

Saturday

0.022 ***

0.108 ***

invMillsRatio


0.001

0.001

0.002

0.043 ***

0.002

0.004

0.045 ***

0.208 ***

0.014 ***

0.019 ***

0.034 ***

0.192 ***

0.019 ***

0.022 ***

0.259 ***


0.092 ***

Weak instruments (P-value)

j¼k < i¼k N s:t: X > > x1i  1 and > > > > i¼1 > : x ˛ f0; 1g; ci; j ˛ f1; 2; /; Ng O  max l ði; jÞ; lD ði; jÞ xij  R; ci; j ˛ f1; 2; /; Ng

(13.3)

xij ¼ 0; ci  j; i; j ˛ f1; 2; /; Ng

(13.4)

(13.2)

where V indicates the benefit obtained by the pool-matching of one pair of passengers and is set to a value that is larger than all lminði;jÞ to enable the algorithm to pool-match as many passengers as possible. Objective (13.1) seeks to maximise the pairs of pool-matched passengers and minimise the overall vehicle routing distance for all matched passengers. Constraints (13.2) guarantee that each passenger k; ck ˛ f1; 2; /; Ng can be pool-matched with a maximum of one passenger. If the kth constraint is binding, then passenger k is pool-matched; otherwise, passenger k is not pool-matched. Constraint (13.3) ensures that the origins and destinations of the pool-matched passengers are within a matching radius R; that is, passengers i and j are only pool-matched if the distances between their origins (destinations) are less than the matching radius R (which can also be regarded as the maximum allowable matching distance). Constraints (13.4) apply to two scenarios: when i ¼ j, they ensure that passengers cannot be pool-matched with themselves; and when i < j, they ensure that only half of the decision variable matrix need to be considered, as aforementioned (Tables 13.1 and 13.2). This pool-matching strategy does not consider en-route pick-up and drop-off events, which simplifies the experimental settings and thus facilitates the repeated experimentation that is

332 Supply and Demand Management in Ride-Sourcing Markets

TABLE 13.1 Decision table for xij, which indicates whether trip i and trip j are matched. Passenger

1

2

3

/

j

/

N

1

0

x12

x13

/

x1j

/

x1N

2

0

0

x23

/

x2j

/

x2N

/

0

0

0

/

/

/

/

i

0

0

0

/

xij

/

xiN

/

0

0

0

/

/

/

/

N

0

0

0

/

0

/

0

TABLE 13.2 Cost table for lminði;jÞ, the minimum vehicle routing distance of four possible pick-upedrop-off sequences for trip i and trip j. Passenger

1

2

/

j

/

N

1

0

lminð1;2Þ

/

lminð1;jÞ

/

lminð1;NÞ

2

0

0

/

lminð2;jÞ

/

lminð2;NÞ

/

0

0

/

/

/

/

i

0

0

/

lminði;jÞ

/

lminði;NÞ

/

0

0

/

/

/

/

N

0

0

/

0

/

0

necessary to discover the underlying laws of such a system. Moreover, as highlighted by Yan et al. (2020), this type of strategy is used for pool-matching in Uber ExpressPool.2 Additionally, we evaluate the robustness of the empirical laws by conducting experiments with an alternative objective function that aims to poolmatch as many passengers as possible and also minimise their total detour distance. The optimisation problem is formulated as follows: ðP2Þ max xij

N N  X X  V Dli ði; jÞ Dlj ði; jÞ xij i¼1

(13.5)

j¼i

s.t. Eqs. (13.2)e(13.4) where Dli ði; jÞ þ Dlj ði; jÞ is the total detour distance 2. A description of Uber ExpressPool can be found at https://www.uber.com/us/en/ride/expresspool/.

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experienced by passengers i and j if they are pool-matched by the algorithm; V is the benefit of pool-matching one pair of passengers, and is set to a value that is larger than all Dli ði; jÞ þ Dlj ði; jÞ, to let the algorithm pool-match as many passengers as possible. Objective (13.5) seeks to maximise the pairs of poolmatched passengers and minimise the overall detour distance of the poolmatched passengers. Like optimisation P1, Constraints (13.2) guarantee that each passenger is pool-matched with a maximum of one passenger, while Constraints (13.3) ensure that the origins and destinations of the pool-matched passengers are within matching radius R. We investigate how the passenger detour distance and ld change with N or the spatial density of passengers by examining the process after a driver picks up paired passengers, i.e., the en-route phase shown in Fig. 13.2. The pick-up distance of a vehicle dispatched to meet the first passenger of a pooled ride, which depends on the spatial densities of passengers and idle vehicles, is also of practical importance but is a separate aspect that has been examined in multiple recent studies on ride-sourcing markets (Ke et al., 2020a; Castillo et al., 2017; Yang et al., 2020). Overall, the empirical laws governing detour distances that are identified in the work described in this chapter, together with the pick-up distance distribution formulas available in the literature, constitute an invaluable toolkit for use in ride-pooling studies.

13.3.3 Random matching without an optimisation objective To verify the generality of our findings, we also perform experiments using a random pooling strategy (i.e., the scenario denoted by P3). This means that a predefined optimisation objective function is not used, such that two passengers are paired randomly provided that both their origins and their destinations are within the maximum R.

13.3.4 Experimental settings and data description We perform a sensitivity analysis to examine the relationships between the three key measuresdP, Dl, and ld e and N. First, we set several groups of matching radii R ranging from 2 km to 5 km with a step-size of 1 km.

FIGURE 13.2 En-route phase of ride-pooling.

334 Supply and Demand Management in Ride-Sourcing Markets

(We do not start from 1 km as (i) our analysis of the data shows that the distance between two matched trips is generally greater than 1 km; and (ii) a matching radius of 1 km may result in an unrealistic solution with limited pairs of pool-matched passengers, which would not enable meaningful laws to be developed. In each experiment and for each group of matching radii, we range N from 10 to 200 with a step-size of 10. Second, for each combination of R and N, we repeat the following experiment: (1) randomly sample N passenger requests from the mobility dataset of historical trip records with origins and destinations; (2) find the optimal pool-matching strategy by solving P1 and P2, under a constraint on R; and (3) store the information (such as ld or Dl) of the pool-matched pairs of passengers in the optimal strategy. After steps (1)e(3) are repeated a sufficient number of times (4000 times in our experiments), Dl and ld are calculated by averaging the detour and routing distances of the pool-matched passengers in all experiments, respectively, while p is estimated as the ratio of the number of all pool-matched passengers to the number of sampled passengers. The abovementioned experimental procedures are conducted on three datasets: on-demand ride-sourcing trip records for Chengdu and Haikou in China, which are obtained from Didi Gaia Open Data3; and yellow taxi trip records4 for Manhattan, NYC, USA, which are retrieved from the NYC Taxi5 and Limousine Commission. Each trip-request record in each dataset contains the pick-up location (i.e., origin, identified by latitude and longitude), drop-off location (destination), and the times at which the passenger was picked up and dropped off. The basic information of the three datasets is presented in TABLE 13.3 Dataset description. City, country

Total number of trips

Chengdu, China

595,151

2016/11/6 00:00:00e2016/ 11/12 23:59:59

w85

Haikou, China

227,883

2017/5/1 00:00:00e2017/5/7 23:59:59

w70

Manhattan (New York city), USA

2,827,464

2016/3/2 00:00:00e2016/3/8 23:59:59

w59.1

Time period

Area (km2)

3. Road networks from Open Street Map: www.openstreetmap.org. 4. Website of Didi Gaiya Open data: https://outreach.didichuxing.com/research/opendata/. 5. Website of New York City taxi dataset: https://data.cityofnewyork.us/Transportation/.

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FIGURE 13.3 Spatial distribution of trip origins and destinations in various road networks; (A) Chengdu; (B) Haikou; (C) Manhattan; (note. For illustrative purposes, only 1000 trips are selected in each city).

Table 13.3, while the corresponding network structures of the three cities are illustrated in Fig. 13.3. Only trips with both origins and destinations located in the downtown regions of the three cities are used in the sampling process. In addition, the ride-sourcing datasets for Chengdu and Haikou do not include ride-pooling trips, and the trends of trip requests during 1 week are depicted in Fig. 13.4 Moreover, abnormal trip records are removed from the datasets, namely, those in which (a) the shortest distance between the origin and the destination is less than 0:1 km or greater than 50 km; (b) the trip duration, i.e., from the pick-up location to the drop-off location, is shorter than 10 s or longer than 2 h (for trips with an origin and destination both within the study area); and (c) the average speed of the trip is greater than 100 km=h (the speed limits in the urban areas of most Chinese cities are typically 30  60 km=h and the speed limit in Manhattan, NYC is less than or equal to 50 mph (w80 km/h)). For computational efficiency, the origins and destinations of the trips are assigned to their nearest nodes in the network, as this enables the Dijkstra

FIGURE 13.4 Hourly variation of trip generation in 1 week; (A) Chengdu; (B) Haikou.

336 Supply and Demand Management in Ride-Sourcing Markets

algorithm to rapidly calculate the shortest path from one point to another (e.g., from origin i to destination j or from origin i to origin j). The spatial distributions of the filtered trips and their origins and destinations in the three cities are displayed in Fig. 13.3, which shows that the network topologies and originedestination distributions of the mobility trips vary between the three cities.

13.4 Empirical laws We conduct the aforementioned experiments using each of the datasets from these three cities, which enables us to obtain the curves for the three measures vs N for various matching radii. It is generally regarded that P increases but Dl and ld decrease with N. However, few studies have investigated the properties of these ascending or decaying curves. Our experiments that solve P1, P2 and P3 (as presented in the last Section 13.3) reveal that Dl decreases linearly with N; that ld decreases logarithmically with N; and that the curve for P vs N exhibits negative-exponential saturation. As the above outcomes of P1, P2 and P3 are similar, we present the results of P1 in this section and the results of P2 and P3 in appendices FeJ and appendices KeO, respectively, to avoid redundancy. The abovementioned similarities indicate that the identified empirical laws are robust with respect to optimisation algorithms with various objective functions. The details are presented below.

13.4.1 Law of passenger detour distance Fig. 13.5 illustrates the probabilistic density distribution of (a) ld and (b) Dl in the simulations with four values of N, i.e., N ¼ 10, N ¼ 50, N ¼ 100, and N ¼ 150, and R ¼ 3 km. As N increases, the mean values of both Dl and ld decrease and the right tails of the distributions become shorter, which indicates that there are fewer long passenger detours and ld decreases. Consequently, if N in the matching pool increases, a pool-matching optimisation algorithm such as that in P1 can identify shorter detour distances. As has been reported in the literature, human mobility and many other natural, social and cognitive phenomena generally exhibit Levy flight characteristics and behaviour that follows distributions of a power law or its variants (Brockmann et al., 2006; Batty, 2008; Song et al., 2010; Liang et al., 2012). However, the fundamental mechanisms underlying such behaviour are yet to be characterised. Similarly, we acknowledge that the empirical laws identified here are useful but lack rigorous theoretical foundations. After substantial trial-and-error efforts, a handful of common functions remain for further examination in our study. As an illustration, Table 13.4 lists a few examples of these for R ¼ 4 km in Manhattan; the inverse proportional

FIGURE 13.5 Probabilistic density distribution of passengers’ average detour distance Dl and drivers’ average routing distance ld for various values of passenger demand N; (A) [drivers’ average routing distance ld ]; (B) [passengers’ average detour distance Dl].

TABLE 13.4 Empirical laws for passenger detour distance Dl in Manhattan with a matching radius of 4 km. Name

Expression

Parameter

r-Squared

Inverse proportional form devised in this worka

Dl l

1 ¼ at Nþb t

at ; bt

0.9525

Inverse logarithm

Dl l

¼b

bd ; ad ; C

0.7565

Exponential

Dl l

¼ Ae le N

A; le

0.3245

Dl l

k

B; k

0.6154

Power law

d

1 Logðad NÞþC

¼ BðNÞ

a In fact, an inverse relationship is also a power law when its exponent equals 1, because a change in one quantity results in a negative change in another.

338 Supply and Demand Management in Ride-Sourcing Markets

FIGURE  13.6 (P1) Curve fitting for passengers’ average detour distance Dl; (A) [Dl= l vs N]; (B)[l Dl vs N].

function is found to best characterise the relationship between Dl and demand in terms of an r-squared value. Next, we examine the mean values of Dl with various values of N as inputs. Surprisingly, the relationship (Fig. 13.6) between Dl and N is well-fitted by the following simple equation: Dl 1 ¼ l at N þ bt

(13.6)

where l is the average non-pooling trip distance (as aforementioned), parameter at has a dimension of time (hour/unit), and bt is a dimensionless parameter. Fig. 13.5 shows the curve fitting results for (a) Dl (denoted the l

l Dl

detour distance ratio) vs N, and (b) (the reciprocal of the detour distance ratio) vs N in Manhattan with a matching radius of R ¼ 3 km. It can be seen that l is linear with respect to N and has a positive intercept. Appendix 13.A. Dl and Appendix 13.C. provide the curve-fittings for experiments in other cities and with other values of R. We further show that the empirical law Eq. (13.6) is universal across cities and for all values of R. Table 13.5 lists the fitted parameters and goodnessesof-fit, i.e., r-squared for Eq. (13.6) in experiments with various values of R in the three cities. In most cases, particularly when R is large, the empirical law fits the relationship between Dl and N well (as shown by the reasonably high r-squared (> 90%)).

13.4.2 Law of average vehicle routing distance To identify the most applicable empirical law to describe the relationship between ld and N, we re-examine the several common functional forms in

R (km)

Chengdu

Haikou

Manhattan

1

at

bt

r-Squared

at

bt

r-Squared

at

bt

r-Squared

2

0.0089

5.2765

0.7986

0.0059

6.6671

0.5609

0.0309

3.5343

0.9944

3

0.0219

3.8355

0.9884

0.0141

4.1537

0.9354

0.0531

2.5723

0.9744

4

0.0336

3.3114

0.9559

0.0243

3.4480

0.9932

0.0646

2.2271

0.9525

5

0.0386

3.2476

0.9103

0.0347

2.9205

0.9697

0.0661

2.3196

0.9395

Some empirical laws of ride-pooling services Chapter | 13

 TABLE 13.5 Empirical law for Dl lwN-values of parameters in curve fitting.

339

340 Supply and Demand Management in Ride-Sourcing Markets

TABLE 13.6 Empirical law for vehicle routing distance l d in Manhattan with a matching radius of R ¼ 4 km. Name

Expression

Parameter

r-Square

Inverse proportional function

ld l

¼

at ; b t

0.6147

Inverse logarithm (devised in this chapter)

ld l

¼b

bd ; ad

0.9968

Exponential

ld l

¼ Ae le N

A; le

0.2457

Power law

ld l

¼ BðNÞk

B; k

0.5146

1 at Nþbt d

1 Logðad NÞ

þ1

Table 13.6. As the inverse logarithmic formula has the highest r-squared and most concise form, this formula is selected to describe the change in ld with N. Next, we explore ld in experiments with various values of N as inputs. We find that the following empirical law effectively describes the relationship between ld and N: ld 1 ¼1 þ bd logðad NÞ l

(13.7)

 where ad and bd are dimensionless parameters; and ld l is the routing distance ratio. Fig. 13.7 depicts the (a) means of the routing distance ratio ld = l, and  (b) the inverse of the routing distance ratio l ld for various values of N and for the curves fitted by Eq. (13.7). It is interesting to find that unlike the relationship between Dl and N, there is not a linear relationship between ld = l and  N; rather, ld l increases with the logarithm of N. Appendix 13.B. and

FIGURE  13.7 (P1) Curve fitting for drivers’ average routing distance ld ; (A) [ld = l vs N]; (B) [l ld vs N].

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Appendix 13.D. contain the figures depicting the relationships between ld and N in the three cities and with various values of R. As shown by the calibrated  formula, ld l approaches 1 as N approaches infinity. This is consistent with the fact that the routing distance approaches the trip distance as the number of waiting passengers in the matching pool approaches infinity. This means that the platform can pool-match passengers with extremely close origins and destinations, such that drivers have few detours to make when picking up and dropping off a second passenger. Through extensive experiments, we also show that the empirical law described by Eq. (13.7) in most cases effectively represents the relationship between ld and N with various values of R (particularly for high values of R) in the three cities. The experiments’ fitted parameters and r-squared values are shown in Table 13.7, where the fairly high r-squared values represent the best fits.

13.4.3 Law of pool-matching probability Finally, we investigate p with various values of N as inputs. We find that the following empirical law can effectively represent the relationship between p and N: p ¼ 1 z expðgNÞ

(13.8)

where g is a parameter with a dimension of 1=unit and z is a positive parameter (0 < z  1) related to the network structure, R, and passengers’ and drivers’ distributions. In particular, if z ¼ 1, Eq. (13.8) reduces to the formula identified in the numerical studies of Yan et al. (2020), i.e., p ¼ 1 expðgqfÞ, where q is the arrival rate of passengers, and f is the length of the batch window. In a batch-matching setting where a platform assigns drivers to all waiting passengers at the end of each batch, qf should be the number of waiting passengers in the matching pool when the platform executes a match. Moreover, Yan et al. (2020) assumed that drivers are required to pick up two passengers at the point intermediate between their respective origins and drop them off them at the point intermediate between their respective destinations. Our formula lacks these strict requirements and is thus a more general form of the formula of Yan et al. (2020). Consequently, compared with the formula of Yan et al. (2020), our formula is more effective for characterising low-demand situations in which there is a certain possibility, dependent on z, that passengers will be matched. Specifically, our formula allows different cities to have diverse values of z, whereas the formula of Yan et al. (2020) assumes that all cities have the same value of z. Fig. 13.8 demonstrates the variation in P in Chengdu, Haikou, and Manhattan for various values of N with R ¼ 3 km, and the corresponding fitted curve (see Appendix 13.L. contains the figures for experiments in these cities with values of R). It can be seen that the ascending curve of P with N resembles a saturation curve that initially increases rapidly and then increases slowly.

[c]2* R (km)

Chengdu

Haikou

Manhattan

ad

bd

r-Squared

ad

bd

r-Squared

ad

bd

r-Squared

2

5Eþ12

0.0197

0.9524

2Eþ25

0.0112

0.6838

3817.07

0.0497

0.9548

3

2Eþ04

0.0445

0.9671

5Eþ06

0.0321

0.9770

80.8712

0.0713

0.9933

4

530.25

0.0596

0.9873

7357

0.0471

0.9465

31.1548

0.0802

0.9968

5

267.03

0.0639

0.9985

319

0.0615

0.9873

28.8668

0.0812

0.9972

342 Supply and Demand Management in Ride-Sourcing Markets

 TABLE 13.7 Empirical law l d lwN for values of parameters in curve fitting.

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FIGURE 13.8 (P1) pool-matching probability P vs passenger demand N; (A) [Chengdu e p vs N]; (B) [Haikou e p vs N]; (C) [Manhattan e p vs N].

The empirical law defined by Eq. (13.8) also can in some cases effectively describe the relationship between p and N with various values of R in the three cities. Specifically, Table 13.8 shows the fitted parameters and r-squared for these results obtained by applying Eq. (13.8), demonstrating that when R is not extremely large, the relationship between P and N is well characterised (r-squared >90%) by Eq. (13.8). In contrast, when R is relatively large (e.g., 5 km), this relationship is not as well characterised (r squared 0 and vM=vN  1 (as shown in Appendix 13.P.). The former inequality shows that the number of vehicles required to serve all ride-pooling passengers increases with N. This implies that as N increases, the resulting increase in P means that the additional vehicles required to serve new passengers cannot be offset by the decrease in the number of required vehicles. The latter inequality shows that due to pool-matching, if N increases by one unit, the additional number of vehicles required is less than one unit. Moreover, an open question in the field of ride-sourcing services is how ride-pooling services affect transit usage and congestion. As mentioned in the Introduction, P, Dl, and ld are the key factors that govern the aggregate effects of ride-pooling services. The empirical laws revealed in this chapter can aid in the estimation of these factors at various levels of N. This can reveal the value of N for ride-pooling services that leads to a winewin situation for all participantsdthat is, the platform, passengers, drivers and governments (which aim to reduce traffic congestion) (Ke et al., 2019c).

13.4.4 Effect of matching radius As discussed in Sections 13.4.1e13.4.3, the maximum allowable R greatly affects the values of the parameters of the empirical laws that describe the relationships between the three key measures and N. It can be seen in Table 13.5 that at increases while bt decreases with R in the empirical law for Dl. By contrast, Table 13.7 shows that ad decreases and bd increases in the empirical law for ld. These monotonic properties inspire us to incorporate R into the empirical laws. First, we show that the following formula effectively describes the dependence of Dl on N and R: Dl 1 ¼  mt l l N þ ut R

(13.12)

where mt and ut are dimensionless parameters. It is clear that Eq. (13.12) has the same form as Eq. (13.6) when R takes a specific value; thus, Eq. (13.12) is a generalised form of Eq. (13.6) that is obtained by incorporating R. Fig. 13.9  demonstrates the variation in Dl l with respect to N in systems with various R. Table 13.9 lists the parameters and goodnesses-of-fit (i.e., r-squared values) of

346 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 13.9 (P1) Empirical law  of average passengers’ detour distance Dl incorporating  matching radius R; (A) [Chengdu e Dl l vs N]; (B) [Haikou e Dl l vs N] (C) [Manhattan e Dl= l vs N].

TABLE 13.9 Values of parameters in curve fitting of average passengers’ detour distance with incorporation of the matching radius. City, country

mt

ut

r-Squared

Chengdu, China

0.0020

9.1015

0.7053

Haikou, China

0.0013

10.4842

0.7460

Manhattan, New York city, USA

0.0036

6.7223

0.6909

the calibrated formula specified by Eq. (13.12). Although the r-squared value for the formula incorporating R is less than that of the formula not incorporating R, it is acceptable. This indicates that the formula in Eq. (13.12) effectively characterises the relationship between Dl, N, and R. It is also consistent with the intuition that a larger R results in a larger Dl. Second, we show that the following formula can approximate the dependence of ld on N and R: ld ¼1 þ l

1   m Nl ud log d R

(13.13)

where ud and md are dimensionless parameters. Clearly, Eq. (13.13) can be

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347

regarded as a generalised form of Eq. (13.7), as R takes a specific value.  Fig. 13.10 displays the curves of ld l vs N in matching systems with various R. Table 13.10 shows the parameters and goodnesses-of-fit (in the form of r-squared values) of the calibrated formula specified by Eq. (13.13). The r-squared values are within an acceptable range, which indicates that this simple formula can characterise the relationship between ld , N, and R. From the formula, we can infer that a larger R results in a longer ld , which is intuitive. Third, we construct the following formula to delineate how N and R affect p:   m  R p ¼ 1 u exp  N (13.14) l where u and m are dimensionless parameters. Clearly, the generalised formula in Eq. (13.14) reduces to Eq. (13.8) when R is given. Fig. 13.11 illustrates the

FIGURE 13.10 (P1)Empirical law of vehicle routing distance ld incorporating matching radius  R; (A) [Chengdu e ld l vs N]; (B) [Haikou e ld l vs N]; (c) [Manhattan e ld l vs N].

TABLE 13.10 Values of parameters in curve fitting of average passengers’ detour distance with incorporation of the matching radius. City, country

md

ud

r-Squared

Chengdu, China

24,917.5869

0.0487

0.6605

Haikou, China

4170.7138

0.0471

0.6847

Manhattan, New York city, USA

534.6560

0.0670

0.7140

Third, we construct the following formula to delineate how N and R affect p.

348 Supply and Demand Management in Ride-Sourcing Markets

P for various values of R as N increases. We calibrate Eq. (13.14) with the actual mobility data in the three cities, and the results are shown in Table 13.11. As can be seen, the r-squared is again high, although it is lower than that in Eq. (13.7). This indicates that this formula effectively describes the relationship between P, N, and R. In addition, it shows that a larger R means a higher P. This is intuitive, as a larger allowable R enables the pool-matching of more passengers in each matching-time interval. To summarise, this subsection discusses three generalised empirical laws that incorporate the key decision variabledRdfor the three key measures: Dl, ld and P. These empirical laws are concise, as they effectively describe the

FIGURE 13.11 (P1) Empirical law of matching probability (P) incorporating the matching radius (R); (A) [Chengdu e p vs N]; (B) [Haikou e p vs N]; (C) [Manhattan e p vs N].

TABLE 13.11 Values of parameters in curve fitting of pool-matching probability with incorporation of matching radius. City, country

m; a parameter in Eq. (13.14)

u; a parameter in Eq. (13.14)

Chengdu, China

0.0020

9.1015

0.7053

Haikou, China

0.0013

10.4842

0.7460

Manhattan, New York city, USA

0.0036

6.7223

0.6909

r-Squared

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349

relationships using only a few parameters. In addition, they reveal the effects of R on the three measures, which are consistent with intuition. However, although R is incorporated into the empirical laws, the values of the calibrated parameters nevertheless vary between cities. Thus, these parameters reflect the geographical characteristics of the cities, such as their network structure, density of roads, and the spatial distribution of their passenger requests. It would be interesting to examine how these exogenous factors affect the empirical laws or the relationships between the key measures, i.e., P and R. However, as we do not have data for a sufficient number of cities, such an examination will be a part of future studies.

13.4.5 Discussion on empirical laws Despite the proposed empirical laws performing well, as shown by the relatively high goodnesses-of-fit, there are some limitations to our experiments under certain settings. For example, when R is extremely small (< 1 km), P decreases substantially and the patterns of ld and Dl become ambiguous and unstable. This is due to the fact that very few passengers are pool-matched in this circumstance as the allowable matching distance is rather short. This results in a large variation in ld and Dl, and thus unstable estimates in curve fitting. Another limitation is related to the network topology: in our experimental settings, for convenience, we attach all trips’ origins and destinations to the road network’s intersections and only consider the arterial roads, which may lead to errors in the calculation of average trip distances. It remains to be determined how much this affects our findings. Moreover, due to data limitations, we only investigate three cities, which prevents us from performing statistical analyses to determine how the values of curve parameters are related to city characteristics, such as the density of demand, the area and shape of a network, and the density of road segments. We leave such an analysis for future studies, when mobility data of more cities become available.

13.5 Conclusions This chapter investigates how three key measuresdP, Dl, and ld e increase or decrease with N. Through extensive experiments, we find that (1) the ratio of passenger detour distance to non-pooling trip distance is inversely proportional to a linear form of N; (2) the ratio of ld to non-pooling trip distance is inversely proportional to a logarithmic form of N; and (3) P increases with N in a negative-exponential fashion. These empirical laws effectively describe the relationships between the three measures and N in three cities and for various values of R. The empirical laws thus exhibit nice properties and could serve as the fundamental building blocks for modelling on-demand ride-pooling markets to determine the effects of ride-pooling services on traffic congestion.

350 Supply and Demand Management in Ride-Sourcing Markets

There are several possible extensions to this work. First, for simplicity, the optimisation models only consider the pool-matching of two passengers, so the pool-matching of more than two passengers could be investigated in future studies. Second, it would be of immense interest to determine the effects of geographical features and spatial patterns of N on Dl, ld , and P. Third, despite the fact that neat and simple empirical laws can effectively fit the curves of the key measures vs N, they lack theoretical foundations and explanations, and thus these aspects warrant exploration. Fourth, it would be interesting to investigate the empirical laws under a continuous ride-pooling strategy, in which drivers may pick up a passenger before dropping off all of their passengers. Fifth, a challenging but important research direction is to evaluate the empirical laws or derive some new laws by theoretically analysing the physical matching process. This chapter is based on one of our recent articles (Ke et al., 2021).

Appendices Appendix 13.A. Probabilistic density distribution of Dl under objective P 1 (P1) Probabilistic density distribution of Dl in simulations with various values of N (N ¼ 10, N ¼ 50, N ¼ 100 and N ¼ 150) and various values of R ((a) 2 km, (b) 3 km, (c) 4 km and (d) 5 km, respectively) for each of the three cities (Fig. 13.A.12).

Appendix 13.B. Probabilistic density distribution of l d under objective P 1 (P1) Probabilistic density distribution of ld in simulations with various values of N (N ¼ 10, N ¼ 50, N ¼ 100 and N ¼ 150) and R ((a) R ¼ 2 km, ðbÞ R ¼ 3 km, (c) R ¼ 4 km and (d) R ¼ 5 km respectively) for each of the three cities (Fig. 13.B.13).

Appendix 13.C. Empirical law of Dl under objective P 1 (P1) Dl vs N with various values of R ((a) R ¼ 2 km, ðbÞ R ¼ 3 km, (c) R ¼ 4 km and (d) R ¼ 5 km, respectively) for each of the three cities (Fig. 13.C.14).

Appendix 13.D. Empirical law of l d under objective P 1 (P1) ld vs N with various values of R ((a) R ¼ 2 km, (b) R ¼ 3 km, (c) R ¼ 4 km and (d) R ¼ 5 km, respectively) for each of the three cities (Fig. 13.D.15).

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FIGURE 13.A.12 (P1) Probabilistic density distribution of average passengers’ detour distance Dl with various values of matching radius R; (A) [Chengdu e pdf of Dl]; (B) [ Haikou e pdf of Dl]; (C) [ Manhattan e pdf of Dl].

Appendix 13.E. Empirical fitting of p under objective P 1 (P1) P vs N with various values of R ((a) R ¼ 2 km, (b) R ¼ 3 km, (c) R ¼ 4 km and (d) R ¼ 5 km, respectively) for each of the three cities (Fig. 13.E.16).

Appendix 13.F. Probabilistic density distribution of Dl under objective P 2 (P2) Probabilistic density distribution of Dl in simulations with various values of N (N ¼ 10, N ¼ 50, N ¼ 100 and N ¼ 150) and R ((a) R ¼ 2 km,

352 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 13.B.13 (P1) Probabilistic density distribution of vehicle routing distance ld with various values of matching radius R; (A) [Chengdu e pdf of ld ]; (B) [ Haikou e pdf of ld ]; (C) [ Manhatten e pdf of ld ].

(b) R ¼ 3 km, (c) R ¼ 4 km and (d) R ¼ 5 km, respectively) for each of the three cities (Fig. 13.F.17).

Appendix 13.G. Probabilistic density distribution of l d under objective P2 (P2) Probabilistic density distribution of ld in simulations with various values of N (N ¼ 10, N ¼ 50, N ¼ 100 and N ¼ 150) and R ((a) R ¼ 2 km, R ¼ 3 km, R ¼ 4 km and R ¼ 5 km, respectively) for each of the three cities (Fig. 13.G.18).

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FIGURE 13.C.14 (P1) Passenger detour distance Dl vs passenger demand N with various values of matching radius R; (A) [Chengdu e Empirical fitting of Dl vs N]; (B) [Haikou e Empirical fitting of Dl vs N]; (C) [Manhattan e Empirical fitting of Dl vs N].

Appendix 13.H. Empirical law of Dl under objective P 2 (P2) Dl vs N with various values of R ((A) R ¼ 2 km, (b)R ¼ 3 km, (c) R ¼ 4 km, ðdÞ R ¼ 5 km, respectively) for each of the three cities (Fig. 13.H.19).

Appendix 13.I. Empirical law of l d under objective P 2 (P2) ld vs N with various values of R ((a) R ¼ 2 km, (b) R ¼ 3 km, ðcÞ R ¼ 4 km and (d) R ¼ 5 km, respectively) for each of the three cities (Fig. 13.I.20).

354 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 13.D.15 (P1) Vehicle routing distance ld with various arrival rates and values of matching radius R; (A) [Chengdu e Empirical fitting of ld vs N]; (B) [Haikou e Empirical fitting of ld vs N]; (C) [Manhattan e Empirical fitting of ld vs N].

Appendix 13.J. Empirical fitting of p under objective P 2 (P2) P vs N with various values of R ((a) R ¼ 2 km, (b) R ¼ 3 km, (c) R ¼ 4 km and (d) R ¼ 5 km, respectively) for each of the three cities (Fig. 13.J.21).

Appendix 13.K. Probabilistic density distribution of Dl under objective P 3 (P3) Probabilistic density distribution of Dl in simulations with various values of N (N ¼ 10, N ¼ 50, N ¼ 100 and N ¼ 150) and R ((a), R ¼ 2 km,

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FIGURE 13.E.16 (P1) Pool-matching probability P with various arrival rates and values of matching radius R; (A) [Chengdu e Empirical fitting of p vs N]; (B) [Haikou e Empirical fitting of p vs N]; (C) [Manhattan e Empirical fitting of p vs N].

(b), R ¼ 3 km, (c), R ¼ 4 km, (d), R ¼ 5 km, respectively) for each of the three cities (Fig. 13.K.22).

Appendix 13.L. Probabilistic density distribution of l d under objective P 3 (P3) Probabilistic density distribution of ld in simulations with various values of N (N ¼ 10, N ¼ 50, N ¼ 100 and N ¼ 150) and R ((a) R ¼ 2 km, (b) R ¼ 3 km, (c) R ¼ 4 km and (d) R ¼ 5 km, respectively) for each of the three cities (Fig. 13.L.23).

356 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 13.F.17 (P2) Probabilistic density distribution of average passengers’ detour distance Dl with various values of matching radius R; (A) [Chengdu e pdf of Dl]; (B) [Haikou e pdf of Dl]; (C) [Manhatten e pdf of Dl].

Appendix 13.M. Empirical law of Dl under objective P 3 (P3) Dl vs N with various values of R ((a) R ¼ 2 km, ðbÞ R ¼ 3 km, (c) R ¼ 4 km and (d) R ¼ 5 km, respectively) for each of the three cities (Fig. 13.M.24).

Appendix 13.N. Empirical law of l d under objective P 3 (P3) ld vs N with various values of R ((a) R ¼ 2 km, (b) R ¼ 3 km, ðcÞ R ¼ 4 km and (d) R ¼ 5 km, respectively) for each of the three cities (Fig. 13.N.25).

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FIGURE 13.G.18 (P2) Probabilistic density distribution of vehicle routing distance ld with various values of matching radius R; (A) [Chengdu e pdf of ld ]; (B) [ Haikou e pdf of ld ] (C) [ Manhattan e pdf of ld ].

Appendix 13.O. Empirical fitting of p under objective P 3 (P3) P vs N with various values of R ((a) R ¼ 2 km, (b) R ¼ 3 km, (c) R ¼ 4 km and (d) R ¼ 5 km, respectively) for each of the three cities (Fig. 13.O.26).

Appendix 13.P. Proof of vM=vN>0 and vM=vN£1 Proof. Let f ðxÞ ¼ 1 þ zðexpðxÞ x expðxÞÞ, where x > 0, 1  z > 0. Then, it can be shown that

358 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 13.H.19 (P2) Passenger detour distance Dl vs passenger demand N with various values of matching radius R; (A) [Chengdu e Empirical fitting of Dl vs N]; (B) [ Haikou e Empirical fitting of Dl vs N]; (C)[ Manhattan e Empirical fitting of Dl vs N].

f 0 ðxÞ ¼ z expðxÞðx  2Þ

(13.A1)

which implies that f 0 ðxÞ > 0 if x > 2, and that f 0 ðxÞ < 0 if x < 2. Therefore, f ðxÞ  f ð2Þ > 0 and f ðxÞ  maxðf ð0Þ; f ðNÞ Þ ¼ 1 þ z  2. This implies that vM=vN > 0 and vM vN  1. This completes the proof.

Appendix 13.Q. Empirical laws for the downtown area of Manhattan To investigate the effect of the spatial distribution of N on our empirical law, we conduct experiments in the Manhattan downtown area, which is shown in

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FIGURE 13.I.20 (P2) Vehicle routing distance ld with various arrival rates and values of matching radius R; (A) [Chengdu e Empirical fitting of ld vs N; (B) [Haikou e Empirical fitting of ld vs N]; (C) [Manhattan e Empirical fitting of ld vs N].

Fig. 13.Q.27. According to our records, approximately 8.2% of all trips (231,972 trips) in the entire dataset have their origins and destinations in the downtown area. The spatial distribution of these trips is displayed in Fig. 13.Q.28. We re-run our experiments based on the new dataset with similar settings and using the algorithm given in Section 3. In particular, as the targeted area is relatively small, we initially set a smaller R, i.e., R ¼ 1 km. Then, to examine the validity of the empirical law more generally, we also set a larger R, i.e., R ¼ 4 km. The results are given in Figs. 13.Q.28e13.Q.30, and are in line with our main findings. Table 13.Q.12 gives the fitting values of parameters with respect to the three corresponding empirical laws.

360 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 13.J.21 (P2) Pool-matching probability P with various arrival rates and values of matching radius R; (A) [Chengdu e Empirical fitting of p vs N]; (B) [Haikou e Empirical fitting of p vs N]; (C) [Manhattan e Empirical fitting of p vs N].

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FIGURE 13.K.22 (P3) Probabilistic density distribution of average passengers’ detour distance Dl with various values of matching radius R; (A) [Chengdu e pdf of Dl]; (B) [Haikou e pdf of Dl]; (C) [Manhattan e pdf of Dl].

362 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 13.L.23 (P3) Probabilistic density distribution of vehicle routing distance ld with various values of matching radius R; (A) [Chengdu e pdf of ld ]; (B) [ Haikou e pdf of ld ]; (C) [ Manhattan e pdf of ld ].

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FIGURE 13.M.24 (P3) Dl vs passenger demand N with various values of matching radius R; (A) [Chengdu e Empirical fitting of Dl vs N]; (B) [Haikou e Empirical fitting of Dl vs N]; (C) [Manhattan e Empirical fitting of Dl vs N].

364 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 13.N.25 (P3) vehicle routing distance ld with various arrival rates and values of matching radius R; (A) [Chengdu e Empirical fitting of ld vs N]; (B) [Haikou e Empirical fitting of ld vs N]; (C) [Manhattan e Empirical fitting of ld vs N].

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FIGURE 13.O.26 Pool-matching probability P with various arrival rates and values of matching radius R; (A) [Chengdu e Empirical fitting of p vs N]; (B) [Haikou e Empirical fitting of p vs N]; (c) [Manhattan e Empirical fitting of p vs N].

366 Supply and Demand Management in Ride-Sourcing Markets

FIGURE 13.Q.27 Downtown area in Manhattan and the spatial distribution of trip origins and destinations; (A) [selected Manhattan downtown area]; (B) [spatial distribution of 1000 representative trips].

FIGURE 13.Q.28 Probabilistic density distribution of average passengers’ detour distance Dl; (a) [pdf of Dl, R ¼ 1 km]; (b) [pdf of Dl km].

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FIGURE 13.Q.29 Probabilistic density distribution of vehicle routing distance ld ; (A) [pdf of ld , R ¼ 1 km]; (B) [pdf of ld , R ¼ 4 km].

FIGURE 13.Q.30 (P1) Manhattan downtown areadEmpirical fitting under different scenarios  with a matching radius R ¼ 1 km and 4 km; (A) [Empirical fitting of Dl l vs N when R ¼ 1 km  and R ¼ 4 km]; (B) [Empirical fitting of ld l vs N when R ¼ 1 km and R ¼ 4 km]; (C) [Empirical fitting of p vs N when R ¼ 1 km and R ¼ 4 km].

 Dl l vs N

[c]2* R (km)

 l d l vs N

p vs N

at

bt

r-Squared

ad

bd

r-Squared

g

z

r-Squared

1

0.0753

2.8695

0.8577

0.7808

0.4658

0.9879

0.0962

0.0187

0.9566

4

0.0886

2.7028

0.9773

0.7328

0.4754

0.9832

0.1003

0.0331

0.8826

368 Supply and Demand Management in Ride-Sourcing Markets

TABLE 13.Q.12 Empirical law values of parameters in curve fitting.

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Glossary of notation N Number of passengers (interchangeable with trips) in the pool to be matched per time unit p Pool-matching probability, i.e., the proportion of the total passengers who are poolmatched l Average non-pooling trip distance Dl Average detour distance experienced by pool-matched passengers ld Average vehicle routing distance involved in serving each pair of pool-matched passengers li Distance of non-pooling trip i lminði;jÞ Minimum vehicle routing distance of four possible pick-upedrop-off sequences for trip i and trip j Dli ði; jÞ Detour distance experienced by passengers i associated with the minimum vehicle routing distance lminði;jÞ Dlj ði; jÞ Detour distance experienced by passengers j associated with the minimum vehicle routing distance lminði;jÞ lO ði; jÞ Distance between the origins of trip i and trip j lD ði; jÞ Distance between the destinations of trip i and trip j xij Binary variable that indicates whether trip i and trip j are matched

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Chapter 14

Summary Jintao Ke1, Hai Yang2, Hai Wang3 and Yafeng Yin4 1

Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong, China; Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Hong Kong, China; 3School of Computing and Information Systems, Singapore Management University, Bras Basah, Singapore; 4Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI, United States 2

In this book, a series of mathematical models are developed to describe the stationary equilibrium states of ride-sourcing markets. These models are then used to analyse the optimal operating strategies for such markets in terms of pricing, wages, and matching strategies to maximise social welfare or platform profit. Ride-sourcing markets are two-sided markets in which demand and supply interact in a complex manner, and are thus challenging to model. On the demand side, passengers choose between a ride-sourcing service and other travel modes depending on the service’s trip fare and service quality. On the supply side, drivers determine whether to enter the market and offer ondemand ride services depending on their hourly income, which is governed by vehicle utilisation and average wage per ride. The major characteristic that distinguishes a ride-sourcing market from a classical economic market is the matching friction between drivers and passengers. These frictions arise from the supply-demand imbalance, imperfect information, and spatial heterogeneity of passengers and drivers. Due to the matching frictions, passenger demand and driver supply affect each other in an endogenous way. The availability of idle drivers governs passengers’ waiting time and thus influences passenger demand, while passenger demand in turn determines the utilization rate of vehicles, which affects drivers’ expected revenue and their participation in the market. In such a market, a platform can leverage the trip fare and wage to affect passengers’ choices, drivers’ entry, matching frictions, and thus the market equilibrium. The models we devise effectively delineate the intricate relationships between platform decision variables (such as the trip fare and wage) and system endogenous variables (such as passengers’ expected waiting time, vehicle utilisation, effective demand, and labour supply). Specifically, several inductive and deductive approaches are developed to approximate the matching Supply and Demand Management in Ride-Sourcing Markets. https://doi.org/10.1016/B978-0-443-18937-1.00003-6 Copyright © 2023 Elsevier Inc. All rights reserved.

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374 Supply and Demand Management in Ride-Sourcing Markets

frictions between waiting passengers and idle drivers, which govern the passenger demand through service quality and driver supply through the vehicle utilisation rate. The inductive approaches directly use a hypothesised matching function to characterise the matching frictions, whereas the deductive approaches approximate the matching frictions based on some presumed matching strategies, such as a first-come-first-served (FCFS) matching mechanism or a batch-matching mechanism. We outline and compare the advantages and disadvantages of the various inductive and deductive approaches in Chapter 2, and conduct a series of simulation studies to evaluate the approximating accuracy of different matching models in Chapter 3. Based on the proposed models, we analytically identify the monopoly optimum (MO; the strategy for maximising the platform profit), the social optimum (SO; the strategy for maximising the social welfare), and the Paretoefficient frontier (along which neither the platform profit nor the social welfare can be increased without decreasing the other). The MO reveals how a platform makes decisions in a market without regulation, whereas the SO represents an ideal scenario in which a platform chooses a strategy to maximise social welfare, which includes the benefits of passengers, drivers, and the platform. We prove that the trip fare and wage at the MO are higher than those at the SO and that the commission (the difference between the trip fare and wage) is also higher at the MO than at the SO. This indicates that if no regulation is imposed, a platform will distort both the trip fare and wage from their efficient levels, and charge a higher commission for each passenger request. Although the SO is generally unsustainable, as it results in a negative platform profit and will be rejected by a business, a government can target an appropriate Pareto-efficient solution that establishes a reasonable balance between social welfare and platform profit. In Chapter 4, we examine what regulations can induce a platform to choose a targeted Pareto-efficient solution, which reveals that most currently used regulations, such as a price cap, a minimum wage, or a maximum utilisation rate, are not effective, as they induce a platform to choose a non-Pareto-efficient strategy. In contrast, a commission-cap regulation (which restrains the commission charged by a platform for each order) and a demand regulation (which requires a platform to serve a certain number of passengers per hour) are shown to effectively induce a platform to choose a targeted Pareto-efficient strategy. In addition, we extend the basic model to analyse a market providing ridepooling services, which allows one vehicle to serve two or more passengers per ride in Chapter 5. We show that the MO and SO trip fares in a ride-pooling market are lower than those in a non-ride-pooling market. This is because a unit decrease in the trip fare in a ride-pooling market not only directly increases passenger demand (due to passengers’ sensitivity to the trip fare) but also reduces the detour time, which leads to an additional increase in passenger demand. As a result, a platform offering a ride-pooling service will

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choose to decrease its trip fare to attract more passengers and thereby maximise social welfare or its profit. Moreover, we examine the equilibrium properties of ride-sourcing markets (with or without ride-pooling services) in the presence of traffic congestion in Chapters 6 and 7. We find that the level of traffic congestion affects the decisions of a platform in such scenarios, as a platform tends to set a higher trip fare to internalise the traffic congestion externalities and thereby maximise the platform’s profit or social welfare. This increase in trip fare is similar to congestion pricing and alleviates the negative effects of traffic congestion on passengers’ travel and waiting time. We also examine and compare the optimal operating strategies for non-ride-pooling services and ride-pooling services in the presence of traffic congestion. We further revise the implications and designs of government regulations in markets with mild, moderate or server traffic congestion in Chapter 8. The modelling framework is also extended and employed to investigate other interesting and important research topics, such as the competition between ride-sourcing platforms and third-party platform integration in Chapter 9. Our model reveals how the competition between platforms influences market fragmentation and the resulting market outcomes in terms of effective demand and supply, and passengers’ and drivers’ waiting time. We also investigate the potential implications of third-party integration on competing platforms’ decisions and the resulting market equilibrium. Furthermore, the model is used to examine the complementary and substitutive effects of ride-sourcing services on public transit usage in Chapter 10. On the one hand, ride-sourcing services complement public transit services by solving the first-mile/last-mile problem experienced by these services. On the other hand, ride-sourcing services may compete with public transit services by absorbing passengers from public transit services. To investigate these opposing effects, we establish a model to determine the situations under which complementary effects dominate and to optimise cooperation or coopetition between ride-sourcing services and public transit services. In Chapter 11, in contrast to the previous chapters that focus on a stationary equilibrium state, we study the effects of the two decision variables (matching time interval and matching radius) on the performance of an on-demand ridematching system in a real-time process, which may not be at equilibrium. We approximate the matching process using a spatial model and then perform a joint optimisation to identify the optimal combination of matching time interval and matching radius. The analyses reveal some valuable managerial and operational insights, such as by defining the situations in which a platform should set a short matching time interval to match arriving passengers with idle drivers as soon as possible, and those in which it should set a long matching time interval to accumulate a sufficient number of drivers and passengers to conduct more efficient matching. We further design a dynamic optimisation strategy that adjusts the matching time interval and matching

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radius in response to real-time supplyedemand conditions, and demonstrate via a simulation study using data from New York that this strategy outperforms strategies with a fixed matching time interval and matching radius. In Chapter 12, we conduct an empirical study to analyse the effects of hourly income rates on drivers’ supply along extensive and intensive margins. This reveals that the participation elasticity and working-hour elasticity of labour supply are positive and significant in the dataset of a ride-sourcing platform. Through simulation-based experiments in Chapter 13, we discover some empirical laws that effectively describe the relationships between the key market measures (the pool-matching probability, the average passengers’ detour distance, and the average drivers’ routing distance) and the passenger demand for ride-pooling services. These empirical laws can be used for theoretical modelling and applications in ride-sourcing markets, such as for evaluating the effects of ride-pooling on transit usage and traffic congestion. This book majorly focuses on equilibrium analysis and the proposed operational and regulatory strategies to manage the supply and demand in ridesourcing markets are based on assumptions that the markets are in stationary equilibrium states. Also, the models proposed in this book are majorly aggregated models that consider the expectation or average of exogenous, endogenous and decision variables. Such an analysis is helpful for determining platform decisions and government regulation policies at a strategical level, but might not be well suitable for decision makings at a tactical level, such as determining the pair-by-pair matchings between drivers and passengers and setting differentiated prices for different passengers and drivers. Moreover, the proposed models are unable to characterize the interactions and transitions among consecutive time periods in a dynamic and stochastic ride-sourcing system.

Glossary of abbreviations Abbreviations

definition

CDF CS IVSI LHS MO NP NS PDF PS RHS RP RS SB SO TNCs VOT

cumulative distribution function consumer surplus idle vehicle shifting interval left-hand-side monopoly optimum non-pooling non-splitting probability density function provider surplus right-hand-side ride-pooling ride-splitting second-best solution social optimum transportation network companies value of time

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Index ‘Note: Page numbers followed by “f ” indicate figures and “t” indicate tables.’

A

C

A-to-B matching sequence, 36e37 Aggregate labour supply, 171 Aggregate models, 3e4, 194 Application (app), 192 Arterial roads, 346e348 Artificial intelligence, 328 Average income per driver per hour, 170 Average time cost, 141

Calibration, 61 California Assembly Bill (AB5), 7e8 Careem, 1 Circle of radius, 264e265 CobbeDouglas production function, 29, 60e61 CobbeDouglas-type matching function, 41, 56 CobbeDouglas-type meeting function, 5e6 Combinatorial optimisation-based online matching strategies, 15 Commission fee, 204 effect of, 217e219 Commission ratio regulation, 99e100 Commission regulation, 98, 168 Comparative static effects of regulatory variables, 116e117 Complements public transit, 241e242 Congestion effect, 142 externalities, 10e12, 142 Constraints, 331e332 Consumer surplus (CS), 43, 90, 141, 168 Cruising, 66 Cumulative distribution function (CDF), 38, 110, 265 Curve fitting, 338, 349

B Background traffic, 133 Balanced scenario, 267, 275e277 Base model, 58e59 Basic model, 304e305 Batch-matching. See also Pool-matching model, 63e64 process, 36e39, 37f setting, 341 Benchmark algorithm, 275, 280 Best-fit models for estimation, 68e72 mcev estimated matching rates for supply edemand relationships, 71f formulas for key market metrics in models, 70t matching rate in markets, 69f passenger pick-up time, 74e76, 75f estimates of pick-up time for supply edemand relationships, 75f passengers’ total waiting time in markets, 77f passengers’ total waiting time, 76 wm, 72e74 estimation of matching time for supply edemand relationships, 73f matching time in markets, 72f Bi-objective maximisation problem, 32e33, 47e48, 123e124, 143, 157 Binary decision, 331e332 Broadcast mode, 15 Bundled mode, 241

D Data sources, 286 Data-driven research, 328 Datasets, 334e336 Decision variables, 117, 278e279 matrix, 331e332 Decision-makers, 252e253 Decision-making process, 214, 376 Deductive approaches, 27, 31e39, 52t, 56 Demand function, 113e115, 134e135, 151 Demand regulation, 101, 188 Demand splitting, 191e192 Destinations, 331e332, 334e336

379

380 Index Detour ratios, 265e266 distribution, 38e39 Detour-constrained scenario, 108, 110, 126e130 Detour-unconstrained scenario, 108, 110e111, 125e126 DiDi, 1, 5 Didi Chuxing, 239 DiDi Express Pool, 8, 107 Difference-in-differences approach, 14 Dimensionless parameters, 340e341, 346e348 Directly controlled decision variables, 32e33 Disaggregate models, 3e4 Dispatch mode, 15 Dominant zone, 37 Double-ended queueing model, 6 Drivers, 167, 261, 283 classification along extensive and intensive margins, 304 detour distance, 326 driverepassenger matching process, 242e243 estimates of labour supply elasticity in presence of driver heterogeneity, 316e317 rationing effects, 185e186 supply, 26e27 Dynamic optimisation strategy, 279e280, 375e376 Dynamic simulation environment, model performance in, 278e280 Dynamic waiting strategy, 111

E Earnings opportunities, 284 Econometric framework, 328 Elasticities of matching function, 28 Empirical analysis, 304e308 basic model, 304e305 consideration of potential sample selection bias, 305e306 identification of outcome equation, 306e308 Empirical laws, 336e349, 376 discussion on, 349 law of 7d, 338e341 law of p, 341e345 law of D7, 336e338 optimisation framework and data descriptions, 328e336 definitions of key measures, 329e330

experimental settings and, 333e336 optimisation algorithms, 330e333 random matching without optimisation objective, 333 effect of R, 345e349 En-route pool-matching scheme, 108e111 general model, 109 probabilistic model, 109e111 Endogeneity, 285 evidence of sample selection and endogeneity bias, 309e314 Endogenous system, 373e374 Endogenous variables, 117, 240e241 Equilibrium. See also Nash equilibrium (NE) analysis, 250e253 of profit-and/or social welfare maximising strategies, 254e256 models, 249, 289, 292e293 outcomes, 159e160 solutions, 60e61 Equilibrium analysis, 113e117, 134e140, 149e154 comparative static effects of regulatory variables, 116e117 demand function, 113e115, 134e135, 151 equilibrium solution, 139e140, 152e154 market equilibrium, 115e116 market measures, 118e124 numerical illustrations, 124e130 detour-constrained scenario, 126e130 detour-unconstrained scenario, 125e126 experimental settings, 124e125 pool-matching schemes, 108e113 speed function, 135e138 supply function, 113e115, 138e139, 151e152 vehicle conservation, 149e150 Euclidean distance, 38e39 Exogenous shocks in natural experiments, 302 Exogenous variables, 240e241 Expected pick-up distance, 38e39, 265e266 Experimental settings and data description, 333e336 Experiments, 66e67, 329 Explicit function, 34, 115 Express, 1 ExpressPool, 1 Extensive experiments, 341 Extensive margin in labour supply model, importance of, 291e292 Externality, 141

Index

F Feeder, 240 First-best solution (FB solution). See Social optimum (SO) First-come-first-served (FCFS), 33e35, 169, 373e374 FCFS-based model, 42e43, 46e47, 49e50, 63, 107e108, 134 matching, 194 mechanism, 5 First-order conditions, 92e93 First-order necessary conditions, 141 First-order partial derivatives of T, 270 First-pickupefirst-dropoff, 113e114 First-pickupelast-drop-off, 113e114 Firstemile, 240 Fleet size control, 168 regulation, 95e96 Fragmentation, 375 Friction, 373

G Gaussian error function, 38e39, 265e266, 271 Government regulations, 167 alternative method to obtain and analyse pareto-efficient solutions, 91e93 numerical studies, 178e188 properties of pareto-efficient solutions, 87e91 theoretical analyses, 168e178 Grab, 1 GrabShare, 8, 107

H Heavy traffic congestion, 168 Heckman sample-selection model, 294 Heckman two-step sample selection model, 308e309 Heterogeneous origins, 330e331 Heterogeneous service, 242 Hitch, 1 Homogeneous reservation rate, 167 Hypercongested flow regime, 136e138 Hypothesised queuing model, 325e326

I Idle drivers, 36, 263 Idle phase, 34

381

Idle vehicle shifting interval (IVSI), 250e251 Idle vehicles, 333 Imbalanced scenarios, 267, 277e278 model properties in, 271e275 effects of matching-time interval, 271e273 properties of optimal matching-time interval, 273e274 Implicit equation, 116 In-trip phase, 34 In-trip vehicles, 149e150 Income accelerator. See Income multiplier Income elasticity, 298 Income multiplier, 300e302 Income rate elasticity, 291 Income regulation, 97e98 Income-targeting theory, 170e171 Inductive approaches, 27e31, 52t, 56 Instant matching, 63 Instrumental variable approach (IV approach), 285 IV-type estimation, 296 validity, 314e316 Intensive margin elasticity, 291 Interior solution, 206

K Key decision variables, 261e262 Key measures, definitions of, 329e330

L Labour supply elasticity, 286 in presence of driver heterogeneity, estimates of, 316e317 on ride-sourcing platform, model of, 297e300 Labour supply model, 283e286, 289e292 importance of extensive margin in labour supply model, 291e292 main contributions, 286 methodology, 285 modelling endogeneity of income rates and self-selected participation in labour force, 292e300 methodological implications of selfselection and endogeneity, 292e297 model of labour supply elasticity on ride-sourcing platform, 297e300 motivation, 284e285 optimal decisions on hours worked based on income targets, 289e291

382 Index Labour supply model (Continued ) related literature, 286e289 research design, 300e308 data description, 302e303 driver classification along extensive and intensive margins, 304 empirical analysis, 304e308 large-scale natural experiment, 300e302 research context, 300 research questions, 285 results, 285e286, 308e318 estimates of labour supply elasticity in presence of driver heterogeneity, 316e317 labour supply in subgroups, 316e317 model estimation, 309e316 Lagrange dual decomposition-based algorithm, 66 Lagrange function, 48 Lagrangian function, 123e124, 157 Large-scale natural experiment, 300e302 exogenous shocks in natural experiments, 302 income multiplier, 300e302 Lastemile, 241e242 Law of 7d, 338e341 Law of p, 341e345 discussions on matching probability, 343e345 Law of D7, 336e338 Learning approach, 328 Leaving, 66 Lerner formula, 41, 119e120, 245e246 Linear traffic-flow model, 135, 158e159 Little’s law, 60, 170 Logarithm-sum formula, 249 Logarithmic formula, 338e340 Luxe, 1 Lyft, 1, 7e8, 147e148, 239 Lyft Line, 8, 107

M M/M/1 queuing system, 32, 61 M/M/1/k queuing model, 61e62 M/M/N queuing model, 62e63 Machine learning, 328 approaches, 263 Macroscopic fundamental diagram (MFD), 11e12, 133e134 Magnitudes, 212e213 of demand and supply, 271

Market equilibrium, 30, 32, 94e95, 115e116, 193, 247 and optimal strategies, 194e206 market with platform integration, 198e206 market without platform integration, 195e198 point, 35 state, 244e245 Market fragmentation, 191e192, 210e211 effect of, 210e214 Market measures, 39e50, 118e124, 140e144, 154e158 monopoly optimum, 40e43, 119e120, 140e141, 154e155 Pareto-efficient solutions, 123e124, 143e144, 157e158 social optimum, 120e123, 141e143, 155e156 Market metrics, 57e64 Market segmentation, 68 Market with platform integration, 198e206 Market without platform integration, 195e198 Markov decision process, 15e16 Matched passenger, 264e265 Matching algorithms, 261 Matching externality, 41 Matching frictions, 27e39, 213, 217 perfect matching function, 27e28 production functions, 28e31 Matching functions, 5e6, 56e64 batch-matching model, 63e64 CobbeDouglas production function, 60e61 FCFS model, 63 M/M/1 queuing model, 61 M/M/1/k queuing model, 61e62 M/M/N queuing model, 62e63 perfect matching function, 59e60 Matching process, 66 Matching radius, 261, 268e269 effect of, 262f optimization of matching radius in ridesourcing markets model properties in imbalanced scenarios, 271e275 modelling and optimising matching process, 262e271 numerical studies, 275e280 research problem, 261e262

Index Matching rate, 57, 263 Matching stability, 15 Matching time, 4, 38e39, 261 Matching window effects, 162e163 Matchingetime interval, 269e271 effects of, 262f, 271e273 optimization of matchingetime interval in ride-sourcing markets model properties in imbalanced scenarios, 271e275 modelling and optimising matching process, 262e271 numerical studies, 275e280 research problem, 261e262 Mathematical models, 5, 56, 193e194, 240e241, 373 Mean absolute percentage error (MAPE), 68 Mean pick-up time, 267 Methodological implications of self-selection and endogeneity, 292e297 modelling endogeneity of hourly income rate, 296e297 modelling self-selected participation in labour force, 293e296 Minibus, 1 Minimum utilisation rate regulation, 100e101 Mining data, 326 Mobile communication technologies, 283 Mobile Internet technologies, 107 Mobility components, 239 system, 323e324 Mobility as a service (MaaS), 12e13 Model description, 241e245 Model estimation, 309e316 evidence of sample selection and endogeneity bias, 309e314 validity of IVs, 314e316 Model performance in dynamic simulation environment, 278e280 Modelling endogeneity of hourly income rate, 296e297 Modelling matching process, 262e271 Modelling self-selected participation in labour force, 293e296 modelling endogeneity of income rates and, 292e300 Monopolistic platform, 191e192 Monopoly markup, 141

383

Monopoly optimum (MO), 6e7, 39e43, 87e88, 107, 119e120, 133, 140e141, 154e155, 168, 172e173, 212e213, 245e246, 374 FCFS-based model, 42e43 production-function-based model, 41 queuing model, 42 Monopoly ride-sourcing platform, 141 Monotonic relationships, 89e90 Motivation, 284e285 Multi-homing, 192 Multimodal transportation system, 191, 239e240

N Nash equilibrium (NE), 193e194 effect of platform integration at, 208e209 effect of vehicle fleet size at, 207e208 Natural experimental environment, 286 Natural experiments, 296, 302 exogenous shocks in, 302 Negative labour-supply elasticity, 170e171 Neoclassical theory, 170e171 Network topologies, 334e336 Network vehicular speed, 151 New York City (NYC), 326 Newly arriving passenger, 272 Non-monotonicity, 126 Non-pooling market, 160e161 Non-ride-splitting (NS), 134, 148 Non-WGC regime, 34e35 Normal flow regime, 136e138 Novel business model, 192 Numerical studies, 158e163, 178e188, 210e219, 275e280 balanced scenario, 275e277 effect of commission fee, 217e219 effects of driver rationing, 185e186 effects of matching window, 162e163 equilibrium outcomes, 159e160 imbalanced scenarios, 277e278 effect of market fragmentation, 210e214 market with drivers with heterogeneous r and no traffic congestion, 179e182 market with drivers with heterogeneous r and traffic congestion, 182e185 model performance in dynamic simulation environment, 278e280

384 Index Numerical studies (Continued ) optimal operating strategies, 160e162 settings, 178e179 effect of vehicle fleet size, 214e217

O Ola, 1 On-demand matching and key decision variables, 14e16 On-demand ride service, 14 On-demand ridesourcing platform, 271 Online matching process, 263 Operating strategies, 147e148 Optimal decisions on hours worked based on income targets, 289e291 Optimal matching process, 331e332 properties of optimal matching-time interval, 273e274 Optimal operating strategies, 160e162 Optimal pricing formula, 142e144, 157 Optimal solutions, 211e212 Optimal strategies, 194e206, 249 design, 245e249 MO, 245e246 SO, 247e248 market with platform integration, 198e206 market without platform integration, 195e198 Optimal system performance, 275e276 Optimisation algorithms, 330e333, 336 framework, 328e336 definitions of key measures, 329e330 experimental settings and data description, 333e336 optimisation algorithms, 330e333 random matching without optimisation objective, 333 model, 329 objective function, 333 problem, 119e120, 206 Optimising matching process, 262e271 expected pick-up distance, 265e266 general model properties, 267e271 effect of matching radius, 268e269 effect of matching-time interval, 269e271 matched passengeredriver pairs, 264e265 online matching process, 263 system performance measure, 266e267 Order generation, 65e66

Order-cancellation behaviours of customers, 327 Ordinary least squares (OLS), 296 regression model, 304e305 Outcome equation, 316 identification of, 306e308

P p function, 112 Pareto-efficient frontier, 47, 87e88, 90e91, 168, 248e249 Pareto-efficient rationing factor, 176e177 Pareto-efficient solutions, 6e7, 47e50, 93, 107, 123e124, 143e144, 157e158, 175e178 alternative method to obtain and analyse, 91e93 properties of, 87e91 Pareto-efficient strategy, 98, 255e256 Partial derivatives, 30e31, 34e35, 92, 110e111, 140 Participation decision, 284 Participation elasticity, 285e286 Passengers, 192, 240, 261 abandonment behaviours, 276e277 average matching time, 57 average pick-up time, 57 average total waiting time, 57 demand, 25e26, 148, 326 detour distance, 325e326 and drivers arrival processes, 263 matching time, 38 Perfect matching function, 27e28, 56, 59e60 P-maximisation problem, 92, 119e120 Pick-up phase, 34 Pick-up time, 4e5, 29, 38e39, 195e196, 265 of passengers, 207e208 Platform competition, 12e13, 192 Platform integration, 12e13, 192 evaluation of performance of, 207e210 effect of platform integration at NE, 208e209 effect of platform integration at SO, 209e210 effect of vehicle fleet size at NE/SO, 207e208 market with, 198e206 market without, 195e198 at NE, effect of, 208e209 at SO, effect of, 209e210 Platform integrators, 192

Index Platform maximisation problem, 32e33 Platform optimisation problem, 267 Platform profit, 39e40, 141 Platforms, 202e203, 275e276, 340e341 Poisson distribution, 274e275 Policy, 104 Pool-matching mechanism, 149 optimisation algorithm, 336 probability, 327e328 process, 9e10, 325e326 schemes, 108e113 comparisons, 112e113 en-route pool-matching scheme, 108e111 pre-assigned pool-matching with meeting points, 111e112 strategy, 148 window, 112, 148 Positive congestion externality, 141 Positive labour-supply elasticity, 170e171 Pre-assigned pool-matching with meeting points, 111e112 Premier, 1 Price-cap regulation, 94e95, 188 Pricing formula, 124 Probabilistic model, 109e111 Probability density function (PDF), 38, 110 Production functions, 28e31 production-function-based model, 41, 44e45, 48e49 Profit-maximisation problem, 245e246 Profit-maximising platforms, 209 Profit-welfare maximising strategies, analysis of, 254e256 Provider surplus (PS), 90, 168 Public transit, 13e14, 241e242 model description, 241e245 numerical case study, 249e256 analysis of equilibrium states, 250e253 analysis of profit-and/or social welfare maximising strategies, 254e256 optimal strategy design, 245e249 services, 242 Python, 278e279

Q Q function, 91e92, 101e102, 124e125, 172, 178e179 Queuing models, 6, 31e33, 42, 45e46, 49, 56, 61

385

R r-squared values, 345e346 Random matching without optimisation objective, 333 Real data, 210 Real-time on-demand dynamic ride-sharing services, 107 Real-time supply, 261e262 Real-time supplyedemand conditions, 273 Reasonable balance, 374 Reasonable model, 241e242 Reciprocal interactions, 107 Reference-dependent preference theory, 285, 289e290 Regular private car users, 135 Regulations, 7e8, 94e103. See also Government regulations commission ratio regulation, 99e100 commission regulation, 98 demand regulation, 101 fleet size regulation, 95e96 income regulation, 97e98 minimum utilisation rate regulation, 100e101 numerical illustrations, 101e103 outcomes of regulatory regimes, 102t price-cap regulation, 94e95 wage regulation, 96e97 Reinforcement learning, 15e16 Research context, 300 Research design, 300e308 data description, 302e303 driver classification along extensive and intensive margins, 304 empirical analysis, 304e308 large-scale natural experiment, 300e302 research context, 300 Research questions, 285 Revenue management, 201 Revenue-maximising problem, 119 Ride sourcing, 13e14 Ride-hailing, 1 Ride-matching system, 375e376 Ride-pooling market, 161e162 routing sequences, 114f services, 8e10, 107, 323e324, 327e328, 374e375 vehicles, 324 Ride-sourcing, 107 market, 2e3, 2f platform, 2e3

386 Index Ride-sourcing (Continued ) platforms, 283, 293, 300e301 model of labour supply elasticity on, 297e300 services, 1 systems, 3, 262e263, 328 theoretical developments, 3e16 vehicles, 136, 242 Ride-sourcing markets, 55e56, 107, 133e134, 138, 373 analysis of experimental results, 68e76 characteristic, 56 equilibrium analyses, 25 driver supply, 26e27 market measures, 39e50 matching frictions, 27e39 passenger demand, 25e26 experimental settings, 65e67 matching functions and market metrics, 57e64 Ride-sourcing services, 55e56, 147, 191, 239 model description, 241e245 numerical case study, 249e256 analysis of equilibrium states, 250e253 analysis of profit-and/or social welfare maximising strategies, 254e256 optimal strategy design, 245e249 Ride-splitting (RS), 134, 147e148 Road network, 241e242 Road segments, 346e348 Robustness of empirical laws, 332e333 Rolling horizon model, 15e16

S S maximisation problem, 92e93, 173e174 Sample selection, 285 and endogeneity bias, evidence of, 309e314 Saturation curve, 341 Second-best solution (SB solution), 148e149, 245 Second-order partial derivatives of T, 270 Selection equation, 309 Sensitivity analysis, 333e334 Sequential decision-making algorithms, 15e16 Shared mobility, 112 Simulation, 57 platform, 275 simulation-based sensitivity analysis, 57 Simulator, 65e66 flowchart, 65f

processes in, 65 Smartphone applications, 239 Social optimum (SO), 6e7, 39e40, 43e47, 87e88, 107, 120e123, 133, 141e143, 155e156, 168, 173e175, 193e194, 245, 247e248, 374. See also Monopoly optimum (MO) effect of platform integration at, 209e210 effect of vehicle fleet size at, 207e208 Social welfare, 32e33, 39e40, 173e174, 203e204, 241, 374 analysis of social welfare maximising strategies, 254e256 maximisation problem, 32e33, 247 Spatial density, 324e325 dataset description, 334t spatial distribution of trip origins and destinations in various road networks, 335f Spatial distribution, 264 Spatial heterogeneity, 373 Spatial Poisson distribution, 265 Spatial probability model, 262e263 Speed function, 135e138 Standard deviation, 279 Standard labour economics theory, 284, 292 Stationary equilibrium, 25e26, 375e376 state, 4e6 of ride-sourcing market, 172 Status of drivers and passengers, 66 Stochastic online matching process, 278e279 Stochastic ride-sourcing system, 376 Street-hailing taxi market, 139 Substitutes for public transit, 241e242 Supply function, 113e115, 138e139, 151e152 SUV, 1 System performance, 278e280 measure, 266e267

T Taxi, 1 Third-party platform integration in ridesourcing markets background, 191e194 evaluation of performance of platform integration, 207e210 market equilibrium and optimal strategies, 194e206 numerical studies, 210e219 Time-conservation condition, 34, 138 Traditional taxi industry, 191, 283

Index Traffic congestion, 133, 147e149, 167, 323e324, 375 Traffic simulators, 57 Transportation network companies (TNCs), 1, 14, 107, 133, 147, 169e170, 239, 323 Travel distance, 330e331 Trip fare, 133 Trip-request record, 334e336 Two-dimensional space of Dt, 275 Two-sided matching market, 292 Two-sided matching model, 5e6 Two-stage least squares (2SLS), 289 regression analysis, 296e297

U Uber, 1, 5, 7e8, 239 Uber Black, 1 Uber Express Pool, 9e10, 112e113, 331e332 Uber services, 147 UberPool, 1, 8, 107 UberX, 1 Univariate function, 34 Universal hornshaped distribution law, 265e266 Urban mobility, 1 systems, 239e240 Urban transportation systems, 239 Utilisation rate regulation, 181t Utility function, 289e290

387

V Vacancy rate, 87e88 Validation, 77 Value of time (VOT), 242 Variable description, 302e303 Vehicle conservation, 149e150 constraint, 200e202 function, 113 Vehicle fleet size, 133 effect of, 214e217 vehicle fleet size at NE/SO, 207e208 Vehicle routing distance, 327e328 Vehicle utilisation, 324, 373e374 rate, 207e208, 215e216 Vehicle-conservation equation, 63e64, 170

W Wage regulation, 96e97 Waiting passengers, 37e39, 266e267 Welfare-maximising problem, 196e197 Wild goose chase regime (WGC regime), 34e35, 116, 139, 152, 197, 326e327 Winewin situation, 148 Wireless communication technologies, 283 Workingehour elasticity, 304e305 WueHausman test, 309e314

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