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Bounded Rational Choice Behaviour: Applications in Transport
 9781784410728, 1784410721

Table of contents :
Bounded Rational Choice Behaviour: Applications in Transport......Page 4
Copyright page......Page 5
Contents......Page 6
List of Contributors......Page 8
Preface......Page 10
Frontiers in Modelling Bounded Rationality in Travel Behaviour Research: Introduction to the Collection of Chapters......Page 12
References......Page 20
Chapter 1 Models of Bounded Rationality under Certainty......Page 22
1.1 Framework......Page 25
1.2 Non-Optimal Decision Mechanisms......Page 27
1.3 Considering a Subset of Influential Attributes......Page 35
1.4 Indifference between Small Differences in Utility or Attribute Values......Page 39
1.5 Considering a Subset of Choice Alternatives......Page 40
1.6 Applications......Page 44
1.7 Conclusions and Discussion......Page 46
References......Page 47
2.1 Introduction......Page 52
2.2.1 The Generalised RRM Model......Page 54
2.2.2 Mixture Model......Page 58
2.3.1 Data......Page 59
2.3.2 Results......Page 60
2.4 Summary and Conclusions......Page 67
References......Page 68
3.1 Introduction......Page 70
3.2.1 General Formulation......Page 72
3.2.2 Alternative-Oriented Relative Utility......Page 73
3.2.4 Endogenous Modelling of Choice Set Generation......Page 75
3.2.5 Reflecting the Non-linearity of Context Dependency......Page 76
3.3 Major Findings of Existing Studies......Page 77
3.3.1 Choices of Destinations and Stop Patterns......Page 78
3.3.2 Dynamic Travel Mode Choice......Page 79
3.3.3 Travel Information and Travel Behaviour......Page 80
3.3.4 Choices of Packaged Tours......Page 82
3.3.5 Choice Set Generation......Page 83
3.4.1 Modelling Improvements......Page 85
3.4.2 Model Estimation......Page 87
3.5 Conclusion......Page 89
References......Page 91
Chapter 4 The Influence of Varying Information Load on Inferred Attribute Non-Attendance......Page 94
4.1 Introduction......Page 95
4.2 Methodology......Page 98
4.3 Data and Model Specifications......Page 101
4.4 Results......Page 105
4.5 Discussion......Page 112
References......Page 114
Chapter 5 The Heterogeneous Heuristic Modeling Framework for Inferring Decision Processes......Page 116
5.1 Motivation......Page 117
5.2.1.1 Preference structure......Page 119
5.2.1.2 Deriving decision heuristics......Page 121
5.2.1.3 Choice of heuristics......Page 122
5.2.1.4 Mental effort......Page 123
5.2.1.5 Risk perception......Page 124
5.2.2 Extension to Comparative Decision......Page 125
5.3.1.1 Model specification......Page 127
5.3.1.2 Results......Page 128
5.3.2.2 Results......Page 130
5.4 Conclusion and Future Work......Page 132
References......Page 133
Chapter 6 Investigating Situational Differences in Individuals’ Mental Representations of Activity-Travel Decisions: Progress and Empirical Illustration for the Impact of Online Alternatives......Page 136
6.1 Introduction......Page 137
6.2.1 Theory and Concepts......Page 138
6.2.2 Measuring Mental Representations......Page 140
6.3.1 Choice Task and Experimental Design......Page 143
6.3.3.1 The complexity of respondents’ mental representations......Page 144
Number of cognitive subsets......Page 146
Ranking of decision variables......Page 147
The frequency of elicited attributes......Page 148
The frequency of elicited cognitive subsets......Page 149
Centrality of variables......Page 150
6.3.4 Conclusion......Page 151
6.4 Conclusions and Discussion......Page 152
References......Page 154
7.1 Introduction to Decision Theory......Page 158
7.2.2 Decision Theory in AB Models......Page 160
7.3.1 General Concepts......Page 162
7.3.2 Parameter Learning......Page 164
7.3.3 Entering Evidences......Page 165
7.3.4 Structural Learning......Page 166
7.4 Bayesian Network Classifiers: Problem Formulation......Page 167
7.5 Towards A New Integrated Classifier......Page 170
7.6.1 Data......Page 172
7.6.2 Design of the Experiments......Page 173
7.7.1 Model Comparison: Accuracy Results......Page 174
7.7.2 Model Comparison in Terms of Model and Individual Rule Complexity......Page 175
7.7.3 Activity Pattern Level Analysis......Page 177
7.7.4 Trip Matrix Level Analysis......Page 178
7.8 Conclusion and Discussion of the Results......Page 179
References......Page 180
Chapter 8 Bounded Rationality in Dynamic Traffic Assignment......Page 184
8.2.1 Problem Definition......Page 185
8.2.2 Example......Page 186
8.2.3 Travel Flow Component......Page 188
8.2.4 Wardrop’s First Principle and its Dynamic Extensions......Page 189
8.3 Bounded Rationality in Traffic Assignment......Page 191
8.4.1 Tolerance-Based Dynamic User Optimal Principle......Page 193
8.4.2 Nonlinear Complementarity Problem (NCP) Formulation......Page 195
8.4.3 Solution Existence and Uniqueness......Page 196
8.4.4 Solution Method......Page 198
8.5 Boundedly Rational Dynamic user Equilibrium Route and Departure Time Choice Assignment......Page 200
8.6 Concluding Remarks......Page 202
References......Page 203
Chapter 9 Incorporating Bounded Rationality in a Model of Endogenous Dynamics of Activity-Travel Behaviour......Page 210
9.1 Introduction......Page 211
9.2 The Model......Page 212
9.2.2 Decision-Making Process......Page 213
9.2.2.1 Cognitive responses......Page 214
9.2.2.3 Choice set formation......Page 215
9.2.2.4 Aspirations and stress......Page 216
9.2.2.5 Habit formation......Page 217
Exploration choice mode......Page 218
Becoming ‘awake’......Page 219
9.3.1 Simulation Settings......Page 220
9.3.2 Basic Case Results......Page 221
9.3.3.1 Effect of λ2 parameter......Page 223
9.3.3.2 Effect of γ parameter......Page 225
9.3.4.1 Effect of α2 parameter......Page 226
9.3.4.2 Effect of α1 parameter......Page 228
9.4 Conclusions and Discussion......Page 230
References......Page 232
Chapter 10 Multidimensional Travel Decision-Making: Descriptive Behavioural Theory and Agent-Based Models......Page 234
10.1 Background......Page 235
10.2 Theory and Models......Page 236
10.2.1 Modelling Imperfect Knowledge......Page 238
10.2.2 Modelling Multidimensional Search......Page 240
10.2.3 Search Rules and Decision Rules......Page 243
10.2.4 Empirical Data Collection......Page 245
10.3 Simulation Results......Page 246
10.4 Discussion and Conclusion......Page 249
References......Page 251
Chapter 11 Prospect Theory and its Applications to the Modelling of Travel Choice......Page 254
11.1 Introduction......Page 255
11.2 Making Travel Choices under Risk: The Assumptions of Expected Utility Theory (EUT) and Prospect Theory (PT)......Page 256
11.3 Cumulative Prospect Theory (CPT)......Page 261
11.4 Numeric Example......Page 263
11.5 Incorporating Prospect Theory in Travel Choice Modelling — Application Areas and Evidence......Page 265
11.6 Prospect Theory and Behavioural Change......Page 268
11.7 Shortcomings and Limitations of Prospect Theory in Modelling Travel Behaviour......Page 269
11.8 Summary and Conclusions......Page 270
References......Page 271
About the Authors......Page 278
Index......Page 286

Citation preview

BOUNDED RATIONAL CHOICE BEHAVIOUR: APPLICATIONS IN TRANSPORT

BOUNDED RATIONAL CHOICE BEHAVIOUR: APPLICATIONS IN TRANSPORT

EDITED BY

SOORA RASOULI Urban Planning Group, Eindhoven University of Technology, Eindhoven, The Netherlands

HARRY TIMMERMANS Urban Planning Group, Eindhoven University of Technology, Eindhoven, The Netherlands

United Kingdom • North America • Japan India • Malaysia • China

Emerald Group Publishing Limited Howard House, Wagon Lane, Bingley BD16 1WA, UK First edition 2015 Copyright r 2015 Emerald Group Publishing Limited Reprints and permissions service Contact: [email protected] No part of this book may be reproduced, stored in a retrieval system, transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without either the prior written permission of the publisher or a licence permitting restricted copying issued in the UK by The Copyright Licensing Agency and in the USA by The Copyright Clearance Center. Any opinions expressed in the chapters are those of the authors. Whilst Emerald makes every effort to ensure the quality and accuracy of its content, Emerald makes no representation implied or otherwise, as to the chapters’ suitability and application and disclaims any warranties, express or implied, to their use. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-1-78441-072-8

ISOQAR certified Management System, awarded to Emerald for adherence to Environmental standard ISO 14001:2004. Certificate Number 1985 ISO 14001

Contents

List of Contributors

vii

Preface

ix

Frontiers in Modelling Bounded Rationality in Travel Behaviour Research: Introduction to the Collection of Chapters

xi

1.

2.

3.

4.

5.

6.

Models of Bounded Rationality under Certainty Soora Rasouli and Harry Timmermans

1

Utility Maximisation and Regret Minimisation: A Mixture of a Generalisation Stephane Hess and Caspar G. Chorus

31

Relative Utility Modelling Junyi Zhang

49

The Influence of Varying Information Load on Inferred Attribute Non-Attendance Andrew T. Collins and David A. Hensher

73

The Heterogeneous Heuristic Modeling Framework for Inferring Decision Processes Wei Zhu and Harry Timmermans

95

Investigating Situational Differences in Individuals’ Mental Representations of Activity-Travel Decisions: Progress and Empirical Illustration for the Impact of Online Alternatives Oliver Horeni, Theo Arentze, Benedict G. C. Dellaert and Harry Timmermans

115

vi

Contents

7. Towards a Novel Classifier for the Representation of Bounded Rationality in Models of Travel Demand Davy Janssens and Geert Wets

137

8. Bounded Rationality in Dynamic Traffic Assignment WY Szeto, Yi Wang and Ke Han

163

9. Incorporating Bounded Rationality in a Model of Endogenous Dynamics of Activity-Travel Behaviour Ifigenia Psarra, Theo Arentze and Harry Timmermans

189

10. Multidimensional Travel Decision-Making: Descriptive Behavioural Theory and Agent-Based Models Chenfeng Xiong, Xiqun Chen and Lei Zhang

213

11. Prospect Theory and its Applications to the Modelling of Travel Choice Erel Avineri and Eran Ben-Elia

233

About the Authors

257

Index

265

List of Contributors

Theo Arentze

Urban Planning Group, Eindhoven University of Technology, Eindhoven, The Netherlands

Erel Avineri

Engineering and Management of Infrastructure Systems, ACITRAL  Afeka Center for Infrastructure, Transportation and Logistic, Afeka  Tel-Aviv Academic College of Engineering, Tel Aviv, Israel

Eran Ben-Elia

Gurion University of the Negev, Department of Geography and Environmental Development, Beer-Sheva, Israel

Xiqun Chen

Department of Civil and Environmental Engineering, University of Maryland, College Park, MD, USA

Caspar G. Chorus

Faculty of Technology, Policy and Management, Delft University of Technology, Delft, The Netherlands

Andrew T. Collins

Institute of Transport and Logistics Studies, The University of Sydney Business School, The University of Sydney, Sydney, Australia

Benedict G. C. Dellaert

Department of Business Economics, Erasmus School of Economics, Erasmus University Rotterdam, Rotterdam, The Netherlands

Ke Han

Department of Civil and Environmental Engineering, Imperial College London, London, UK

David A. Hensher

Institute of Transport and Logistics Studies, The University of Sydney Business School, The University of Sydney, Sydney, Australia

viii

List of Contributors

Oliver Horeni

Verkehrsverbund Oberelbe Gmb, Dresden, Germany

Stephane Hess

Institute for Transport Studies, University of Leeds, Leeds, UK

Davy Janssens

Transportation Research Institute, Hasselt University, Diepenbeek, Belgium

Ifigenia Psarra

Urban Planning Group, Eindhoven University of Technology, Eindhoven, The Netherlands

Soora Rasouli

Urban Planning Group, Eindhoven University of Technology, Eindhoven, The Netherlands

WY Szeto

Department of Civil Engineering, The University of Hong Kong, Hong Kong

Harry Timmermans

Urban Planning Group, Eindhoven University of Technology, Eindhoven, The Netherlands

Yi Wang

Department of Civil Engineering, The University of Hong Kong, Hong Kong

Geert Wets

Transportation Research Institute, Hasselt University, Diepenbeek, Belgium

Chenfeng Xiong

Department of Civil and Environmental Engineering, University of Maryland, College Park, MD, USA

Lei Zhang

Department of Civil and Environmental Engineering, University of Maryland, College Park, MD, USA

Junyi Zhang

Graduate School for International Development and Cooperation, Hiroshima University, Higashi Hiroshima, Japan

Wei Zhu

Department of Urban Planning, Tongji University, Shanghai, China

Preface

The development of scientific disciplines has all the properties of man-made artificial systems. Although one would expect that scientific evidence is the main driver of the survival and perseverance of theories and models, academic networks are institutionalised in terms of journals, conferences and other means of dissemination. Quality tends to be peer-reviewed, but the process is subjective or inter-subjective at best. Like in any social system, highly respected scholars serve as sources of inspiration, but at the same time tend to be the gatekeepers of the historical development of the discipline and acceptance standards. For very good reasons, new approaches are typically critically assessed under much or too much scrutiny, implying they may not receive the attention they deserve. There are signs of self-selection as chances of acceptance may decrease if one deviates too much from the state of the art. Incremental contributions tend to be applauded; divergent views need more convincing. Although the transport community is known for its balance between accumulative research within long-standing modelling approaches, supported and sustained by continuous training and dissemination practices, and constructive openness to new ideas, some fundamental foundations of transport research were largely left unchallenged or were never put on the agenda for decades. The notion of equilibrium and the principle of rational choice behaviour have been the cornerstones of the disciplines for the last 40 years. Without any doubt, these concepts have played a pivotal role in the development of the models that have become commonly used in transportation planning practice. In turn, accepted practice cannot be disentangled from these basic principles. At the same time, however, the principle of fully rational behaviour lacks behavioural realism. Nevertheless, compared to other disciplines, attempts to explore the possibilities of formulating alternative models of activity-travel behaviour, derived from principles of bounded rationality, have been limited in number in the travel behaviour community. In part, this may be because transportation is primarily an applied engineering science, and as such less concerned with more fundamental explanations of observed behaviour. However, the very nature of the decisionmaking processes underlying activity-travel behaviour, characterised by a relatively stable of antecedent conditions and instrumental in kind, may not need a more subtle and varied set of behavioural principles and mechanisms. In any case, although models of bounded rationality have been around in travel behaviour research since its inception, they never have played a central role in this

x

Preface

research community. This book, based on a special session organised at the Transportation Research Board Annual Meeting and subsequent invited chapters, represents an attempt to put the spotlight on promising models of bounded rationality. There are several reasons why these models should be put on the centre stage. First, we have the feeling that conventional theories and discrete choice models, based on principles of fully rational behaviour have entered the last stage of their lifecycle. They have fully matured and there is little evidence of much further progress. Second, the application of models of transport demand forecasting has gradually shifted from long-term investment in infrastructure policy decisions to shortterm transport management decisions. It implies a refocusing of modelling approaches on concepts such as uncertainty, adaptation and inertia, which do not particularly lend themselves the basic premises of classic economic theories. Third, choice behaviour itself is changing in the sense that increasingly more transportrelated choices have become instantaneous choices made under time pressure. It changes the very nature of the underlying decision-making process and may necessitate different modelling frameworks. Fourth, the still rapidly increasing computer power and availability of varied, real time data sets in principle no longer limits to the specification and conceptual richness of models. This incentive to focus attention to models of bounded rationality does not reflect any claim that such models are necessarily better than conventional utilitymaximising models. Our position is that the relevance of any model depends on the processes that it is supposed to represent and how much value is attached to face and construct validity versus its predictive performance. Ultimately, the transportation and travel behaviour community is served when systematic model comparisons are made and debate challenges the limitations of any particular model as opposed to uncritically cherishing its merits. The challenge is to develop models of bounded rationality with equal rigour and if that turns out impossible to discuss the implications for underlying methodological issues. We trust that this book contains sufficient food for thought and will contribute to additional future work on making models of bounded rationality full competitors of our currently dominant models. Soora Rasouli and Harry Timmermans Eindhoven, August 2014

Frontiers in Modelling Bounded Rationality in Travel Behaviour Research: Introduction to the Collection of Chapters

The core business of transportation planners and engineers is to design, engineer and maintain infrastructure and transportation policies that reflect the needs of people and firms, meet particular norms and costs requirements and achieve particular societal objectives related to the environment (noise, emissions, etc.). The assessment of these design and policies requires travel demand forecasting models and models that predict traveller response to changing land use and transportation systems. Behavioural forecasts can then be turned into the performance indicators that are deemed relevant to evaluate a design or policy. It is no surprise, therefore, that in light of the relevance of travel demand forecasting models in design processes transportation research has a rich history in developing and applying various types of travel forecasting models. For long, the four-step model has dominated travel demand forecasting. This approach predicts travel demand according to four separate and independent steps: trip generation, trip distribution, transport mode choice and trip assignment. More recently, several types of cross-sectional activity-based models of travel demand have gradually replaced the four-step model in academic research (Rasouli & Timmermans, 2014a). Practice has followed, although the rate of dissemination of activity-based models varies considerably from country to country. On balance, at this moment in time, the four-step models still dominate transportation practice, while academia has moved to develop the next generation of activity-based models of travel demand: dynamic activity-based models. Zooming in on the kind of models that underlie these competing approaches, the trip distribution module of four-step models has traditionally been based on production-constrained or doubly-constrained spatial interaction models. Basically, production-constrained models assume that the probability of a trip arriving at a particular destination is proportional to the attractiveness of the destination, negatively proportional to some travel distance or travel costs function and negatively proportional to the sum of attractiveness/impedance ratios of competing destinations. Doubly constrained models are based on the same set of assumptions, but are scaled such that both the production and attraction of trips for all origins and all

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Introduction

destinations are equal to the observed total number of trips departing from the origins and arriving at the destinations. Many different specifications of the attractiveness and deterrence function exist, but a detailed discussion of the development of spatial interaction models is beyond the scope. Useful introductions and reviews of spatial interaction models can be found in Hayes and Fortheringham (1984) and O’Kelly (2009). One of the criticisms of four-step models, and consequently against spatial interaction models, concerned their lack of behavioural foundations. The models were copied from physics and represent in statistical terms macroscopic aggregate relationships between spatial units (zones, districts). Although the mathematical expressions have been given various economic interpretations (e.g. Anderson, 2011), zones do not make any decisions, and the total number of trips is not the outcome of an individual travel decision. Thus, spatial interaction models describe regularities in aggregated decision outcomes of individuals, not the decisions of individuals themselves. Based on the argument that models capturing individual and household decisions processes and choice behaviour are superior forecasting tools compared to models that describe statistical regularities in aggregate distributions, developments in categorical data analysis led to the formulation of models of individual choice behaviour. The multinomial logit model soon became the benchmark in modelling transport mode, destination and route choice decisions. Many more advanced discrete choice models followed to relax the limiting assumptions underlying the MNL model, allowing for substitution effects. Although it should be noted that the mathematical expression of the MNL can logically be derived from several different, even fundamentally conflicting, theoretical constructs, the MNL model and many of its variants have been predominantly linked to random utility theory. Random utility theory assumes that individuals derive a utility from the chosen alternative. This utility consists of a deterministic part and an error term. Consequently, individuals are assumed to have stochastic preferences. In addition, they are assumed to maximise their utility when choosing a single alternative from the available options. Assumptions about the error terms of the utility functions then, ceteris paribus, dictate the probability of choosing a particular alternative. Random utility theory can be viewed as an example of rational decision-making. The term ‘rational’ has received multiple definitions and interpretations, but in the context of travel demand forecasting, it is commonly been used to indicate that the concept of utility maximisation refers to the best or optimal choice. Rational means that the decision-maker will systematically evaluate all available choice alternatives and select the best, based on reason (i.e. a cognitive process), from the possible choices. Models based on the principle of rationality assume that an individual will define the set of attributes that is important to the decision-making problem. Next, an individual will cognitively assess the outcomes of his possible decision for each alternative in the choice set and choose the best option. The cognitive decisionmaking process involves processing the various attributes and arrive at some overall judgement by integrating the evaluation of the various attributes according to some

Introduction

xiii

integration function. The notion of cognitive processing of attributes implies the negation of affective responses in decision-making. Implicitly, models of rational choice behaviour assume that individuals have no limitations in processing attributes and choice alternatives. A true optimal choice can only be made if an individual has full and perfect information of all relevant attributes of all alternatives in his/her choice set. A more realistic set of assumptions, however, would state that individuals have partial, imperfect, biased cognitive representations of reality. Of course, they can still act rationally and maximise the utility based on their subjective beliefs and mental representation of reality. On the other hand, an individual is said to demonstrate bounded rational behaviour if he/she does not systematically consider all attributes deemed relevant for the decision problem at hand, does not systematically consider all relevant choice options and/or does not choose the best choice alternative. Such simplified representation and limited processing may occur due to time budget constraints, low involvement in the decision problem, relying on habits or too high mental effort. Choice models have been developed for decision under conditions of certainty and under conditions of uncertainty. In transportation and travel behaviour research, the vast majority of choice models have assumed that individuals have full and perfect information about the choice alternatives and their attributes. Even though attributes may be inherently uncertain, single attribute values have been used in the choice models. Only recently, the travel behaviour community has slightly increased its interest into decision-making under conditions of uncertainty. The equivalent to the random utility model of rational behaviour is the expected utility model, which is based on a similar set of assumptions and adds the assumption that individuals weigh their attribute utility with the (subjective) probability of states of the world and that they choose the alternative that provides the maximum expected utility. Because the results of many experiments violated the model predictions, several other models for decision-making under uncertainty have been advanced in economics, social psychology and decision sciences, and some of these have also found their way into travel behaviour research. Prospect theory and more recently regret theory have been most popular in this regard. A recent overview of studies based on prospect theory can be found in Li and Hensher (2011), while Rasouli and Timmermans (2014b) summarised a wider set of modelling frameworks. In the models, the notion of bounded rationality has focused on the inclusion of reference points and the curvature of the utility or value function. We realise that a more fundamental stance on rational versus bounded rational behaviour can be taken. However, the notion of simplifying the choice task serves well to position different modelling approaches that have tried to develop models of decision-making and choice that can be viewed as alternatives to the dominantly used random utility models and their underlying premises. This volume contains a set of chapters that describe the latest developments in a particular model or modelling approach of bounded rationality. We have organised this book into two main sections. First, models of bounded rationality for decisionmaking under conditions of certainty will be presented. Next, in a smaller section,

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Introduction

the focus of attention will shift to alternatives to expected utility-maximising models of decision-making under uncertainty. To set the stage, we provide a review of the history of models of bounded rationality in urban planning and transportation research, which has addressed choice and decision problem under certain conditions. This chapter serves to discuss modelling approaches and model specifications that are not discussed elsewhere in this book. Consequently, this chapter also allows readers to position and value the approaches and models that are discussed in the various chapters, against this earlier literature. In line with this introduction, the chapter uses the different ways of simplifying the choice problems as the organisational principle to discuss the models of bounded rationality. Regret-based models have been developed as an alternative to classic utilitymaximising models, both for conditions under certainty and uncertainty. These models are based on the premise that individuals minimise regret when choosing between alternatives. Under conditions of certainty, it implies that regret associated with a choice alternative is a function of attribute differences between the considered choice alternative and one or more other alternatives in the choice set. Another important development in choice modelling is the concept of behavioural mixing: the notion that different individuals may employ different decision rules when arriving at a choice. In the second chapter of this book, Hess and Chorus present the results of a modelling approach that combines the notion of behavioural mixing with the most recent generalised version of the random regret model, which has the random utility-maximising model, random regret minimisation model and hybrid models as special cases. Thus, their model accounts for heterogeneity in decision rules across individuals and attributes. A latent class structure is estimated, in which the classes represent different decision rules. Results support the potential value of the suggested approach. The generalised random regret minimisation model expresses regret in terms of a function of attribute differences between choice alternatives. It has this feature in common with relative utility models, which were introduced in travel behaviour about a decade ago. It raises the question about the similarity of these modelling approaches. Zhang addresses this issue in his contribution to this volume. After discussing the motivation behind the formulation of relative utility models, the original model specification and the formulation of elaborated models, he shows how not only regret minimising models, but also other context-dependent choice models and prospect theoretic models can be accommodated in this modelling framework. Results of examples of the application of various specifications of relative utilitymaximising models show that these models often outperform classic random utilitymaximising models, but that overall differences in explanatory power tend to remain limited. To some extent, this may reflect the insensitivity of our current apparatus to detect differences in model performance, but it also expresses the fact that critical differences in predicted choices between different models tend to be small. As discussed, bounded rational behaviour can be reflected in individuals simplifying the choice task by considering only a subset of attributes when making choices. Recently, advanced choice models addressing the issue of attribute non-attendance

Introduction

xv

have been developed. While much of this work has been focused on information processing in stated preference and choice experiments, there is no reason to assume that similar reduction of task complexity will not be operant in real-world choice and decision-making. Collins and Hensher provide a detailed review of the historical evolution of various attribute non-attendance models that have been suggested, primarily in the transportation and in the environmental economics literature. They present and illustrate a random parameters attribute non-attendance model to simultaneously infer attribute non-attendance and handle preference heterogeneity. Using stated choice data on route choice of commuters under travel time uncertainty and one or more time and cost attributes, their results indicate that attribute non-attendance becomes more prevalent with an increasing number of attribute levels, a decreasing number of choice alternatives and an increasing number of attributes. Zhu and Timmermans also address the problem that individuals may not consider all potentially relevant attributes when making a decision. Rather than assuming a single threshold, they define a series of successive activation levels. In addition to the use of activation thresholds, defined at the attribute levels, an overall threshold is estimated, which differentiates the choice alternatives into accepted and rejected alternatives. Different overall thresholds then represent different non-compensatory decision rules, such as disjunctive, conjunctive and lexicographic rules. For this reason, they call their model a heterogeneous decision rule framework. The probability that a particular rule will be used is a probabilistic function of mental effort, risk perception and expected outcome. This approach is unique for travel behaviour research where choices are usually modelled in terms of some performance measure of decision outcomes and not in terms of cognitive processes. Differences in mental effort occur because the different non-compensatory decision rules involve a different degree of processing the attributes. Risk perception depends on the setting of the threshold. Little mental effort may imply some opportunity costs related to the expected regret that results from making an inferior decision. Shannon’s entropy measure is used to represent risk perception. Finally, expected outcome measures the extent the use of a decision rule leads to preferred outcomes. Results of applications of the model to aspect of pedestrian movement show that it represents observed data slightly better than utility-maximising multinomial logit models. As indicated, attribute non-attendance models have been predominantly developed in the context of stated choice experiments. Although it is likely that individuals also apply simplifying decision heuristics in real-world settings, some differences between real-world decision-making and decisions in quasi-laboratory settings prevail. In stated choice experiments, subjects have to understand the experimental task, relate it to their personal decision context, process the attributes and their levels, and the choice alternatives and choice sets, and then try to apply their internalised preference structures to the reconstructed experimental task. Selectivity and representation bias may occur in this process. By contrast, when faced with a decision to be made, in real-world settings individuals need to apply their preference functions to attribute levels of the choice alternatives that are retrieved from their memory, which holds a cognitive representation of the

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Introduction

environment, build up over time as a function of experiencing the outcomes of previous decisions and possibly other active and/or passive sources of information. Contextual circumstances, such as the degree of the involvement in the decision and the available amount of time to make the decision, will dictate the amount of retrieval from memory, leading to simple, highly reduced or quite detailed mental representations of the decision problem. In their chapter, Horeni, Arentze, Dellaert and Timmermans sketch a conceptualisation of this problem, develop a computer-based tool to measure mental representations and, based on a case study on shopping choice behaviour, provide evidence that mental representations vary significantly between individuals and choice contexts. In addition to providing a framework for attribute non-attendance and corresponding mental representations, which has been addressed mainly from a technical perspective in the literature, another key element of their approach concerns the representation of benefits and causal relationships between attributes. It suggests that in addition to heterogeneity in observed characteristics and decision rules, additional heterogeneity due to different, contextdependent mental representations of reality and the specific decision problems should be addressed in models of activity-travel behaviour. Utility-based choice theories and models are based on the postulate that individuals derive some utility from the attribute levels of the choice alternatives and then choose the alternative that will maximise their utility. Thus, observed choice outcomes are interpreted to reflect maximum utility; it is assumed that valid preference functions can be derived from observed choices. Regardless of the question whether this claim is justified, by contrast the theoretical considerations underlying computational process models of travel behaviour highlight the notion that by making repeated choices individuals learn their environment and experience which decisions are more satisfying and which are less satisfying under a given set of circumstances. Over time, individuals are believed to develop a set of decision heuristics, which indicate which decision or action to take under a set of conditions. In principle, different formalisms can be used to represent these conditional action or decision rules. In transportation, decision tables have been predominantly used. Janssens and Wets suggest a novel and improved approach by combining commonly used decision tables and Bayesian Belief Networks. More specifically, their proposal is not to derive the decision tables from the observations as is usually done, but rather from the Bayesian network, which is built upon the original data. The potential advantage is that the model is more stable because the Bayesian network already captures the correlations among the conditions triggering the choice. The authors apply the suggested approach to the original Albatross data and find mixed results. The derived decision tree indeed turned out to be structurally more stable and less vulnerable to the variable masking problem. However, at a more detailed level, the classic decision table extraction approach has benefits. From an activity-based perspective of travel demand, models of travel demand forecasting predict the (combined) choice of activity, travel party, destination, transport mode, departure time, activity duration and route. The limited number of studies, grounded on principles of bounded rationality, has typically examined problems of individual choice behaviour for one of these choice facets. Slightly

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more scholars have examined the problem of dynamic route assignment from the perspective of bounded rationality. The notion of bounded rationality in this domain of study has also been subject of varying and often too vague definitions, missing mathematical rigour. Szeto, Wang and Han deliver a good introduction to the dynamic traffic assignment, the alternative meanings of the notion of bounded rationality in this field of study and the latest developments. Bounded rationality in route choice implies that the travel times of all selected routes between the same origin-destination are all the same within some defined acceptable tolerance threshold from the minimum travel time. They present (heuristic) solution methods for this objective and discuss existence and uniqueness of solutions. Finally, an extension to the joint departure time route choice problem is discussed. The challenge of these approaches is the find a close form mathematical specification that is consistent with the attempted behavioural principles and which at the same time can be estimated. Consequently, there are limits to these kinds of model in general, and particularly in modelling complex dynamic processes and systems. To enrich the models, some agent-based model systems of decision-making processes that are based on principles of bounded rationality have been suggested. Two of these are included in this volume. First, Psarra, Arentze and Timmermans outline an agent-based model and illustrate its properties using numerical simulations that simulate dynamic choice behaviour in response to endogenous and exogenous change. Agents learn about their environment when making choices. Consequently, agents become aware of the choice alternatives in their environment, develop choice sets and build up context-dependent cognitive representations about the attributes of the alternatives in their choice set. It leads to dynamically updated beliefs about the state of the world. Over time, if a choice alternative has not been visited forgetting is also simulated, implying that choice alternatives have different activation levels. In addition to this cognitive mechanism, agents build up affective beliefs, which are defined as a function of the discrepancy between expected and experienced utility and act on those. At the same time, agents have context-dependent aspirations, which may also change over time if after trying different behaviour they cannot be met. Endogenous change is triggered as a function of stress, which builds up if experienced utility is lower than the corresponding aspiration level. The agentbased system thus is capable of simulating very different dynamic behavioural trajectories of activity-travel behaviour, depending on the parameters setting. It will simulate the emergence of habitual behaviour from a state of complete unawareness of the environment if the agent’s environment allows a balance between aspiration levels and the utility that can be derived from the environment. It may also simulation lowering of aspirations levels or a change of residential and/or job locations if the current long-term decisions do not allow achieving aspirations levels associated with their activity-travel behaviour. The model system incorporates several mechanisms that assume agents do not maximise their utility and have perfect knowledge, but rather act in a bounded rational way. The numerical simulations reported in their chapter focus on the impact of memory-activation parameters, habit strength and the strength of emotional response. Results illustrate the effect of trade-offs between past and recent emotional experiences, and between cognitive and affective

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Introduction

responses. They indicate that higher dependence on emotional responses results in more exploration and decreasing aspiration values. Similarly, relying only on recent emotional experiences and ignoring accumulated past experiences leads to more disappointment and consequently exploratory behaviour. Xiong, Chen and Zhang present another computational process that also departs from rationality assumptions of classic models of activity-travel behaviour by imitating travel behaviour in terms of information acquisition, learning, adaptation and decision heuristics. Similar to Psarra et al., agents learn by acquiring information from different sources. Travel experiences reinforce positive behaviour and not visiting particular location leads to decay and forgetting of alternatives. An interesting feature of their model is that information acquisition and other mental efforts are explicitly modelled in terms of perceived search costs that are judged against subjective search gains to direct search behaviour. Agents apply a set of heuristics to activate their knowledge and identify alternatives. Principles of Bayesian updating are used to simulate learning and forgetting based on the recentness and representativeness of past experiences. Prior beliefs are assumed to follow a Dirichlet distribution. Different production rules, derived by applying various machine learning algorithms, are used to direct short-term departure time and route search, while long-term travel mode search is simulated using a hidden Markov process. The behaviour of the computation process model is illustrated for a small hypothetical network using stated adaptation data. Results witness the richness of the model in the sense that quite different dynamics can be simulated. Both these computational process models, however, also clearly demonstrate that the enhanced richness of the models and the inclusion of various process mechanism also imply that the impact of any particular variable cannot be directly assessed. Computational process models imply different causation regimes and may lead to quite different dynamics (from chaotic behaviour, via bifurcations to habitual behaviour) dependent on parameter settings. All above chapters relate to choice behaviour under conditions of certainty. Although it should be noted that relative utility and regret theory have also been developed for decisions under conditions of risk and uncertainty and that the theoretical foundations of particularly regret theory may appear stronger for that case, the number of studies in travel behaviour research applying these models to decision-making under risk and uncertainty is very limited indeed (Rasouli & Timmermans, 2014b). The equivalent of utility maximisation for the case of decision-making under conditions of uncertainty is maximisation of expected utility. Interestingly, because in many different fields of study, a huge body of empirical evidence has accumulated showing that the principle of maximisation of expected utility is often not congruent with human decision-making, it is often viewed as a normative theory of decision-making and alternative descriptive theories and models have been suggested. The most widely applied theory in this context is (cumulative) prospect theory. It asserts that outcomes of decision-making processes depend on the framing of the decision problem, that individuals differentiate between gains and losses and that risk attitudes work out differently in these two regimes. Consequently, models have a reference point and need non-linear specifications to

Introduction

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account for the typical violations of expected utility maximisation that have been documented in the literature. Seminal work on prospect theory in travel behaviour can be traced back to Avineri and his co-authors in their attempts to operationalise the key concepts of prospect theory in a travel behaviour context and judge the relevance of prospect theory for route choice departure choice and other choice problems in travel behaviour research. In this book, Avineri and Ben-Elia provide an excellent overview of the theoretical foundations of (cumulative) prospect theory, discuss the model specifications that have been applied and give a detailed account of the design and results of accumulated research in travel behaviour research that is based on these theoretical foundations that deviate from rational behaviour under conditions of risk and uncertainty. The potential of prospect theory for particular decision-making in travel behaviour research is clearly articulated, but limitations are also identified, leading to further research needs. This collection of chapters represents the frontier in travel behaviour research in endeavours to increase the behavioural realism of our model apparatus that is used to predict transport demand. The different approaches and models witness, all in their own right, how principles of bounded rationality can be incorporated into theories and models of choice and decision-making, both under conditions of certainty and uncertainty, as they are related to the different facets of activity-travel behaviour. These contributions, however, also evidence that increased realism tends to come with increased complexity. The number of parameters tends to increase. Moreover, while conventional models come with performance indicators such as willingness to pay and consumer surplus and straightforward equations for calculating (cross-)elasticities, for some of the models discussed in this volume, equivalent equations will be difficult or impossible to generate. Moreover, as discussed, some of these models of bounded rationality violate properties of classic models such as regularity, which the travel behaviour research community seems to have embraced, regardless of empirical evidence on the contrary. Furthermore, the estimation of some models of bounded rationality is far from standard, and may require dedicated software development. The lack of software to estimate a model of course should never be an excuse for not accepting or further exploring it, but it does indicate that substantial investment in the development, dissemination and discussion of alternative modelling approaches is needed. Soora Rasouli Harry Timmermans Editors

References Anderson, J. E. (2011). The gravity model. Annual Review of Economics, 3, 133160. Hayes, K. E., & Fotheringham, S. (1984). Gravity and spatial interaction models. Thousand Oaks, CA: Sage.

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Li, Z., & Hensher, D. (2011). Prospect theoretic contributions in understanding traveller behaviour: A review and some comments. Transport Reviews, 31, 97115. O’Kel, M. (2009). Spatial interaction models. International Encyclopedia of Human Geography, 2009, 365368. Rasouli, S., & Timmermans, H. J. P. (2014a). Activity-based models of travel demand: Promises, progress and prospects. International Journal of Urban Sciences, 18, 3160. Rasouli, S., & Timmermans, H. J. P. (2014b). Applications of theories and models of choice and decision-making under conditions of uncertainty in travel behavior research. Travel Behaviour and Society, 1(3), 7990.

Chapter 1

Models of Bounded Rationality under Certainty Soora Rasouli and Harry Timmermans

Abstract Purpose  This chapter reviews models of decision-making and choice under conditions of certainty. It allows readers to position the contribution of the other chapters in this book in the historical development of the topic area. Theory  Bounded rationality is defined in terms of a strategy to simplify the decision-making process. Based on this definition, different models are reviewed. These models have assumed that individuals simplify the decision-making process by considering a subset of attributes, and/or a subset of choice alternatives and/or by disregarding small differences between attribute differences. Findings  A body of empirical evidence has accumulated showing that under some circumstances the principle of bounded rationality better explains observed choices than the principle of utility maximization. Differences in predictive performance with utility-maximizing models are however small. Originality and value  The chapter provides a detailed account of the different models, based on the principle of bounded rationality, that have been suggested over the years in travel behaviour analysis. The potential relevance of these models is articulated, model specifications are discussed and a selection of empirical evidence is presented. Aspects of an agenda of future research are identified. Keywords: Lexicographic models; attribute non-attendance; choice set composition; regret models; decision rules

The study of travel behaviour concerns the description, analyses and modelling of decision processes related to multi-faceted travel behaviour. It aims at better understanding and predicting travel choices and how these co-vary with the decisionmakers’ personal traits and characteristics, attributes of the choice alternatives and

Bounded Rational Choice Behaviour: Applications in Transport Copyright r 2015 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISBN: 978-1-78441-072-8

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Soora Rasouli and Harry Timmermans

context. The focus may be on cross-sectional analysis or on dynamics as it relates to scripts, variability in activity-travel patterns and adaptation. Decision-makers can be individuals, households, friends, business partners, etc. Facets concern departure time, destination, day of the week, travel party, transportation mode(s) and route. Generally, the attributes depend on the facet or combination of facets being modelled, while context may relate to the urban setting, economic conditions, weather, time pressure, etc. The study of travel behaviour ranges from a focus on a single facet to the modelling of dynamic comprehensive activity-travel patterns. Forecasting travel behaviour is an essential component of feasibility and impact studies. The purpose of a feasibility study is to assess whether a project can achieve (financially or otherwise) a targeted performance. Performance measures often require a forecast of the number of people using the planned infrastructure or facility. Unless the new project attracts more people or expenditure from a larger market area than the minimum required to achieve feasibility, it will compete with the existing facilities in the market area. In that case, it is relevant to assess not only the feasibility, but also the impact the project will have on each of the competing facilities. Although under such conditions, the impact will be negative, the policy question is how negative and how the effects are distributed across the existing facilities. In this forecasting setting, if one is not satisfied with a statistical analysis only, but it is felt important to base the model on a theory of decision-making, preference or choice, then the researcher has to (i) select or formulate a theory of decisionmaking that is assumed valid for the problem at hand, (ii) translate the general theory into an operational model that mathematically expresses the functional relationship between decision outcomes and the set of individual and household characteristics, attributes of competing choice alternatives and context conditions. An adequate reproduction of observed decision outcomes is then seen as a validation of the behavioural postulates underlying the model. In turn, models based on behavioural theories are often implicitly or explicitly perceived as being superior to aggregate statistical models. A good example is the shift from aggregate spatial interaction models, embedded in four-step travel demand forecasting models, to activity-based travel demand models, which are based on behavioural postulates and mechanisms (Rasouli & Timmermans, 2014a). While we generally agree with the contention that models capturing behavioural processes are better capable of predicting the effects of policies than statistical models that are only based the outcomes of decision processes than on decision processes themselves, particularly if the policies violate the antecedent conditions that have led to observed aggregate patterns, the issue is more complicated than often suggested in the literature. Firstly, it is common practice to estimate variations of a model that are based on the same underlying behavioural postulate. For example, one may be satisfied with estimating a multinomial logit model only because the literature has suggested it to be robust. Alternatively, one may compare the performance of a multinomial model against a model allowing for different variances and/or covariances, but both are based on the same behavioural postulate of utility-maximizing behaviour. The more

Models of Bounded Rationality under Certainty

3

complex versions of the model may better reproduce the observed choices, but this approach does not give any guidance whether the assumed utility-maximizing decision process is the best representation of the decision-making process. Secondly, one should realize that the same mathematical expression, depicting the functional relationship between the dependent and the set of independent variables, can often be derived from different conflicting behavioural theories. For example, the multinomial logit model can be deducted from Luce’s choice theorem and random utility theory, which fundamentally differ with respect to the nature of preferences (deterministic vs. stochastic) and the nature of the decision process (probabilistic vs. deterministic). Regret-based choice models which define regret as a linear function of attribute difference between the best non-chosen and the chosen alternative are mathematically equivalent to the multinomial logit model; yet the principle of regret-minimization is fundamentally different from the principle of utility-maximizing behaviour. To make matters even more complicated, the mathematical expression of the multinomial logit model can also be derived from the quantum response model, which is a theory of decision-making under uncertainty rather than a theory of riskless choice. This equivalence implies that any satisfactory fit of the model to the data is just a necessary but not a sufficient condition for validating the behavioural principles and mechanisms underlying the mathematical model. When developing behavioural models, it is important to critically consider which theory seems most valid for the decision-making process under investigation. Unfortunately, the travel behaviour community, unlike for example the marketing community, does not have a rich tradition of developing, let alone systematically comparing, alternate theories of choice and decision-making. The vast majority of studies on various facets of travel behaviour has been based on discrete choice models, which in turn have been interpreted as representations of random utility theory. This theory can be seen as an example of a theory of rational decision-making. Individuals are assumed engaged in a high involvement decision process in which they have full information about the set of attributes, characterizing the choice alternatives in their choice set, from which they derive a utility. The behavioural principle of utility-maximizing behaviour then leads to a set of probabilities of choosing the alternatives in an individual’s choice set. The approach negates any emotional considerations. Although the literature in travel behaviour research on models of bounded rationality is relatively small, travel behaviour researchers have occasionally explored the formulation and application of such models. The purpose of this chapter is to provide an overview of these models, allowing readers to better understand the contribution of the specific papers, included in this book. This chapter is organized as follows. Firstly, we will present a general framework for positioning various models and theories of choice and decision-making. Based on this framework, we will continue the conditions under which we would consider the decision-making process evidencing bounded rationality. These conditions are used in the remainder of the chapter to organize existing research in mainly transportation and urban planning research on bounded rationality. More specifically,

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we will first discuss models that do not necessarily lead to an optimal choice. Next, we will discuss models that involve simplifications of the decision process by not considering all relevant attributes. This is followed by a discussion of models, which assume that individuals ignore small attribute or alternative differences, and therefore are indifferent between those choice alternatives that only differ marginally. Finally, we will discuss modelling attempts aimed at mimicking how individuals ignore choice options to reduce their consideration set. The chapter is completed with a conclusion, discussion and agenda of further research.

1.1.

Framework

Although models of decision-making and choice behaviour differ substantially, they have in common (part of) the following approach. In travel behaviour analysis and related disciplines, the aim of the modelling approach is to predict the choices of different segments of the population, which are assumed to represent observed heterogeneity, related to a particular facet (destination, travel mode, route, departure time, etc.), mostly to support planning processes. Because planning is primarily related to the physical attributes of the transportation and urban system, essentially choice probabilities are predicted as a function of the physical attributes of the spatial setting. Physical attributes may be augmented with other, for example, economic attributes. Let us first introduce basic notation. The urban-transportation system can be described in terms of a set of i = 1; 2; …; I ∈ C choice alternatives, making up choice set C. Each choice alternative i is characterized by a non-empty set of Ki attributes. To allow that choice sets differ between individuals, let us denote the choice set Cn of individual n. This set of attributes may be identical for the different choice alternatives; for example, a set of generic attributes for shopping centres. It may also (partly) be specific for different alternatives; for example, attributes of transport modes. Let Xik describe the attribute value of alternative i on attribute k. Finally, let the characteristics of decision-makers n = 1; 2; …; N be described as Znj ; j = 1; 2; …; J. In some modelling approaches, attributes of the choice alternatives are directly linked to choices observed in the real world fXik g→ pi ; i ∈ C. In this case, the researcher has to decide on the set of attributes fKi g assumed to influence the choice behaviour of interest. In addition, the choice set C has to be identified. In case of transport mode and departure time choice, this modelling step is trivial. In contrast, identification of the choice set in case of destination and route choice is a challenging and non-trivial task. More detailed approaches involve one or more of the following decision-making steps: 1. Mapping of objective, physical space into a cognitive space, measuring individuals’ perception: fXik g→fxnik g ∀ i ∈ C; ∀ n; k ∈ Kni . This step acknowledges that individuals may have partial ðk ∈ Kni Þ and imperfect knowledge xnik ≠ Xik about the environment surrounding them. Moreover, they may not be familiar with all

Models of Bounded Rationality under Certainty

5

choice alternatives ðCn ∈ CÞ. It also contends that individuals base their decisions on their cognitive representations of reality as opposed to reality itself. The notion of imperfect knowledge will be picked up by the mapping function. Any non-linear function or linear function, not running at 45 degrees angle through the origin will describe transformations of objective attribute values into cognitive values. The mapping can be described as xnik = fnk ðXik Þ ∀ i ∈ Cn ; ∀ n; k ∈ Kni where fnk is any function applied to attribute k. The cognitive representation of choice alternative i of individual n with respect to its attributes can thus be described by vector xni = ðxni1 ; xni2 ; …; xniKni Þ. 2. The transformation of the cognitions into a set of value judgments fVni g. The evaluation of attribute values may be based on processing of the absolute values of the attributes only, it may be based on processing of the values of the attributes relative to one or more attribute values of other choice alternatives, or it may be based on processing of the values of attributes relative to some exogenous reference point. This process of integrating the evaluation can be described as Vnik = gnk ðxnik Þ ∀ i ∈ Cn ; ∀ n; k ∈ Kni , where gnk is any valuation function applied to attribute k. It may be algebraic (linear or non-linear) or Boolean. This representation allows the valuing function to be individual-specific. Existing models have typically assumed a homogeneous function (in specification and parameters) across all individuals, a function that is homogeneous in specification but not in parameters, or a function that depends on latent classes (specification and/or parameters). 3. The integration of these value judgments into an overall judgment that constitutes the basis for ordering the choice alternatives in order of preference. This process of integrating the evaluation can be described as Uni = hn ðVnik Þ ∀ i ∈ Cn ; ∀ n; k ∈ Kni , where hn is any integration function applied to attribute values Vnik . The integration function defines the functional form of the integration, which represents the assumed underlying decision-making process in mathematical terms, and the weights or importance of the various attributes. It results in an ordered set of overall value judgments (utility, satisfaction, etc.) Uni ; ∀ i ∈ Cn on some preference scale. Function h may be algebraic (linear or non-linear) or Boolean. 4. Deciding which choice to make, considering the preferences and the choice set. The ultimate choice process involves a function qn operating on fUni g ∀ i ∈ Cn , which maps the value judgments some final set of choice probabilities fpni g; ∀ i ∈ Cn . The choice may depend on the absolute value of Uni ; in other cases, it may depend on a comparison of all choice alternatives in the choice set. This framework should be instrumental to understanding both rational decisionmaking processes and processes based on bounded rationality. For example, the commonly applied multinomial logit model assumes that fk and gk are linear for all Ki attributes; h is a linear additive function, while q is a logit transformation. Bounded rationality implies that individuals consider only a subset of the potentially influential attributes, and/or in comparing choice alternatives do not differentiate

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between asymptotically small differences in attribute values, and/or do not consider all alternatives in the choice set, and/or do not seek the optimal choice, based on maximizing their overall judgments. The models of bounded rationality, discussed in this chapter, have concepts such as utility, attribute importance and thresholds in common. In general, these have been applied in two diverging modelling approaches. Some studies are based on explicit and direct questioning of respondents who are asked to indicate their utility for particular attributes of choice alternatives, threshold and/or attribute importance. Other studies are based on econometric modelling, where one or more of these concepts are treated as parameters that need to be estimated. The remainder of this chapter discusses a cross-section of existing studies that has addressed these issues. We will first discuss alternatives to classic utilitymaximizing behaviour, based on non-compensatory decision rules that do not necessarily lead to optimal choices according to the traditional view. Next, we continue by summarizing attempts of modelling how individuals’ decisions may only include a subset of all attributes. Then, we discuss models that assume individuals simplify the decision-making process by ignoring small attribute/alternative differences. Finally, we will consider models that address the problem how individuals reduce a large consideration set into a much smaller consideration set. This is followed by a discussion of some results of empirical applications. Finally, we draw some conclusions.

1.2.

Non-Optimal Decision Mechanisms

The notion of an optimal choice is usually associated with a trade-off between attribute values. Some specific combination of attribute values generates the maximum overall value judgments across all choice alternatives. xni = ðxni1; xni2; …; xniKni Þ can be viewed a vector that positions choice alternative i in the cognitive space of individual n with xnik representing the projections of i on the attribute dimension k. UðxÞ transforms the cognitive space into utility space. An optimal choice can be viewed as finding the location in this space that maximizes the utility, that is the highest point in this multidimensional space. Any other point in this space that would be chosen represents a non-optimal choice, according to this definition. Several non-optimal decision mechanisms have been suggested in the literature. A conjunctive decision rule involves a choice mechanism in which an individual sets minimum threshold values for each attribute utility. If an alternative fails to meet these minimum threshold values, it will not be chosen/deemed viable. Fundamentally, it implies that an individual judges a choice alternative on its least performing attribute. If the choice alternative passes this judgment, it constitutes a viable alternative; if it does not, it fails. Note that higher evaluation scores of one or more of the remaining attributes cannot compensate for any low judgment of an attribute. A conjunctive decision rule thus describes a case of non-compensatory decision-making. It implies an attribute-by-attribute processing of the choice alternatives. If the utility of one or more of the attributes that do not meet the threshold

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7

value would be positive, the choice under a conjunctive decision rule will not be the optimal, utility-maximizing choice. In case of multiple choice alternatives, conjunctive decision rules do not necessarily lead to a single choice. Rather, they lead to a partitioning of an individual’s choice set into subsets of viable and non-viable (satisfying/non-satisfying) choices, each of which, but not both, may be empty. Conjunctive choice behaviour may be expressed as:  1; Unik ≥ μnk ∀ k ∈ Kin pni = ð1:1Þ 0; otherwise where μnk is an individual minimum utility threshold value, defined for each attribute k to make choice alternative i acceptable. A pure conjunctive rule would require an individual to set a threshold utility value for each attribute and judge all attributes to decide whether it is an acceptable or non-acceptable choice. However, from a process perspective, as soon as any attribute fails to meet the corresponding threshold value, the alternative can be rejected. Wright (1975) called this the sequential elimination decision rule. It involves not only the minimum thresholds, but also a specific order in which the attributes are judged. This decision rule implies that the thresholds are endogenously defined. Simon’s theory of aspiration levels and satisficing states that individuals seek to achieve a minimum acceptable utility and stop their search for better alternatives once this minimum has been achieved (Simon, 1955). It should be noted, however, that this principle is not necessarily confined to conjunctive decision rules. Eq. (1.1) defines acceptance in utility space. Conjunctive rules may also be expressed in objective or subjective, cognitive space. In that case, the threshold parameters should be redefined. In case of a continuous monotonically increasing attribute, with a monotonically increasing underlying utility function, a minimum attribute value is defined (e.g. minimum number of parking places). Correspondingly, if the underlying utility function is monotonically decreasing, a maximum attribute value is defined (e.g. maximum acceptable travel time). If the attribute is monotonically decreasing and the underlying utility is monotonically increasing, a maximum attribute value is defined; otherwise with a monotonically decreasing utility, a minimum value is defined. If the attribute is defined in terms of a set of ordered values, the same principles apply, except that a threshold level as opposed to a value is used. Finally, if the attributes are binomial or multinomial, thresholds are replaced with disjoint subsets, which split attribute categories into a subset of acceptable categories and a subset of non-acceptable categories. Some studies have relied on explicit measurement of thresholds, whereas other studies have estimated thresholds, typically assuming some distribution for the aggregate level. A second non-optimal decision rule is the disjunctive rule. It asserts that choice alternatives are evaluated on the basis of their maximum rather than their minimum utility values. A choice alternative that satisfies a maximum threshold value on at

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least one attribute is considered viable, regardless of its utility values on any other attribute. Thus, as the conjunctive rule, disjunctive rules partition choice sets into subsets of acceptable and non-acceptable choice alternatives. Disjunctive choice behaviour may be expressed as:  1; Unik ≥ ϱnk ∃ k ∈ Kin pni = ð1:2Þ 0; otherwise where ϱnk is an individual’s maximum utility threshold value, defined for each attribute k, at least one of which should be met to make choice alternative i acceptable. Eq. (1.2) defines the conditions to make a choice alternative acceptable in utility space. Disjunctive rules may, however, also be expressed in objective or cognitive space. When defined in objective or cognitive space, the threshold parameters represent maximum attribute values in case of a continuous monotonically increasing attribute, and a monotonically increasing underlying utility function. Similarly, if the underlying utility function is monotonically decreasing, a minimum attribute value is defined. If the attribute is monotonically decreasing and the underlying utility is monotonically increasing, again a minimum attribute value is defined. If the attribute is defined in terms of a set of ordered values, the same principles apply, except that a threshold level rather than a value is used. Finally, if the attributes are nominal or categorical, disjoint subsets, which split attribute categories into a subset of acceptable categories and a subset of non-acceptable categories is used. Similar to the conjunctive decision rule, some studies have relied on explicit measurement of thresholds, whereas other studies have estimated thresholds, typically assuming some distribution for the aggregate level. Einhorn (1970, 1971) suggested non-linear approximations of the conjunctive and disjunctive choice rules. The conjunctive model can be approximated as: β

Uni = ∏ xnikk

ð1:3Þ

k

where Uni is the judgment (utility) about alternative i by individual n and βk is a weight parameter. By taking the log, the following model is estimated: log Uni =

X

βk log ðxnik Þ þ ɛni

ð1:4Þ

k

Note that Eq. (1.4) represents a parabolic response curve, implying that low attribute values cannot be compensated by higher value on one or more of the other attributes. A non-linear approximation of the disjunctive model is given by:  βk 1 ð1:5Þ Uni = ∏ αik − xnik k

Models of Bounded Rationality under Certainty

9

By taking the log, this approximation can be estimated as: log Uni = −

X

βk log ðαik − xnik Þ þ ɛni

ð1:6Þ

k

βk is a parameter and αik is a reference point above the asymptotic value, for example, αik > maxi xik ∀i; k. Note that Eq. (1.6) represents a hyperbolic response curve. The conjunctive and disjunctive models are based on the assumption that individuals define a set of endogenous reference points or thresholds and that the choice alternatives are compared against these reference points in utility or attribute space. Another set of decision mechanisms is based on exogenous defined reference points. Examples include the maximin, maximax and minimax regret models. Usually, these models are based on explicit measurements of utilities/satisfaction. The maximin model assumes that individuals identify the least satisfactory attribute of each choice alternative, and then choose the alternative with the highest of these minimum utility values. In contrast, the maximax model calls for the identification of the most satisfactory attribute of each choice alternative, and assumes that an individual will choose the alternative with the highest of these maximum values. The minimax regret model assumes that an individual identifies the attribute with the largest utility difference and chooses the alternative with the highest utility on this attribute, irrespective of its utility on the other attributes. Mathematically, the maximin model can be expressed as follows. Let k ← mink ðUnik Þ; ∀ k ∈ Kni ; ∀ i; i0 ≠ i ∈ Cn . Then,  1; Unik > maxi0 ðUni0 k Þ; ∀ i; i0 ≠ i ∈ Cn pni = ð1:7Þ 0; otherwise Similarly, the maximax model can be expressed as follows. Let k ← maxk ðUnik Þ; ∀ k ∈ Kni ; ∀ i; i0 ≠ i ∈ Cn :  ∀ i; i0 ≠ i ∈ Cn 1; Unik > maxi0 ðUni0 k Þ; pni = ð1:8Þ 0; otherwise Finally, let k ← maxk ðUnik − Uni0 k Þ; ∀ k ∈ Kni ; regret model is defined as follows:  pni =

1; 0;

Unik > Uni0 k ; otherwise

∀ i; i0 ≠ i ∈ Cn . Then, the minimax ∀ i; i0 ≠ i ∈ Cn

ð1:9Þ

These equations assume that all alternatives can be perfectly ordered on all attributes. If ties exist, that is choice alternatives cannot be perfectly ordered on one or more attributes, the choice probabilities should be adjusted for such ties. Models of strong rational behaviour assume that preferences are invariant across context. If that would not be the case, a different model would need to be

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estimated for every different situation. Nevertheless, the contention and empirical evidence that choice behaviour depends on the context has received some attention in the choice literature. While conventional utility-maximizing models of rational behaviour assume that utility is derived directly from the attribute levels of a choice alternative, independent of context, several other context-dependent choice models often based on different behavioural principles have been assumed. The multinomial logit model, the most commonly used utility-maximizing discrete choice model, is characterized by the so-called IIA or Independence from Irrelevant Alternatives property, which states that the odds of choosing an alternative over another alternative is independent of any other alternative and its attributes in the choice set. To avoid this potential limitation, most scholars have allowed varying variances and/or covariances in the error terms of the utility function, reflecting the notion that choice alternatives may share some unobserved variables and these unobserved terms could have had different expected values (see Timmermans & Golledge, 1990 for an overview of these models). Others have suggested to represent the context explicitly in the specification of the utility function, for example, by including measures of similarity of choice alternatives or by estimating the effects of the existence and the attributes values of competing alternatives on the utility of the considered choice alternative. While the mathematical specification of the utility function thus differs from the conventional additive linear-in-parameters specification, the utility function is still maximized and in that sense one may argue these context-dependent choice models still represent rational behaviour. Classic utility-maximizing models exhibit particular properties such as transitivity (if A is preferred to B and B is preferred to C, then A is by definition preferred to C) and regularity (the fact that the introduction of a new choice alternative logically can only reduce the choice probabilities of the existing choice alternatives before the introduction of the new choice alternative). Some context-dependent models violate these properties that may be viewed as essential evidence of rational behaviour. Behaviourally, these context-dependent models allow for the possibility that choice alternatives are evaluated relative to the position of the other choice alternatives in attribute space. One may assume that if individuals only have weak preferences, they do not necessarily choose extreme choice alternatives, but rather choose alternatives that are easier to justify to themselves and others (Simonson, 1989; Simonson & Tversky, 1992). Compromise alternatives would be a good example. This may be seen as evidence of bounded rationality in the sense that more extreme alternatives are less likely considered. In some cases, the model is based on different behavioural postulates such as maximization of relative advantage, maximization of relative utility or minimization of regret. On the other hand, considering the specification of these models, the processing of the information and decision rules assumes a more elaborated information processing and decision-making process and therefore a rational decision style. Thus, although we acknowledge that the question whether context-dependent models are examples of models of bounded rationality is debatable, we include a short review of the development of these models in the current chapter.

Models of Bounded Rationality under Certainty

11

Batsell (1981) introduced a discrete choice model with an extended utility function that accounted for the similarity of choice alternatives. More specifically, his model can be expressed as: P  P P 1 exp k βk Xik þ I i0 k θ k jXik − Xi0 k j P  P P pi = P 1 i0 0 exp k β k Xi 0 0 k þ I i0 k θ k jXi0 0 k − Xi0 k j

ð1:10Þ

Meyer and Eagle (1982) suggested a different specification by multiplying the expectation of the utility function with a measure of similarity. Borgers and Timmermans (1988) showed that if this measure of similarity is expressed as an exponential function of attribute differences and the error terms are identically and independently Gumbel distributed, the M_E model becomes equivalent to Batsell’s model. Although these models do capture the effect of similarity on choice probabilities, they cannot fully capture pure competition in the sense that the market share of the perfect substitutes would be perfectly split, and the market share of the other alternatives be unaffected. Borgers and Timmermans (1988), therefore, suggested yet another specification. P  ∏k Rθikk exp k βk Xik pi = P  P  θk i0 ∏k Ri0 k exp k βk Xi0 k where

#K1 1 X jXik − Xi0 k j Rik = I − 1 i0

ð1:11Þ

"

ð1:12Þ

and 0 ≤ θk ≤ 1 indicates the degree of substitution. This specification can theoretically represent the effects of perfect competition on choice probabilities for up to three choice alternatives correctly. It means that the quest for a general specification is still an unsolved problem. If the similarity relates to the location of alternatives, spatial choice is represented and the similarity compares the average distance to competing destinations. In that case, the model resembles Fotheringham’s competing destination model, except that the latter assumes a full multiplicative model and is based on a spatial interaction modelling as opposed to discrete choice modelling framework (Fotheringham, 1983, 1986). After a period of lack of attention for context-dependent choice models, the problem recently re-emerged on the research agenda in connection with the formulation of regret-based models. These models assume that individuals minimize regret when choosing a choice alternative. Regret ψ ik associated with attribute k is defined as a function of the difference between the attribute value xik of the considered choice alternative i and attribute value xi0 k of one or more of the remaining choice alternatives P i0 ≠ i ∈ C.PTotal (possible) regret when choosing alternative i is equal to Ri = k ψ ik = k f ðxi0 k − xik Þ ≥ 0: In line with utility-maximizing models, total regret

12

Soora Rasouli and Harry Timmermans

is assumed to consist of a deterministic component and an error term: RRni = Rni þ ɛni . Consequently, the probability that individual n will choose i is equal to pni = pðRRni < RRi0 Þ ∀ i0 ≠ i ∈ C. Assuming the error terms are identically and independently Gumbel distributed, the probability pni jC of individual n choosing alternative i from choice set C is given by expð − μRni Þ pni jC = P i″ expð − μRni″ Þ

ð1:13Þ

where μ is a scaling factor. An operational regret-based model requires a specification of function g. Two different functions have been suggested in the literature. The best alternative only linear regret specification (Chorus, Arentze, & Timmermans, 2008a, 2008b) assumes that regret is defined as a linear function of the attribute difference between alternative i 0 and the best forgone alternative: ψ ΛB ik = maxi0 ≠i ðmax½0; βk ðxi0 k − xik ÞÞ ∀ i; i ≠ i ∈ C. By contrast, the paired comparisons linear regret model (Chorus, 2010) assumes that regret is defined as a linear function of the attribute difference between alternative n and all other foregone alternatives in the choice set: ψ ΛΡ ik =

X

max½0; βk fxi0 k − xik g

ð1:14Þ

i0 ≠ i ∈ C

Because the discontinuity in these specifications may cause some difficulty in estimation, Chorus (2010) suggested a logarithmic transformation. In that case, the best alterative only logarithmic regret specification is defined as REB i = ln½1 þ expðβk fðxi0 k − xik ÞgÞ, while the paired comparisons logarithmic regret specification can be expressed as: ψ EP ik =

X

ln½1 þ expfβk ðxi0 k − xik Þg

ð1:15Þ

i0 ≠i∈C

This new logarithmic transformation is asymptotically identical to the original specification if attribute differences are large. However, as shown by Rasouli and Timmermans (2014b), the new formulation violates the concept of regret for some attribute differences, and systematically over-estimates regret for small attribute differences. In the chapter in this volume, Hess and Chorus (2015; see also Chorus, 2013) formulated a generalised function of the following form: ψ EP ik =

X

ln½γ þ expfβk ðxi0 k − xik Þg

ð1:16Þ

i0 ≠i∈C

Note that if γ = 1, Eq. (1.15) is obtained; if γ = 0, Eq. (1.16) collapses into Eq. (1.14). Note that these models do not take the similarity between choice alternatives into account.

Models of Bounded Rationality under Certainty

13

In addition to these pure regret-based models, a number of hybrid models have been formulated, combining aspects of conventional utility maximization and regret minimization behaviour. Chorus et al. (2013) suggested to divide the set of attributes {K} be divided into subset fkg∈fKg of attributes that are processed in a regretbased fashion and subset fk0 g∈fKg; fkg∩fk0 g = ∅; fkg∪fk0 g = fKg of attributes that are processed in a utility-maximising manner. Their hybrid utility function is then equal to: Ui =

X

βk0 ðxik0 Þ −

X

X

lnð1 þ expðβk ðxi0 k − xik ÞÞÞ

ð1:17Þ

fkg ∈ fK g i0 ≠ i ∈ C

fk0 g∈fKg

Leong and Hensher’s (2012) formulated a relative advantage model, originally suggested by Tversky and Simonson (1993) and Kivetz, Netzer, and Srinivasan (2004). This model differentiates between advantage and disadvantage and then calculatesPrelative advantage as a substitute for regret. Advantage can be expressed as Aii0 = k Akii0 , with  Akii0 = Disadvantage Dii0 =

P k

βk xik − βk xi0 k if βk xik > βk xi0 k 0; otherwise

ð1:18Þ

Dkii0 is defined in a similar way. 

Dkii0 =

βk xi0 k − βk xik ; βk xi0 k > βk xik 0; otherwise

ð1:19Þ

Relative advantage is defined as: RAii0 =

Aii0 A þ Dii0

ð1:20Þ

ii0

The utility function is then written as a linear combination of the classic utility component and relative advantage: Ui = β0;i þ

X k

βk xik þ

X

RAii0

ð1:21Þ

i ≠ i0

These models have in common the use of one or more reference points to position the choice alternatives in utility space. In that sense, most of these models can be viewed as special cases of relative utility models. Zhang, Timmermans, Borgers, and Wang (2004) introduced the principle of relative utility maximization in the travel behaviour literature. It assumes that individuals will choose the alternative in their choice sets that provides the maximum relative utility. In this context, relative can mean relative to other choice alternatives, relative to previous time periods,

14

Soora Rasouli and Harry Timmermans

relative to members of a social network, etc. The basic relative utility model can be expressed as: P exp ðγ ni i0 ðνni − νni0 Þ P ð1:22Þ pni = P i0 0 exp ðγ ni00 i0 ðνni0 0 − νni0 Þ where γ ni is a relative interest parameter and νni − νni0 represents the effect of utility differences of alternative i and i0 in the choice set.

1.3.

Considering a Subset of Influential Attributes

While conjunctive and disjunctive rules define thresholds for each attribute and in that sense assume that individuals process all attributes, lexicographic models assume that attributes are considered sequentially and perhaps only partially. Usually, these models are based on some explicit measurement of attribute importance, on the basis of which the alternatives are ordered in terms of decreasing attribute importance. Firstly, alternatives are evaluated on the most important attribute and the best is identified. If there are ties, that is an individual is indifferent between two or more alternatives on that attribute, choice alternatives are evaluated on the second most important attribute. This process continues until a choice can be made or until all attributes have been considered. Note that unlike the conjunctive and disjunctive rules, the lexicographic rule involves an explicit comparison of choice alternatives. Lexicographic choice behaviour may be expressed as follows. Assume that the attributes are ranked in order of importance. Then,  pni =

1; 0;

if Unik > Uni0 k ∧ Unik = Uni0 k ; otherwise

k = 1; …; k − 1;

k ∈ Kni ;

∀ i; i0 ≠ i ∈ Cn ð1:23Þ

Note that the model describes bounded rational behaviour in the sense that a subset of attributes is not considered if choice alternatives can be ranked in terms of more important attributes. The only exception is that two or more choice alternatives are identical on all attributes and the best on the more important and possibly subsequent attributes. In this case, they split the market share among them equally. At the same time, lexicographic models represent non-optimal behaviour in the sense that they do not guarantee that the first ranked alternative is the one with the maximum utility, based on the combination of attribute levels. Pure lexicographic rules indicate that individuals value some attributes so much that they are not willing to make any trade-offs. For example, if an individual want to be on the first commercial flight to the moon, no other attribute will be taken into consideration and no price will be too high. Consequently, it is impossible to construct a utility function, representing lexicographic preferences over multiple

Models of Bounded Rationality under Certainty

15

real-valued attributes (Varian, 1984). The reason is that any multi-attribute utility function is associated with indifference curves in attribute space that express the marginal rate of substitution between pairs of attributes. Because lexicographic preferences imply an infinite marginal rate of substitution, a utility function cannot be constructed. However, lexicographic preferences functions may exist over discrete attributes (e.g. Kohli & Jedidi, 2007; Martignon & Schmitt, 1999). Alternatively, lexicographic rules could be applied to net utility differences higher than zero (e.g. Kawamoto & Setti, 1992). ˇ Le´on, and Hanemann (2008) considered two additional heuristics. The Arana, first is the complete ignorance heuristic. It represents those individuals who are not aware of the influence of the attributes or do not care about the consequences of their responses. It describes the decision-making process of individuals who choose based on a completely random process. A second heuristic is the satisfactory heuristic, originally proposed by Simon (1955). It describes a process in which an individual first selects the choice alternatives that meet the minimum requirements on all attributes. Subsequently, the choice is made at random between the selected candidate alternatives. Thus, this heuristic can be viewed as a combination of the conjunctive and ignorance heuristic, in which the former serves to delineate the consideration set and the latter serves to describe random choice among the remaining choice alternatives. Analogously, individuals may also set a threshold below which an alternative is rejected. In case of nominal attributes, indifference among attribute levels or rejection of alternatives can reflect absence of preference over a subset of attribute levels. The number of indifference classes is between 1 and the number of attribute levels. A single indifference class signals the lack of preference across attribute levels, while the number of indifference classes being equal to the number of attribute levels represents the standard lexicographic model. The strict lexicographic model has been criticized for the fact that it assumes perfect discrimination and perfectly reliable information. The lexicographic semi-order model and the minimum difference lexicographic models have been formulated to relax these rigorous assumptions. The lexicographic semi-order model assumes that individuals consider the second important attribute if two or more choice alternatives, ranked ordered in utility space, do not differ more than some minimum threshold on the most important attribute. That is, given attributes are ranked in order of importance. Then,  pni =

1; 0;

if Unik > Uni0 k þ Δ1 ∧ Unik = Uni0 k ; otherwise

k = 1; …; k − 1; ∀ i; i0 ≠ i ∈ Cn ð1:24Þ

where Δ1 > 0. The minimum difference lexicographic model constitutes a generalization of the lexicographic semi-order model by assuming that individuals consider the next important attribute if two or more choice alternatives, ranked ordered in utility space, do not differ more than some minimum threshold on the currently

16

Soora Rasouli and Harry Timmermans

considered attribute. That is, given attributes are ranked in order of importance. Then,  pni =

1; 0;

if Unik > Uni0 k þ Δk ∧ Unik = Uni0 k ; otherwise

k = 1; …; k − 1;

∀ i; i0 ≠ i ∈ Cn ð1:25Þ

where Δk > 0; ∀ k ∈ Kni . A special case is the ‘just noticeable difference’ lexicographic model (Russ, 1972), which is defined at the level of (cognitive) attribute differences. That is,  pni =

1; if ðxnik − xni0 k Þ > Δk ∧ ðxnik − xni0 k Þ ≥ Δk 0; otherwise

ð1:26Þ

where k = 1; …; k − 1; ∀i; i0 ≠ i ∈ Cn Δk > 0; ∀k ∈ Kni . If this condition is not met at any level of importance, choices are made in a compensatory manner in the sense that all attributes are taken into consideration. Although these two models can be equivalent, in principle the threshold at the perception or cognition level is not necessarily consistent with utility differences. Recker and Golob (1979) also suggested combining a lexicographic attribute valuation model, jointly with thresholds. These thresholds were defined as the percentage deviation of the best alternative for the attribute under consideration. Foester (1977) formulated the ‘just effective difference’ lexicographic model. Rather than setting conditions at the utility or attribute levels, he assumed that choice will involve a compensatory process, unless the importance differences satisfy the condition: Γnk − Γn;k þ 1 ≥ Ψ;

∃ k

ð1:27Þ

where Γ is the importance and Ψ is the effective difference threshold. Kohli and Jedidi (2007) discussed another variant of the lexicographic model: the binary lexicographic model. This model relaxes the assumption of an attribute-byattribute evaluation of choice alternatives. It assumes that individuals first classify the choice alternatives into two classes. One class consists of the choice alternatives with the most preferred attribute level across attributes, while the other class consists of the remaining choice alternatives. Each class is then further partitioned in the same manner for the second most preferred attribute level across attributes. Another interesting model is Tversky’s elimination-by-aspects model (Tversky, 1972). This model assumes that individuals first select a discriminating aspect or attribute and eliminate all choice alternatives that do not have this attribute. Unlike the assumed sequential consideration of lexicographic models, attributes are selected with some probability that is equal to the ratio of the utility of that attribute and the total sum of utilities of all discriminatory attributes. The probability of choosing a choice alternatives is then equal to the sum across discriminating attributes of the probability that that attribute is being selected multiplied by the probability that the choice

Models of Bounded Rationality under Certainty

17

alternatives is chosen among the alternatives that possess the attribute. Although the original model has been formulated in terms of dichotomous attributes, the formulated can also be used for more general conditions that eliminate certain choice alternatives. Manrai and Sinha (1989) formulated an extension, while Batley and Daly (2003) showed the equivalence with generalized extreme value models. Lexicographic models are extreme in the sense they assume that in lieu of any ties choices are based just on a single attribute. More general approaches have recently been developed in travel behaviour analysis and environmental economics. A stream of publications has emerged, which has been labelled as ignoring of attributes (Hensher, 2006, 2010) and attribute non-attendance (Hole, 2011; Scarpa, Zanoli, Bruschi, & Naspetti, 2012). In addition to the notion that consumers may ignore particular attributes due to lack of time, selective information processing (Cameron & DeShazo, 2011; DeShazo & Fermo, 2004) or low involvement, non-attendance of particular attributes may be related to their lack of any inherent utility (Collins, Rose, & Hensher, 2013) to an individual, who will therefore ignore these attributes. In the context of stated preference and choice experiments, unrealistic attribute levels and/or trade-offs (e.g. Alemu, Morkbak, Olsen, & Jensen, 2013; Hensher, Collins, & Greene, 2012) may cause attribute non-attendance. Failure to capture attribute nonattendance may lead to biases in model forecasts (Hensher, Rose, & Greene, 2005), and errors in the signs of random parameter coefficients (Hensher, 2007). Two approaches can be distinguished in the literature to address attribute nonattendance: (i) directly asking respondents which attributes they did not consider in their responses or real-word decision-making processes; (ii) identifying attribute nonattendance using econometric approaches. The first approach involves asking respondents to identity the attributes that were systematically varied in a stated choice experiments but that they ignored (Hensher et al., 2005). Although this is a straightforward and easy way of identifying attribute non-attendance, it is questionable that respondents can rationalize their response strategy. Consequently, the reliability of direct statements of non-attendance has been criticized (e.g. Carlsson, Kataria, & Lampi, 2010; Hess & Hensher, 2010; Hess & Rose, 2007). Moreover, directly asking for non-attribute attendance is inconsistent with the very nature of stated preference and choice experiments. These models have been developed as an alternative to models that were based on direct measurement of part-worth utilities and importance weights and the composition of overall utility based on explicitly and independently measured part-worth utilities and weights because empirical evidence casted doubt on the validity and reliability of such models. In other words, directly asking for attribute non-attendance would introduce measurements error into an alternative modelling approach that it tried to avoid from its very beginning. As an alternative to direct measurement of attribute non-attendance, several econometric models have been suggested. In particular, the following econometric approaches can be distinguished. The so-called attribute non-attendance (ANA) model (Hess & Rose, 2007; Scarpa et al., 2012) is a latent class model, where each class consists of different attendance/non-attendance profiles. For each class, the coefficients of the non-attended attributes are constrained to zero. It implies, and this is a disadvantage of this model, that the number of classes and the number of

18

Soora Rasouli and Harry Timmermans

class membership parameters increase exponentially with an increasing number of attributes. Hole (2011), therefore, assumed that attribute non-attendance rates between attributes are not correlated, implying that the number of parameters to be estimated increases only linearly with the number of attributes. This may be a strong assumption because one would expect that non-attendances of attributes pertaining to the same underlying higher order decision construct are correlated. This independent attribute non-attendance (IANA) model is a special case of the correlated attribute non-attendance (CANA; Scarpa et al., 2012) model, which allows correlations between non-attendance rates across attributes. Others have estimated latent threshold values (Hess & Hensher, 2010; Mariel, Hoyos, & Meyerhoff, 2011). Hensher (2010) used thresholds to model whether attributes with a common metric (e.g. free flow travel time and congestion time) are processed as separate attributes (if their squared difference exceeds a threshold) or are summed and processed as a single common metric attribute with the same utility weight (if their squared difference does not exceed a threshold). The thresholds are assumed exponentially distributed in the population. In addition to these fixed effects attribute non-attendance models, in an attempt to incorporate taste heterogeneity, several random effects models have been suggested. Campbell, Hensher, and Scarpa (2012) estimated point masses for cost only. Hensher et al. (2012) estimated a random parameters attribute non-attendance model (RPIANA). In principle, the random parameters model can be estimated under the assumption of independent attributes and under the assumption of correlated attributes. Hess, Stathapoulos, and Daly (2012) employed a latent class structure, and estimated continuously distributed random parameters. Collins et al. (2013) argued that full correlation of non-attendance across attributes does not only involve many parameters and computation time, but that the assumption of full correlation may also be too strong. They, therefore, developed a generalized latent class structure that allows independence across subsets of attributes, whilst allowing attribute non-attendance to be correlated.

1.4.

Indifference between Small Differences in Utility or Attribute Values

Random utility-maximizing models assume that individuals select from a set of competing alternatives the one with the maximum utility. Implicitly or explicitly the assumption typically made is that individuals perceive and respond to differences in utilities, however small they may be. However, when applying some simplifying decision strategy, individuals may not perceive small attribute differences and/or may be insensitive to small (perceived) attribute differences. Only when differences exceed a certain threshold, a utility difference will be noticed and/or a choice will be made; otherwise the individual will be indifferent. This concept of threshold difference has a long history outside of transportation research. Quandt (1956) argued that an indifference band surrounds the utility

Models of Bounded Rationality under Certainty

19

derived from a commodity. Small differences in the quantity of a product consumed will not affect an individuals’ utility. Similarly, Georgescu-Roegen (1958) suggested that attribute differences between any two commodities are considered only when some ‘necessary minimum’ is exceeded. In psychology, perception or sensorial thresholds have for long been recognized as a central characteristic of human response to stimuli. For example, Thurstone (1927) argued human’s inability to discriminate between very small differences between stimuli. More recently, Krishnan (1977) formalized thresholds as ‘minimum perceivable differences’ between the utilities of the alternatives compared. According to his formalization, an individual prefers one alternative to another if its utility exceeds the utility of the other alternative by at least a positive constant Δ, the minimum perceivable difference. Assuming continuous functions over the possible values of the attributes, if the utility difference is less than this threshold level, the individual is assumed to be indifferent. Krishnan incorporated the threshold parameter into a binary logit model and tested the derived model using data on transport mode choice. Results appeared to support the existence of a positive threshold. This model acknowledges the possible existence of a band of indifference in discriminating utility differences. Individuals only maximize their utility outside this indifference band. However, the postulate leads to intransitivity of indifference when generalized to multiple alternatives. Lioukas (1984) extended Krishnan’s model to the multinomial case. His so-called δ-logit model can be expressed as follows: pni = P

exp ðVni Þ P exp ðV Þ ni þ i i0 exp ðVni0 þ δÞ

∀ i; i0 ≠ Cn

ð1:28Þ

where δ > 0 is a parameter to be estimated. Cantillo, Heydecker, and de Dios Ortu´zar (2006) formulated a similar threshold model, albeit in a dynamic context. Their model assumed that individuals would only perceive differences in utility to the extent that attribute differences over time are larger than attribute-specific thresholds. Variation in thresholds was treated as a function of socio-demographic variables. Otherwise, their model was not fundamentally different from many earlier formulations.

1.5.

Considering a Subset of Choice Alternatives

Another simplifying decision approach is to restrict the number of considered choice alternatives, and focus on that subset only. Sometimes, this subset has been called the consideration set, whereas the (universal) choice set is comprised of all available choice alternatives (e.g. Hauser & Wernerfelt, 1990; Shocker, Ben-Akiva, Boccara, & Nedungadi, 1991). Models that are based on the assumption that individuals simplify the choice task and in that sense evidence bounded rationality have typically been developed in the context of the so-called choice set formation problem. It assumes that individuals apply a two-stage decision process, during which they

20

Soora Rasouli and Harry Timmermans

first select a subset of choice alternatives from the universal choice set, and then choose the alternative they like best from this reduced subset. Different ideas have been used to reduce the choice set. Individuals may not consider all available options because some cannot be reached within a reasonable time, or given spacetime prism. Another reason may be that individuals first apply thresholds on the attributes of the choice alternatives so that a smaller set of alternatives remains. Another consideration is that individuals may not be aware of particular alternatives or that some may not belong to their activity space. Manski (1977) suggested the following formulation: X pni = pðnijC Þ  pðnC jC þ Þ ð1:29Þ C ∈ C þ

where C  is the consideration set and C þ is the set of all possible consideration sets. Note that if choice set C þ consists of I choice alternatives, there are 2I − 1 possible consideration sets. Thus, this approach becomes problematic in case of a large number of choice alternatives. To avoid this problem, several simplifying approaches have been suggested in the literature. Morikawa (1995) compared alternatives in pairs and derived a computationally tractable form even for a large number of alternatives. Most approaches, however, suggest applying screening heuristics to define the consideration sets. Choice probabilities then become equal to pi = Pr ðVi þ ɛ i > Vi0 þ ɛi0 ∧IðM i ; ℋÞ = 1Þ

∀ i; i0 ∈ C 

ð1:30Þ

where Vi is the deterministic utility value of choice alternative i and IðM i ; ℋÞ = 1 is an indicator function with generic argument M i and condition ℋ. Together, these define whether choice alternative i will be considered. M i may be defined in attribute space, cognitive space or utility space. ℋ may be a given value, a set of categories or a parameter to be estimated from the data. Gilbride and Allenby (2004) discussed some examples of screening rules. A compensatory screening rule states that the deterministic part of the utility of the choice alternative must exceed a threshold value to be acceptable. Thus, IðVi > ΔÞ = 1. Similarly, a conjunctive screening heuristic can be represented as ∏ Iðxik > Δk Þ = 1; Δk > 0

k ∈ Ki

∀ k ∈ Ki

ð1:31Þ

Consideration sets formed by a conjunctive decision rule require that a choice alternative to be included is acceptable on all relevant attributes. Hence, the multiplication of the indicator variables should be equal to one to satisfy this condition. A disjunctive screening heuristic stating that at least one attribute should be acceptable for a choice alternative to be included in the consideration set can be represented as X Iðxik > Δk Þ ≥ 1; Δk > 0 ∀ k ∈ Ki ð1:32Þ k ∈ Ki

Models of Bounded Rationality under Certainty

21

Seminal work on choice set composition involved deterministic thresholds. Recker, Mc.Nally, and Root (1983) suggested applying constrained combinatorial scheduling algorithms to generate a set of feasible activity-travel patterns. Next, pattern recognition procedures were applied to identify clearly distinct patterns. After eliminating all inferior patterns, the consideration set is derived. Ben-Akiva, Bergman, Daly, and Ramaswamy (1984), in the context of route choice behaviour, launched the idea of defining a set of labelled paths, found by applying different choice criteria. By identifying these paths, the choice set is dramatically reduced. It goes without saying that the validity of such deterministic thresholds or approaches in general may be questioned (e.g. Huber & Klein, 1991). Thresholds are likely stochastic because individuals have varying perceptions, or differ in terms of their judgments of what is considered acceptable. Moreover, researchers have limited knowledge about the choice set generating process (Swait & Ben-Akiva, 1987). Finally, the urban and transportation system themselves are stochastic (Rasouli & Timmermans, 2012). Several authors have developed models that allow for such stochastic thresholds. Assume that the screening process requires that a choice alternative will only be included in a choice set if its attributes are equal to or larger than a set of thresholds. Let Δn denote the vectors of thresholds of individual n. Then, the probability space is represented by joint density function Ω ðδÞ with mean EðΔn Þ = Δ and covariance matrix Σ. The probability pn; i ∈ Cn that choice alternative i will be included in the choice set of individual n is then equal to: Z ∞Z ∞ Z ∞ pn;i∈Cn = pfXni ≥ Δn g = … ΩðδÞdδ1 dδ2 …dδk ð1:33Þ Xni1

Xni2

XniK

The probability that choice alternative i will not be included in the choice set of individual n equals: pn;i∉Cn = 1 − pn;i ∈ Cn

ð1:34Þ

The probability that an individual’s choice set is empty is equal to the probability that none of the choice alternatives satisfies the inclusion criteria: I

pn;Cn = ∅ = Π pn;i∉Cn i=1

Thus,

I

pn ðC  jC þ Þ = Π

i=1

n o δi pδn;i i ∈ Cn · p1n;−i∉C n

ð1:35Þ ð1:36Þ

where C  = ½δ1 ; δ2 ; …; δi; …; δI , with δi = f0; 1g ∀ i. This equation includes the probability of an empty choice set, which would imply a non-choice. If one wishes to correct for this, expression (1.22) can be reformulated as: o I n 1 δi pn ðC  jC þ Þ = Π pδn;i i ∈ Cn · p1n;−i∉C ð1:37Þ n 1 − pn; Cn = ⊘ i = 1

22

Soora Rasouli and Harry Timmermans

Assume that the choice process in the second stage is a utility-maximizing process based on the multinomial logit model, then choice probabilities are equal to: pni =

1

X

1 − pn; Cn = ⊘ C ∈ C þ

( I

Π

i=1

n

pδn;i i ∈ Cn

δi · p1n;−i∉C n

o

exp ðVni Þ P i0 ∈ Cn exp ðVni0 Þ

) ð1:38Þ

In principle, both stages can be substituted with alternative screening and choice models. It is no surprise, therefore, that various authors suggested replacing the simple multinomial logit model with more complex choice models. For example, Kaplan, Bekhor, and Shiftan (2009) allowed for correlated ordered-response thresholds, while Kaplan, Bekhor, and Shiftan (2010a, 2010b) assumed multinomial thresholds. Kaplan and Prato (2010) examined the use of hazard-based thresholds, while Kaplan, Bekhor, and Shiftan (2011a, 2011b) developed a model allowing for the selection of multiple independent ordered-response thresholds related to individuals’ characteristics. All these models assumed IID error choice alternatives at the choice stage. To relax this assumption, Kaplan, Shiftan, and Bekhor (2012) incorporated a nested correlation pattern across choice alternatives and random taste variation across the population. Technically, their model jointly represents the conjunctive heuristic with a multidimensional mixed ordered-response probit model and the choice mechanism with an error components logit model. This most general function is very demanding. To some extent, these formulations reflect increasing computing power and the formulation of more complex discrete choice models in general. In earlier work, several simplifying assumptions have therefore been introduced to formulate less computationally demanding models. For example, Borgers, Timmermans, and Veldhuisen (1986) and Swait and Ben-Akiva (1987) assumed that the thresholds were independent, so that Ki

pn;i ∈ Cn = Π pnik k=1

ð1:39Þ

If the thresholds are normally distributed with mean Δk and variance σ 2k , the probability that alternative i meets the selection criteria on attribute k is given by:  Δk − Xnik pnik = 1 − Φ σ 2k

ð1:40Þ

where Φ(*) is the standard cumulative normal distribution function. Swait and Ben-Akiva (1987) also considered the case where choice is limited to a subgroup of choice alternatives and identified the set of assumptions under which this leads to the logit model (Gaudry & Dagenais, 1979). Swait (2001a, 2001b) included in the utility function a linear penalty function for violating thresholds points with lower and upper bounds. Ceteris paribus, with an increasing penalty, the probability of choosing a choice alternative that violates one or more of the thresholds is reduced. Swait and Ben-Akiva (1987) incorporated random constraints

Models of Bounded Rationality under Certainty

23

in their model of choice set generation. According to their model an individual will include a choice alternative in his/her choice set if and only if all its attributes are within their respective thresholds. Several applications have used exogenous covariates to explain choice set formation. Thresholds are assumed to vary as a function of socio-demographic and economic profiles of individuals. However, as argued by Horowitz and Louviere (1995), choice sets may be just another expression of underlying utilities as opposed to separate constructs. Swait (2001a, 2001b), therefore, suggested a generalized extreme value model in which inclusion in the consideration set is a function of the expected maximum utility derived from the alternatives in the choice set. His approach has the advantage that as the expected utility of an alternative is increasing, all choice sets including that alternative have a higher probability of being chosen. Cascetta and Papola (2001) extended the utility function by including a cutoff factor. Martı´ nez, Aguila, and Hurtubia (2009) proposed the Constrained Multinomial Logit model (CMNL), which extends previous models by imposing multiple thresholds.

1.6.

Applications

Compared to disciplines such as marketing, the development and application of semi-compensatory and non-compensatory models of choice behaviour has received much less attention in transportation research and similar disciplines. Foerster (1979) was one of the first studies, comparing the performance of additive, lexicographic, maximum, conjunctive-additive and conjunctive-lexicographic choice processes in the context of transportation mode choice decisions in Chapel Hill, North Carolina. The collected data contained information about mode choice, perceptions of speed and costs of car and transit for different trips and importance ratings of time and cost in choosing a transportation mode. He applied the models both at the level of attributes and at the level of subjective ratings. The results of the comparisons, based on Cochran’s generalization of McNemar two sample correlated proportions test, indicated that transportation mode choice decisions are not made according to compensatory decision rules. Conjunctive and lexicographic models outperformed additive models. The subjective variants tended to be better than the objective variants. In contrast, Timmermans (1983) found evidence of underperformance of noncompensatory decision rules compared to additive and multiplicative models in the context of choice of shopping centre in the Netherlands. Choice sets consisted of four to five shopping centres, while 11 attributes were selected. Several studies have examined the application of hybrid compensatory, noncompensatory models in the context of housing choice. Borgers et al. (1986) modelled the screening phase using conjunctive decision rules, while a multinomial logit model was used to predict housing choice within each choice set. In a similar vein, Kaplan et al. (2012) represented the conjunctive heuristic with a multidimensional mixed ordered-response probit model and the utility-based choice with an error

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components logit model. In both these studies, the hybrid model performed well. Rashidi, Auld, and Mohammadian (2012) used a very similar two-step approach in which housing alternatives are evaluated and screened based on household priorities, lifestyle and housing preferences. The consideration set is modelled based on household average work distance using a hazard-model (Weibull distribution). A multinomial logit model was used to predict the probability of finding a new residential location, considering the simulated consideration set of the decision-maker. The authors concluded that their modelling approach is capable of generating highly accurate consideration sets. Young and Richardson (1983) addressed the same problem, but applied the Elimination-by-Aspects model. Earlier, they applied this model for modelling freight modal choice (Young, Richardson, Ogden, & Rattray, 1982). Morikawa (1995) in a study of vacation choice found that a model with probabilistic choice sets, in which the screening stage was based on a set of logistically distributed thresholds outperformed a classic choice model. Basar and Bhat (2004) modelled airport choice and excluded an airport from the choice set if its utility was below some threshold. They allowed this threshold to vary across individuals, assuming the random threshold to be standard logistically distributed. They found that their model outperformed a classic multinomial logit model for both the estimation and a validation sample. Cantillo and de Dios Ortu´zar (2005) compared their semi-compensatory model with correlated attribute thresholds against the multinomial logit model using two data sets: a simulated data set, and an SP survey on route choice for car trips between Santiago and Valparaiso, Chili. They concluded that fully compensatory models can lead to serious errors in prediction and estimation, and therefore marginal rates of substitution, if individuals do use non-compensatory decision-making processes. Their study indicates that the effect of allowing for correlation between thresholds is marginal. Allowing thresholds to be a function of socio-demographic characteristics and choice conditions improved model performance. Zheng and Guo (2008) also applied a two-stage model structure, but their problem concerned destination choice behaviour. They assumed that distance constraints define the feasible area for destination choice and that individuals perceive a spatial choice set as a contiguous areas around their home (or trip origin). They used an ordered probit model for the choice set generation stage, with all zones within an individual’s distance threshold being included in the consideration set. A multinomial logit model is used to predict the probability that a destination within an individual’s consideration set is chosen. The model was estimated to predict the destination choice of shopping, restaurant, and recreational trips in Napa County, California. The authors argue that their model appears to capture behavioural principles that are not accounted for by the MNL model. However, they did not find a significant difference between the two models in terms of goodness-of-fit. Zhu and Timmermans (2008, 2010, 2011) suggested a model, which they called the heterogeneous heuristic model, which includes several of the mechanisms discussed in this chapter. First, a set of accumulative thresholds is defined for each attribute to vary degrees of filtering and attribute attendance. Next, based on the attributes that

Models of Bounded Rationality under Certainty

25

are considered, a conventional valuation equation is applied. Third, a further simplifying mechanism is that the outcome utilities are classified, and using an overall threshold, choice alternatives are distinguished into acceptable and non-acceptable alternatives. The process automatically generates different heuristics. The final step is then concerned with the choice of heuristic, which is modelled as a function of the amount of mental effort, risk perception and expected outcome. Variations of this model have been applied to different aspects of pedestrian movement.

1.7.

Conclusions and Discussion

In this chapter, we discussed the history of the development and application of models of bounded rationality with special reference to travel behaviour research and decision-making under conditions of certainty. We defined bounded rationality as decision-making processes and choice behaviour in which individuals do not seek the optional choice and/or consider only a subset of the potentially influential attributes, and/or in comparing choice alternatives do not differentiate between asymptotically small differences in attribute/alternative values, and/or do not consider all alternatives in the choice set, based on maximizing their overall judgments. These conditions were used to structure the chapter. Attempts to model consideration of a subset of potentially influential factors are generally confined to the formulation of non-compensatory choice models and decision rules. These models have in common that after some screening process, only a single or a subset of attributes remains. In principle, these models can be formulated in utility space, in attribute space and in cognitive space. Empirical evidence with regard to the predictive performance of these models vis-a`-vis the performance of compensatory choice model varies. Some studies have suggested that noncompensatory decision rules outperform compensatory models, but there is also evidence on the contrary. Models that consider the formation of choice sets have received more interest. Although this point of view seems to have received limited support, the distinction between choice set formation and subsequent choice seems arbitrary, artificial and therefore not very appealing. While it would be imaginable that as part of a process model of high involvement decision-making under uncertainty individuals would sequentially reduce the number of alternatives considered as they collect and process additional information, it seems plausible that preference structures both drive the alternatives considered and the actual choice. Hence, the very few models attempting to address both processes in parallel warrant further study and expansion. Recently, travel behaviour research has started to formulate reference-dependent models, in which the references are either endogenously (choice set composition) or exogenously defined. Regret-based and relative utility models are examples of that kind. Current regret models have some issues of definition and specification that need to be addressed. Moreover, these models can be expanded in the sense that many more alternative to the random utility models can be formulated. In addition, work on mixtures of decision rules, not only to represent heterogeneity in

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choice behaviour among individuals but also to capture different processes that may operate in a single decision-making process, should have high priority in the research agenda. Finally, virtually all applications of models of bounded rationality under certainty (and uncertainty for that matter) have been restricted to static individual choice problems. Attempts to expand the scope of these models to problems of joint decisions, group (e.g. social networks) decisions and dynamic choice problems should be applauded.

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Chapter 2

Utility Maximisation and Regret Minimisation: A Mixture of a Generalisation Stephane Hess and Caspar G. Chorus

Abstract Purpose  This chapter proposes a new mixture model which allows for heterogeneity in sensitivities and decision rules across decision makers and attributes. Theory  A new mixture model is put forward in which the different latent classes make use of different decision rules, where the use of generalised random regret minimisation kernel allows for within class heterogeneity in the decision rules applied across attributes. Findings  Our theoretical developments are supported by the findings of an empirical application using data from a typical stated choice survey. Originality and value  Existing work has looked at heterogeneity in decision rules and sensitivities across respondents. Other work has focused on the possibility that different decision rules apply to different attributes. This chapter puts forward a model that combines these two directions of research and does so in a way that lets the optimal specification be driven by the data rather than being imposed by the analyst. Keywords: Generalised random regret model; compromise effect; mixture model; random utility model; route choice

2.1.

Introduction

Recent years have witnessed a rapidly growing number of studies that aim to incorporate decision rules other than the conventional linear-additive random utility maximisation (RUM) rule in discrete choice models of travel behaviour. Motivated

Bounded Rational Choice Behaviour: Applications in Transport Copyright r 2015 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISBN: 978-1-78441-072-8

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by the wish to increase the behavioural realism of travel demand models, as well as their empirical performance, decision rules like symmetric relative advantage maximisation (e.g. Leong & Hensher, 2014a), reference dependent utility maximisation (e.g. Stathopoulos & Hess, 2012), random regret minimisation (RRM, e.g. Chorus, 2010) and decision tree approaches (e.g. Arentze & Timmermans, 2007) have been proposed and applied in recent travel behaviour studies. Earlier work tended to focus on comparing the properties and empirical performance of models based on these alternative decision rules with those of models based on the conventional linear-additive utility maximisation rule. While such comparisons have generated interesting insights, more recent work has adopted a different perspective and puts more emphasis on capturing potential heterogeneity in applied decision rules. Two main strands of this more recent research can be distinguished: first, it is increasingly being acknowledged that it is unrealistic to assume that every individual applies the same decision rule; rather it makes more sense to allow for different groups of individuals to use different decision rules. This conceptual idea can be operationalised using mixture models of the type put forward by Hess, Stathopoulos, and Daly (2012) — see Hess and Stathopoulos (2013) and Boeri, Scarpa, and Chorus (2014) for other recent examples. These models incorporate different latent classes, each with their own decision rule (and set of taste parameters). Estimation results show large improvements in fit compared to models that assume one and the same decision rule for the entire population, where the work by Hess and Stathopoulos (2013) also provides further insights into who might be making choices in what way. A second and related body of papers questions the assumption that every attribute is processed using the same decision rule. The resulting (often called hybrid) model structures allow for different attributes to be processed using different decision rules; an example being the hybrid utility-regret model put forward by Chorus, Rose, and Hensher (2013) and employed by, amongst others, Leong and Hensher (2014b); this model allows for some attributes to be processed in a regret minimisation fashion while others are processed in a utility maximisation fashion. An obvious next step would be to combine these two approaches; that is, allowing for different individuals to use different decision rules, for different attributes. In principle, a combined hybrid-discrete mixture model would be able to accommodate for both types of decision rule heterogeneity simultaneously. However, there is an important caveat that so far has hampered progress in this direction: when the number of attributes is nontrivial, a large number of latent classes is needed to capture all possible combinations of decision rules. For example, hybrid utility maximisation-regret minimisation models for a four-attribute choice context would already result in 16 latent classes. Resulting mixture models are generally not well behaved due to the large number of classes, precluding successful estimation on empirical data. This chapter builds on a recent advance in regret minimisation modelling to circumvent this combinatorial explosion. The recently proposed generalised RRM model (Chorus, 2014) estimates a regret-weight for each attribute; if equal to 1,

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conventional regret minimisation behaviour — that is as in Chorus (2010) — is obtained. If equal to 0, conventional utility maximisation behaviour is obtained. Values between 0 and 1 imply regret minimisation behaviour, but with a smaller degree of nonlinearity in the regret function than is the case in conventional regret minimisation models. Since the regret-weight can be estimated on empirical data, this provides an opportunity to infer from the data, for each attribute, if it is processed using a utility maximisation or regret minimisation rule. That is, rather than having to estimate all possible hybrid utility-regret combinations to find the optimal constellation, the generalised RRM model allows one to directly infer the best fitting decision rule for every attribute. When estimating hybrid mixture models of the type discussed above, the generalised random regret model provides a clear conceptual and operational advantage over previously used hybrid utility-regret models: conceptually, it is much more elegant to estimate attribute-specific decision rules from the data as opposed to these rules being imposed by the researcher. Operationally, the generalised approach limits the number of latent classes, as one no longer needs to allow for every possible combination of attribute-specific decision rules. This results in better behaved and more manageable models. Nevertheless, the model still clearly allows for the possibility of different classes making use of a different combination of RUM/RRM parameters. This chapter is the first to combine the generalised RRM model and the discrete mixture paradigm; as such, it is the first attempt that we know of, to simultaneously allow for heterogeneity in decision rules across individuals and attributes in a discrete choice model of traveller behaviour. The remainder of this chapter is structured as follows. Section 2.2 presents the model structure put forward in our chapter. Section 2.3 presents empirical analysis based on a stated route choice dataset. Conclusions and directions for further research are put forward in Section 2.4.

2.2.

Model Structures

This structure gives an overview of the various model structures used in this chapter. We first discuss the generalised RRM model, which can simplify to both standard RUM and RRM structures, before discussing mixture models. 2.2.1.

The Generalised RRM Model

This section draws heavily from the recently published paper (Chorus, 2014) which puts forward the Generalised RRM (or G-RRM) model. For reasons of space limitations and to avoid repetition, we do not present and discuss the conventional RRM model and its properties; for further information on the RRM model, the interested reader is referred to Chorus (2012), with applications for example in Boeri, Longo, Doherty, and Hynes (2012), Boeri, Longo, Grisolia, Hutchinson, and Kee (2013), Kaplan and Prato (2012) and Prator (2014). The same

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applies for the standard RUM model, which is well covered elsewhere. Instead, we directly start with introducing the G-RRM model; the G-RRM model assumes that discrete choice behaviour is driven by minimisation of the following objective function: XX RRi = Ri þ υi = ln ðγ m þ exp ½βm ⋅ðxjm − xim ÞÞ þ υi ð2:1Þ j≠i

m

where denotes the random (or total) regret associated with a considered alternative i, Ri denotes the ‘deterministic’ regret associated with i, υi denotes the ‘unobserved’ regret associated with i, its negative being distributed i.i.d. Extreme Value Type I with variance π 2 =6, βm denotes the estimable taste parameter associated with attribute xm , xim ; xjm denote the values associated with attribute xm for, respectively, the considered alternative i and another alternative j, γm denotes the regret-weight for attribute xm . RRi

The probability of choosing alternative i out of J is then given by: Pi = P J

− Ri

j=1

− Rj

ð2:2Þ

that is the minimisation of the deterministic regret component. To illustrate the role of regret-weight γ m , consider the so-called attribute regret function ln ðγ m þ exp ½βm ⋅ðxjm − xim ÞÞ, which gives the regret that is associated with comparing considered alternative i with another alternative j, in terms of attribute xm . Note that when γ m equals one, the conventional attribute regret function as put forward in Chorus (2010) is obtained. By varying γ m from 0 to 1, and plotting the resulting attribute regret function for ðxjm − xim Þ ranging from −5 to 5 (keeping βm fixed at unity), the role of the regret-weight becomes immediately clear; the left hand panel of Figure 2.1 shows the effect on the attribute regret function of a step-wise variation in γ, and the right hand panel shows the effect of a continuous change in γ. Both graphs show that when the regret-weight becomes smaller and starts to approach zero, the convexity of the regret function and the resulting reference dependent asymmetry (or nonlinearity) in preferences vanishes. When γ = 0, there is no asymmetry anymore, implying that the impact on regret of a change in an alternative’s attribute is no longer dependent on the alternative’s initial performance in terms of the attribute, relative to its competition. Intuitively, the resulting regret function with symmetric preferences (i.e. with γ = 0) looks like a function generated by a linear-in-parameters RUM model, and indeed it can be shown (Chorus, 2014) that if γ = 0 for a particular attribute, the G-RRM model generates the same choice behaviour as does a linear-in-parameter RUM model, for that attribute.

Utility Maximisation and Regret Minimisation: A Mixture of a Generalisation

ln(γ + exp( βm·(xjm – xim)))

35

ln(γ + exp(βm·(xjm – xim)))

4

4

2

2 γ=1 γ = 0.5

0 –4

–2

0

γ = 0.25 γ = 0.1

–2

2 xjm – xim

4

–2 –4 1

γ = 0.01

–4

γ

0.8

0.6

0.4

0.2 0 –4

–2

0

2

4

xjm – xim

γ=0

Figure 2.1: Impact of variation in regret-weight (γ) on attribute regret (βm = 1). As noted by Chorus (2014), the coefficient values obtained when γ = 0 for a particular attribute are scaled down by a factor of J compared to the estimation of a RUM model, which is immediately obvious from Eq. (2.1) which shows that the coefficient is used J times per G-RRM function, instead of just once as in a RUM function. Clearly, depending on the values of γ m for different attributes, different choice models arise; as such, the G-RRM model can be seen as a generic formulation which nests various types of choice models: the conventional RRM model is a special case which is obtained when γ m = γ = 1 ∀m, and the conventional linear-inparameters RUM model is a special case which is obtained when γ m = γ = 0 ∀m. Furthermore, hybrid RUM-RRM models of the type proposed in Chorus et al. (2013) are obtained when γ m ∈f0; 1g ∀m. A more subtle model structure arises when γ m ∈ 0; 1½ for one or more attributes: as can be seen when inspecting Figure 2.1, for these in-between values of γ, the regret/utility function is still convex, but the degree of nonlinearity is not as high as for a conventional RRM model (or G-RRM with γ = 1). Nonetheless, given the presence of some asymmetry and reference-dependency in the regret function for values of γ ∈ 0; 1½, the resulting or implied behaviour should be considered as regret minimisation as opposed to linear-in-parameters utility maximisation behaviour. As a consequence, for the G-RRM model with γ ∈ 0; 1½, previously derived properties of the conventional RRM model hold, but these properties are less pronounced than for the conventional RRM model. As an illustration of how values of γ influence the salience of key properties of the RRM model, Figure 2.2 presents a numerical simulation in the context of the G-RRM model (with two different regret-weights). The numerical example refers to the existence of a compromise effect, which states that consumers tend to prefer alternatives with a reasonable performance on each attribute, as

36

Stephane Hess and Caspar G. Chorus

P

P(B)

0.40

0.35 P(A)

P(C ) 0.30

0.25

0.20 –1

–0.5

0

0.5

1

Δ

Figure 2.2: Compromise effect for the conventional RRM model and for G-RRM with regret-weights 0.1 (the latter in dotted lines).

opposed to alternatives with a strong performance on some, and a poor performance on other attributes. The fact that the RRM model predicts such a compromise effect has been reported in various theoretical (Chorus, 2010) and empirical (Chorus & Bierlaire, 2013) studies. As these papers show, it is the convexity of the regret function in the RRM model that generates the compromise effect: deterioration of an attribute with an already poor performance generates a lot of additional regret, and this cannot be compensated by the relatively small decreases in regret associated with further improvement of an already strong attribute — this is easily verified when inspecting the regret curve for γ = 1 in Figure 2.1’s left panel. As a consequence of this semi-compensatory behaviour implied by the convexity of the regret function, alternatives with a reasonable performance on all attributes gain a market share bonus in the context of RRM models. Note that, given the symmetric and reference independent treatment of attributes in linear-in-parameter RUM-models, the latter do not feature the compromise effect (as, for example, highlighted and empirically verified in Chorus & Bierlaire, 2013). Since RRM’s accommodation of the compromise effect is a consequence of the convexity of its regret function, the expectation is that for a G-RRM model with 0 < γ < 1 this compromise effect, while still present, will be less pronounced than in a conventional RRM model (i.e. a G-RRM model with γ = 1). The following numerical example — which has been used in previous publications to illustrate how RRM generates compromise effects, and as such forms a good benchmark — serves to illustrate and verify this expectation. Assume a choice

Utility Maximisation and Regret Minimisation: A Mixture of a Generalisation

37

situation between three alternatives, each defined in terms of two quality attributes (x1 and x2 ) that are equally important to the decision-maker (higher values are preferred over lower ones; β = 1 for each of the two attributes): A = {1, 3}, B = {2 + Δ, 2 − Δ}, C = {3,1}. In words, where A and C take on relatively extreme positions on the two attributes, B is a compromise alternative to the extent that Δ is close to 0. Figure 2.2 plots P(A), P(B) and P(C) as a function of Δ, for G-RRM models with γ = 1 for both attributes, and for G-RRM models with γ = 0:1 for both attributes, respectively (the former in solid lines, the latter in dotted lines).1 Figure 2.2 clearly shows the presence of a compromise effect for alternative B: as long as its attributes remain close to 2 (i.e. Δ close to 0), it receives a choice probability bonus at the cost of the two extreme alternatives. Secondly, and this is more relevant in the context of this chapter, the compromise effect is still present for the case where γ = 0:1, but less pronounced than for the case where γ = 1 (i.e. the conventional RRM model). This decreasing salience of the compromise effect for the G-RRM model with the lower regret-weight follows directly from the decreasing difference between the sensitivity to attribute changes in the domain of poor performance versus in the domain of strong performance. This decreasing difference in turn follows directly from the decreased level of nonlinearity of the regret curve, which is a direct consequence of the lower regret-weight. In sum, regret-weights with values between 0 (linear-in-parameters RUM) and 1 (conventional RRM model) still generate regret minimisation behaviour and as such still result in key properties of the conventional RRM model, but with a reduced salience. For estimation purposes, it is pragmatic to parameterise the regret-weight in terms of a binary logit function: γ m = exp ðδm Þ=ð1 þ exp ðδm ÞÞ. For (large) negative values of δm , a RUM specification is approached for the attribute, and for (large) positive values, a RRM specification is approached. When δm is estimated to lie inbetween these two extremes, for example when it is estimated to be insignificantly different from 0, implying γ m = 0:5, regret minimisation behaviour is obtained but with less emphasis on regret than is the case for the conventional RRM model.

2.2.2.

Mixture Model

The mixture model introduced by Hess et al. (2012) is a simple generalisation of a latent class structure where the differences between classes are not just in parameters of the same underlying model structures but also in the use of different model structures in different classes.

1. Note that choice probabilities generated by a RUM’s linear-in-parameters Logit-model would of course be insensitive to changes in Δ as the model would assign equal choice probabilities of 0.33… for all three alternatives, irrespective of the value for Δ; these probabilities are hence not plotted, for clarity of communication.

38

Stephane Hess and Caspar G. Chorus

A general specification of a model allowing for different decision rules within a latent class framework is given by: LCn =

S X

π n;s LCn;s

ð2:3Þ

s=1

where LCn is the contribution to the likelihood function of the observed choices for respondent n. This probability of observed choices is given by a weighted average over S different types of models, where LCns is the probability of the observed sequence of choices for person n if model s is used, and π n;sPis the weight attached to model s (representing a specific decision process), where Ss = 1 π n;s ∀n. The mixing of models is performed at the level of individual respondents rather than individual choices. In the existing work, the above specification has been used to combine models such as RUM, RRM and elimination by aspects (EBA). In the present chapter, we extend on this by making the individual classes use G-RRM models, thus allowing not just for mixtures between pure RUM and RRM classes, but also mixed RUMRRM classes and classes with intermediate specifications for individual parameters.

2.3.

Empirical Analysis

2.3.1.

Data

The data used in the present chapter is the same as that in Chorus and Bierlaire (2013). The data collection effort focused on route choice behaviour among commuters who travel from home to work by car. A total of 550 people were sampled from an internet panel maintained by IntoMart in April 2011. Sampled individuals were at least 18 years old, owned a car, and were employed. The sample was representative of Dutch commuters in terms of gender, age, and education level. Of the 550 sampled individuals, 390 filled out the survey (implying a response rate of 71%). Respondents to the survey were asked to imagine the hypothetical situation where they were planning a new commute from home to work (either because they had recently moved, or because their employer had recently moved, or because they had started a new job). They were asked to choose between three different routes that differed in terms of the following four attributes, with three levels each: average door-to-door travel time (45, 60, 75 minutes), percentage of travel time spent in traffic jams (10%, 25%, 40%), travel time variability (±5, ±15, ± 25 minutes) and total costs (h5.5, h9, h12.5). Using the Ngene-software package, a so-called ‘optimal orthogonal in the differences’ design of choice sets was created to ensure a statistically efficient data collection. This design resulted in nine choice tasks per respondent and 3510 choice observations in total. Table 2.1 shows one of these tasks.

Utility Maximisation and Regret Minimisation: A Mixture of a Generalisation

39

Table 2.1: An example route choice-task.

Average travel time (minutes) Percentage of travel time in congestion (%) Travel time variability (minutes) Travel costs (Euros) YOUR CHOICE 2.3.2.

Route A

Route B

Route C

45 10% ±5 h12.5 □

60 25% ±15 h9 □

75 40% ±25 h5.5 □

Results

A large number of different models were estimated for this study, gradually increasing the number of classes within the latent class structure discussed in Section 2.2, with each class making use of a G-RRM model, where, in varying classes, a priori structures were either imposed (e.g. pure RUM, pure RRM) or the γ parameters were estimated. All models were coded and estimated in Ox (Doornik, 2001), making use of multiple runs with different starting values to reduce the risk of inferior local maxima. The models presented here are generally those with any additional constraints on γ imposed which arose from the modelling work, where this relates to cases where δ became either very small or very large. The models are compared in terms of the Bayesian Information Criterion (BIC), which incurs a higher penalty for increases in the number of parameters than traditional likelihood ratio or adjusted ρ2 measures. We begin with Table 2.2, which shows three models with a single class each, that is S = 1. In models 1 and 2, we a priori impose a RUM and RRM structure, respectively, that is constraining γ = 0 in model 1, and γ = 1 in model 2, across attributes. The results show that the pure RRM structure (model 2) is preferred to the pure RUM structure (model 1), with both producing four significant negative coefficient estimates. In model 3, we initially estimated all four δ parameters freely, but the values for δ(perc. in congestion) and δ(travel cost) reached very large positive values which indicated that γ = 1 for these two attributes, meaning a pure RRM treatment. For the remaining two parameters, the δ estimates are negative albeit not significantly different from zero, indicating a value for γ < 0.5. As can be seen from Figure 2.1, this is still closer to full RRM treatment than to full RUM treatment, a finding that is also in line with the small difference in fit between models 2 and 3, with the BIC measure indicating that the additional two parameters in model 3 do not justify the improvement in log-likelihood. The findings from the first three models hence point towards a pure RRM treatment for all four parameters. Even though the G-RRM model in the end thus ‘collapses’ to a simpler RRM model, it highlights the advantage of the structure as a diagnostic tool, rather than having to estimate all possible combinations of RUM and RRM treatment for the different parameters.

β(travel time) β(perc. in congestion) β(travel time variability) β(travel cost) δ(travel time) δ(perc. in congestion) δ(travel time variability) δ(travel cost) γ(travel time) γ(perc. in congestion) γ(travel time variability) γ(travel cost)

Log-likelihood Parameters Adj ρ2 BIC

Table 2.2: Single class models. Model 2 −2604.96 4 0.3234 5235.22 est. rob. t-rat. −0.0469 −20.57 −0.0181 −14.44 −0.0210 −11.80 −0.1128 −15.59 + inf (fixed a priori) + inf (fixed a priori) + inf (fixed a priori) + inf (fixed a priori) 1 1 1 1

Model 1 −2613.45 4 0.3212 5252.20 est. rob. t-rat. −0.0224 −22.17 −0.0091 −15.21 −0.0105 −11.83 −0.0576 −16.83 − inf (fixed a priori) − inf (fixed a priori) − inf (fixed a priori) − inf (fixed a priori) 0 0 0 0

−2603.26 6 0.3233 5244.47 est. rob. t-rat. −0.0330 −3.91 −0.0188 −14.78 −0.0147 −0.68 −0.1166 −16.66 −0.3622 −0.27 + inf (fixed) −0.4656 −0.05 + inf (fixed) 0.41 1 0.39 1

Model 3

40 Stephane Hess and Caspar G. Chorus

Utility Maximisation and Regret Minimisation: A Mixture of a Generalisation

41

We next move to four separate models using two classes each in Table 2.3. This includes one model (model 4) with two pure RUM classes, one (model 5) with two pure RRM classes, one with one pure RUM class and one pure RRM class (model 6) and one with two G-RRM classes (model 7). All four models show major improvements over the single class models in Table 2.2. They also all show a split into one class with a roughly 1/3 probability, and one with a roughly 2/3 probability, where a very consistent pattern emerges across the four models. In the second class, the relative importance of travel time and travel time variability is reduced substantially, especially the former, while the importance of congestion and to a lesser extent travel cost is increased. Across the four models, the impact of congestion also only attains low levels of significance in the first class. In terms of model structure, we first note that model 5 outperforms model 4, that is a structure with two pure RRM classes is preferred to a structure with two pure RUM classes. However, model 6, which a priori imposes one pure RUM class and one pure RRM class, obtains an even better BIC measure, suggesting heterogeneity across respondents not just in sensitivities but also in decision rules. Model 7 relaxes the assumption in model 6 about within class homogeneity of the decision rule for different attributes. The findings for the second class remain the same as the imposition of a pure RRM treatment for all attributes in model 6. After additional constraints for δ1(% cong.), δ1(tt var) and δ1(cost), the picture in the first class is slightly different from model 6. We still see a pure RUM treatment for congestion and travel time variability. However, for travel time, we see a γ value of around one third, indicating a treatment that is not pure RUM and actually suggests a fairly regret-based treatment, while, for cost, the treatment is purely regret based. Despite these differences, the log-likelihood for models 6 and 7 is essentially the same, where the one additional parameter (δ1(trav. time)) in model 7 leads to a higher (worse) BIC. The recommendations from the two class structures would thus be a further constrained version of model 7, using the G-RRM model once again as a specification search tool, leading to a mixed RUM-RRM treatment in one class, and a pure RRM treatment in the second class. Table 2.4 contains our final set of four models, where models with more classes than those presented in Table 2.4 gave a worse BIC. To allow for a generic format in the presentation of results, Table 2.4 makes use of six classes throughout, where the first two are pure RUM classes, followed by two pure RRM classes, with the fifth and sixth being freely estimated G-RRM models (subject to additional constraints). Not each model makes use of each class, as detailed in the table. Model 8 combines a pure RUM class (class 1) with a pure RRM class (class 3) and a G-RRM class (class 5). This model outperforms the best two class models from Table 2.3, where the findings suggest that, in the third class, all parameters except travel time variability should have a pure RRM treatment, where the estimate for travel time variability is however also not statistically significant in this class. This third class obtains around one quarter of the overall weight in the model, with a slightly higher probability for the other (a priori) pure RRM class than for

β1(trav. time) β1(% cong.) β1(tt var) β1(cost) β2(trav. time) β2(% cong.) β2(tt var) β2(cost) δ1(trav. time) δ1(% cong.) δ1(tt var) δ1(cost) γ1(trav. time) γ1(% cong.) γ1(tt var) γ1(cost) δ2(trav. time) δ2(% cong.) δ2(tt var) δ2(cost)

Log-likelihood Parameters Adj ρ2 BIC

Model 5 −2416.78 9 0.3709 4890.49 est. rob. t-rat. −0.1582 −5.44 −0.0052 −1.35 −0.0510 −4.54 −0.0864 −4.36 −0.0314 −11.23 −0.0266 −12.37 −0.0182 −8.27 −0.1451 −14.05 + inf (fixed a priori) + inf (fixed a priori) + inf (fixed a priori) + inf (fixed a priori) 1 1 1 1 + inf (fixed a priori) + inf (fixed a priori) + inf (fixed a priori) + inf (fixed a priori)

Model 4

−2431.59 9 0.3671 4920.11 est. rob. t-rat. −0.0559 −10.15 −0.0025 −1.44 −0.0261 −6.33 −0.0437 −4.63 −0.0146 −12.81 −0.0131 −13.38 −0.0088 −7.92 −0.0725 −15.72 − inf (fixed a priori) − inf (fixed a priori) − inf (fixed a priori) − inf (fixed a priori) 0 0 0 0 − inf (fixed a priori) − inf (fixed a priori) − inf (fixed a priori) − inf (fixed a priori)

Table 2.3: Two class models.

−2412.92 9 0.3719 4882.77 est. rob. t-rat. −0.0559 −9.71 −0.0030 −1.65 −0.0260 −6.20 −0.0404 −4.16 −0.0310 −12.23 −0.0275 −12.74 −0.0180 −7.88 −0.1495 −13.75 − inf (fixed a priori) − inf (fixed a priori) − inf (fixed a priori) − inf (fixed a priori) 0 0 0 0 + inf (fixed a priori) + inf (fixed a priori) + inf (fixed a priori) + inf (fixed a priori)

Model 6

−2412.83 10 0.3717 4888.91 est. rob. t-rat. −0.0558 −9.77 −0.0034 −1.91 −0.0259 −6.22 −0.0808 −4.16 −0.0309 −12.19 −0.0276 −12.71 −0.0180 −7.87 −0.1496 −13.76 −0.6918 −4.10 − inf (fixed) − inf (fixed) + inf (fixed) 0.33 0.00 0.00 1.00 + inf (fixed) + inf (fixed) + inf (fixed) + inf (fixed)

Model 7

42 Stephane Hess and Caspar G. Chorus

γ2(trav. time) γ2(% cong.) γ2(tt var) γ2(cost) δs1 δs2 π1 π2

−0.6917 0

33.36% 66.64%

0 0 0 0 −4.18 

−0.77489 −3.88 0  31.54% 68.46%

1 1 1 1 −0.6956 0 33.28% 66.72%

1 1 1 1 −4.14 

−0.6918 0 33.36% 66.64%

1.00 1.00 1.00 1.00 −4.10 

Utility Maximisation and Regret Minimisation: A Mixture of a Generalisation 43

β1(trav. time) β1(% cong.) β1(tt var) β1(cost) β2(trav. time) β2(% cong.) β2(tt var) β2(cost) β3(trav. time) β3(% cong.) β3(tt var) β3(cost) β4(trav. time) β4(% cong.) β4(tt var) β4(cost) β5(trav. time) β5(% cong.) β5(tt var) β5(cost) β6(trav. time) β6(% cong.) β6(tt var) β6(cost)

Log-likelihood Parameters Adj ρ2 BIC

Model 9 −2322.86 19 0.3927 4765.90 est. rob. t-rat. −0.5172 0.00 −0.2440 0.00 −0.1769 −0.35 −4.0859 −0.41 −0.0548 −10.78 −0.0041 −2.06 −0.0261 −6.85 −0.0341 −4.05 −0.0151 −4.07 −0.0271 −8.66 −0.0211 −6.08 −0.0810 −6.20 −0.1007 −4.91 −0.0319 −7.35 −0.0167 −4.27 −0.3873 −6.04        

Model 8

−2347.19 15 0.3874 4789.26 est. rob. t-rat. −0.0531 −11.35 −0.0043 −2.18 −0.0250 −7.09 −0.0354 −4.29     −0.0654 −3.75 −0.0306 −8.00 −0.0158 −4.54 −0.3049 −4.79     −0.0115 −1.91 −0.0270 −6.71 −0.0134 −0.26 −0.0652 −2.83    

Table 2.4: Three to six class models.

−2301.42 25 0.3967 4760.98 est. rob. t-rat. −0.0223 −2.47 −0.0347 −3.92 −0.0129 −1.84 −0.0630 −2.16 −0.0598 −7.90 −0.0063 −1.34 −0.0135 −2.67 −0.0370 −1.68 −0.0982 −2.07 −0.0135 −0.39 −0.0989 −4.70 −0.1308 −1.46 −0.0783 −2.77 −0.0257 −5.26 −0.0150 −3.72 −0.3980 −4.32 −0.0084 −1.71 −0.0157 −2.50 −0.0108 −0.17 −0.0692 −1.91    

Model 10

−2284.89 30 0.3997 4759.54 est. rob. t-rat. −0.0040 −1.57 −0.0119 −2.54 −0.0104 −3.73 −0.0222 −1.73 −0.0363 −2.38 −0.0277 −4.24 −0.0085 −0.62 −0.0442 −0.78 −0.6426 −0.88 −0.0653 −0.61 −0.0359 −1.36 −1.0933 −0.62 −0.1529 −4.37 −0.0288 −1.93 −0.0226 −3.19 −0.5812 −6.04 −0.0520 −1.98 −0.0036 −0.80 −0.0400 −4.24 −0.0953 −3.03 −0.0243 −2.69 −0.0282 −1.68 −0.0128 −1.70 −0.2781 −3.78

Model 11

44 Stephane Hess and Caspar G. Chorus

γ1(generic) γ2(generic) γ3(generic) γ4(generic) δ5(trav. time) δ5(% cong.) δ5(tt var) δ5(cost) γ5(trav. time) γ5(% cong.) γ5(tt var) γ5(cost) δ6(trav. time) δ6(% cong.) δ6(tt var) δ6(cost) γ6(trav. time) γ6(% cong.) γ6(tt var) γ6(cost) δs1 δs2 δs3 δs4 δs5 δs6 π1 π2 π3 π4 π5 π6

0 (fixed to pure RUM)  1 (fixed to pure RRM)  + inf (fixed) + inf (fixed) −1.2144 −0.05 + inf (fixed) 1.00 1.00 0.23 1.00         0.3460 0.97  0.4757 0.99  0   35.14%  40.00%  24.86% 

0 (fixed to pure RUM) 0 (fixed to pure RUM) 1 (fixed to pure RRM) 1 (fixed to pure RRM)                 −2.8410 −6.95 0.0226 0.11 −0.0383 −0.16 0    1.92% 33.61% 31.62% 32.86%  

0 (fixed to pure RUM) 0 (fixed to pure RUM) 1 (fixed to pure RRM) 1 (fixed to pure RRM) + inf (fixed) + inf (fixed) −2.1617 −0.03 + inf (fixed) 1.00 1.00 0.10 1.00         −0.2268 −0.40 0.3144 0.64 −0.4835 −0.78 0.4541 0.81 0   14.88% 25.56% 11.51% 29.39% 18.66% 

0 (fixed to pure RUM) 0 (fixed to pure RUM) 1 (fixed to pure RRM) 1 (fixed to pure RRM) −1.9107 −0.49 + inf (fixed) + inf (fixed) + inf (fixed) 0.13 0.00 0.00 1.00 + inf (fixed) + inf (fixed) + inf (fixed) + inf (fixed) 1.00 1.00 1.00 1.00 0.2737 0.35 0.1060 0.13 0.1725 0.30 0.4091 0.67 0.1591 0.27 0  18.03% 15.25% 16.29% 20.64% 16.08% 13.71%

Utility Maximisation and Regret Minimisation: A Mixture of a Generalisation 45

46

Stephane Hess and Caspar G. Chorus

the pure RUM class. These findings are overall compatible with those from model 7, albeit that congestion and travel time now have a pure RRM treatment outside the pure RUM class. The remaining three models all make use of two RUM classes and two RRM classes, where models 10 and 11 add in one, respectively two, additional G-RRM classes. While model 9 provides an improvement over previous structures, this is solely a result of allowing for taste heterogeneity within the RRM segment of the models (i.e. classes 3 and 4) as the additional RUM class (class 1) obtains a near zero weight (1.92%) with no significant parameter estimates. The fact that further improvements in fit are obtained in models 10 and 11 highlights the value of the G-RRM approach in capturing attribute-specific treatments of decision rules. Firstly, by allowing for these additional classes, we are able to capture some of the within decision rule heterogeneity for the RUM classes, where we now see a more even split in probability across RUM classes, along with some significant effects in both classes. The majority of the weight remains with a RRM treatment of attributes, whether in the pure RRM classes or the RRM treatment within G-RRM classes. Additionally however, we see that more pure RRM classes are needed (given that one additional G-RRM class in model 11 becomes pure RRM) before we can recover a class with a stronger mixture between RUM and RRM, namely class 5 in model 11.

2.4.

Summary and Conclusions

There is growing interest in the notion that different decision rules may work differently well in explaining the choices observed in data used for travel behaviour analysis. Going further, there is now growing evidence that different decision-makers may well be making their choices based on different rules. Finally, there are also results that suggest that the optimal decision rule may in fact vary across attributes within a given dataset. The present chapter has brought these different notions together by putting forward a latent class approach which not only allows for different decision rules across classes, but also differences in the decision rules used across attributes within a given class. This would clearly lead to a very large number of different possible combinations of rules across classes, and, within the context of two popular paradigms, RUM and RRM, we have proposed to address this through the use of a G-RRM model within individual classes. This allows the optimal specification in terms of split between RUM and RRM within a given class to be revealed by the data during estimation, rather than needing to be imposed by the analyst. Initial findings on a standard stated choice dataset are promising and show how a rich pattern of taste heterogeneity, and decision rule heterogeneity across respondents and attributes can be revealed. It has also highlighted that, while in many cases, the G-RRM models collapse to either RUM or RRM for individual coefficients, this provides a very useful diagnostic approach rather than needing to estimate each possible combination of RUM and RRM separately.

Utility Maximisation and Regret Minimisation: A Mixture of a Generalisation

47

References Arentze, T. A., & Timmermans, H. J. P. (2007). Parametric action trees: Incorporating continuous attribute variables into rule-based models of discrete choice. Transportation Research Part B, 41(7), 772783. Boeri, M., Longo, A., Doherty, E., & Hynes, S. (2012). Site choices in recreational demand: A matter of utility maximization or regret minimization? Journal of Environmental Economics and Policy, 1(1), 3247. Boeri, M., Longo, A., Grisolia, J. M., Hutchinson, W. G., & Kee, F. (2013). The role of regret minimization in lifestyle choices affecting the risk of coronary heart disease. Journal of Health Economics, 32(1), 253260. Boeri, M., Scarpa, R., & Chorus, C. G. (2014). Stated choices and benefit estimates in the context of traffic calming schemes: Utility maximisation, regret minimisation, or both? Transportation Research Part A, 61, 121135. Chorus, C. G. (2010). A new model of random regret minimisation. European Journal of Transport and Infrastructure Research, 10(2), 181196. Chorus, C. G. (2012). Random regret-based discrete choice modeling: A tutorial. Heidelberg: Springer. Chorus, C. G. (2014). A generalized random regret minimization model. Transportation Research Part B, 68, 224238. Chorus, C. G., & Bierlaire, M. (2013). An empirical comparison of travel choice models that capture preferences for compromise alternatives. Transportation, 40(3), 549562. Chorus, C. G., Rose, J. M., & Hensher, D. A. (2013). Regret minimization or utility maximization: It depends on the attribute. Environment and Planning Part B, 40(1), 154169. Doornik, J. A. (2001). Ox: An object-oriented matrix language. London: Timberlake Consultants Press. Hess, S., & Stathopoulos, A. (2013). A mixed random utility  Random regret model linking the choice of decision rule to latent character traits. Journal of Choice Modelling, 9, 2738. Hess, S., Stathopoulos, A., & Daly, A. (2012). Allowing for heterogeneous decision-rules in discrete choice models: An approach and four case-studies. Transportation, 39(3), 565591. Kaplan, S., & Prato, C. G. (2012). The application of the random regret minimization model to drivers’ choice of crash avoidance manoeuvres. Transportation Research Part F, 15(6), 699709. Leong, W., & Hensher, D. A. (2014a). Relative advantage maximisation as a model of context dependence for binary choice data. Journal of Choice Modelling, 11, 3042. Leong, W., & Hensher, D. A. (2014b). Contrasts of relative advantage maximisation with random utility maximisation and regret minimisation. Journal of Transport Economics and Policy (JTEP), 48(3). Prato, C. G. (2014). Estimating random regret minimization models in the route choice context. Transportation, 41(2), 351375. Stathopoulos, A., & Hess, S. (2012). Revisiting reference point formation, gain-loss asymmetry and non-linear sensitivities with an emphasis on attribute specific treatment. Transportation Research Part A, 46, 16731689.

Chapter 3

Relative Utility Modelling Junyi Zhang

Abstract Purpose  The chapter outlines the principles underlying relative utility models, discusses the results of empirical applications and critically assesses the usefulness of this specification against commonly used random utility models and other context dependence models. It also discusses how relative utility can be viewed as a generalisation of context dependency. Theory  In contrast to the conventional concept of random utility, relative utility assumes that decision-makers derive utility from their choices relative to some threshold(s) or reference points. Relative utility models thus systematically specify the utility against such thresholds or reference points. Findings  Examples in the chapter show that relative utility model perform well in comparison to conventional utility-maximising models in some circumstances. Originality and value  Examples of relative utility models are rare in transportation research. The chapter shows that several recent models can be viewed as special cases of relative utility models. Keywords: Relative utility; multiple context dependency; prospect; regret minimisation; advantage maximisation

3.1.

Introduction

In the early 2000s, the author with his colleagues introduced in the travel behaviour research community a discrete choice model based on the concept of relative utility (Zhang, Timmermans, Borgers, & Wang, 2004). Several considerations motivated this new model development. First, the validity of discrete choice models, such as

Bounded Rational Choice Behaviour: Applications in Transport Copyright r 2015 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISBN: 978-1-78441-072-8

50

Junyi Zhang

the multinomial logit model, that exhibit the independence of irrelevant alternatives (IIA) property was questionable. With few exceptions (e.g. Borgers & Timmermans, 1987) the travel behaviour research community tried to relax this property by allowing for variance differences and covariances in the error term structures of the utility function. It seemed that there was a ‘culture’ in the choice modelling community that advanced choice models must include complicated error term structures. In other words, complicated error term structures had become synonymous with advanced choice models. However, we felt it is next to impossible to judge whether assumptions about error terms are correct or not. More importantly, from the perspective of supporting policy decisions, it is reasonable to argue that modelling innovation should focus on observed factors rather than on unobserved factors. Second, although it has been long argued that utility is a relative concept, the ways this notion is reflected in the modelling process were questionable. Looking at modelling practice, conventional utility-based choice models mainly assume that at least one of constant terms in the utility functions is zero. With this assumption, utilities of other alternatives in choice set can be quantitatively measured. In theory, any constant term can be treated as a reference and changing this reference, in case of the most popular linear utility function, does not affect choice behaviour. However, as argued by Kahneman and Tversky (1979), Tversky and Kahneman (1992) and Tversky and Simonson (1993), choice behaviour depends on status quo or reference point(s) and a change of reference point might lead to preference reversal. In this contention, models need to accommodate the existence of various reference points. Third, decision-makers may not treat all alternatives in a choice set equally. Possible reasons include lack of information, tendency of simplifying complex decision-making tasks due to limited information processing capabilities, personal liking or taste, etc. Thus, the guiding argument was that choice behaviour is context dependent. The importance of incorporating context dependency into choice models has been recognised for about half a century. Earlier studies, dealing with spatial choice behaviour (e.g. Rushton, 1969), argued that individual preference does not exist independently of the environment where the decision is made. Since then, various researchers confirmed the existence of context dependency with respect to various types of human decisions, such as Lichtenstein and Slovic (1971), Borgers and Timmermans (1988), Oppewal and Timmermans (1991), Tversky and Kahneman (1992), Tversky and Simonson (1993), Kokinov and Petrov (2001) and McFadden (2001). In particular, existing studies have repeatedly shown that context-dependent preferences are not mere artifacts but robust features of actual behaviour (e.g. Swait et al., 2002). Thus, the formulation of context-dependent models was felt necessary and it has remained an active field of development ever since. A unified and widely acknowledged definition of context, however, does not exist. The concept has been described inconsistently in different disciplines and even in different studies in the same discipline. Zhang et al. (2004) attempted to provide a unified definition by identifying three types of context: alternativespecific, individual-specific and circumstantial contexts. Furthermore, they proposed adopting the concept of relative utility to represent context dependency in a

Relative Utility Modelling

51

systematic way. They argued that an individual evaluates an alternative in a choice set by comparing it with other alternatives (alternative-oriented relative utility), perhaps with the alternatives the individual chose in the past (time-oriented relative utility), and/or with the alternatives chosen by other individuals (decisionmaker-oriented relative utility). Relative utility argues that utility is only meaningful relative to reference point(s), which is consistent with prospect theory (Kahneman & Tversky, 1979; Tversky & Kahneman, 1992). By contrast, relative utility allows the inclusion of multiple reference points in a systematic way. The influence of multiple reference points reflects general features of human decisions (e.g. Lin, Huang, & Zeelenberg, 2006; Koop & Johnson, 2012). When people make choices, they may have limited information and/or may not have enough time (i.e. bounded rationality may exist). In such situations, setting some reference point(s) may be useful to simplify the decision. Single reference point may be acceptable, but multiple reference points are probably much better because they may lead to a higher degree of satisfaction with the decision. In the remaining part of this chapter, first relative utility modelling efforts under the principle of relative utility maximisation will be discussed. Next, results of empirical studies will be summarised to assess the usefulness of relative utility models against commonly used random utility models and clarify how relative utility can be viewed as a generalisation of context dependency models. Finally, major findings from both empirical and conceptual perspectives will be summarised and future research issues will be discussed.

3.2.

Relative Utility Modelling

3.2.1.

General Formulation

To represent context dependencies in a comprehensive and systematic way, Zhang et al. (2004) defined three types of relative utility ðUnit Þ: Unit = f ðvnit jðvnjt : ∀j ≠ iÞÞ þ enit

ð3:1Þ

Unit = f ðvnit jðvnjt0 : t0 ≠ t and ∀jÞÞ þ enit

ð3:2Þ

Unit = f ðvnit jðvn0 jt : jÞÞ þ enit ; n0 ∈ social reference group

ð3:3Þ

where n (or n0 ) i (or j), and t (or t0 ) refer to individual, alternative, and time, respectively, enit is an error term, vnit and vnjt represent the influences of attributes of alternatives i and j in the choice set, respectively. Eqs. (3.1)(3.3) define alternative-oriented relative utility, time-oriented relative utility and decision-maker-oriented relative utility, respectively. The three types of

52

Junyi Zhang

relative utility reflect the influence of different types of context dependencies. In Eq. (3.1), the relative utility of alternative i is defined by referring to the existence of other alternatives in choice set; in Eq. (3.2), the reference point for the relative utility is the alternative that was chosen in the past and/or will be chosen in the future; and the alternatives chosen by other persons serve as reference points (i.e. social reference group) for the relative utility in Eq. (3.3). With the above definitions, the principle of relative utility maximisation was proposed to replace the conventional random utility maximisation principle. The principle of relative utility maximisation argues that a decision-maker chooses an alternative with the highest relative utility from his/her choice set. 3.2.2.

Alternative-Oriented Relative Utility

This chapter mainly focuses on alternative-oriented relative utility. Note that suffix t is omitted for simplicity. Uni = Vni þ eni = γ ni

X

ðvni − vnj Þ þ eni

ð3:4Þ

j≠i

Here, Vni is the deterministic term of Uni , γ ni is a relative interest parameter for accommodating the idea that people may not equally evaluate different alternatives in a choice set. By using the term vni − vnj , observed similarities among alternatives in choice set are explicitly incorporated into the utility function. Assuming that eni follows an identical and independent Weibull distribution, one can obtain the following relative utility-based multinomial logit (r_MNL) model, which is clearly a non-IIA choice model.

P exp γ ni j ≠ i ðvni − vnj Þ

P Pni = P ð3:5Þ 0Þ exp γ ðv − v 0 nj nj nj j j ≠j In theory, γ ni can take on any real value. For parameter identification, one of the γ ni must be fixed during model estimation. For ease of interpreting the estimation results, one can assume that, X 0 ≤ γ ni ≤ 1 ; γ ni = 1 ð3:6Þ i

When determining the relative utility of an alternative, an individual may not equally deal with comparative alternatives in the choice set. To reflect such a decision-making mechanism, Eq. (3.4) can be expanded as, Uni = Vni þ eni = γ ni

X j≠i

wnij ðvni − vnj Þ þ eni

ð3:7Þ

Relative Utility Modelling 0 ≤ wnij ≤ 1;

X

wnij = 1

53 ð3:8Þ

j≠i

where wnij is a weight parameter reflecting the influence of alternative j on the choice of i. It is obvious that the influence of alternative j on the choice of alternative i is expressed as − γ ni wnij (see Eq. (3.7)), which are summarised in Table 3.1 for all alternatives (Zhang & Fujiwara, 2004). For example, the influence of alternative i on alternative 1 is − γ n1 wn1i , but the opposite is − γ ni wni1 , implying that the mutual influence is not symmetric. Thus, Eq. (3.7) accommodates not only unequal but also asymmetric choice structures. Since different individuals may show different interests in alternatives and attach different weights on alternatives when comparing with each other, these two types of parameters can be defined as a function of individual attributes and/or other factors, respectively, to reflect the influence of heterogeneity, as follows:

P exp θ z iq niq q

P γ ni = P ð3:9aÞ 0 0 exp θ z 0 j q jq njq P  exp s θ ijs znis P  wnij = P j exp s0 θ ijs0 znis0 P  exp s θ ijs znis P  wnij = P j exp s0 θ ijs0 znis0

ð3:9bÞ

where zniq znis are explanatory variables with parameters θiq and θijs , respectively. To estimate relative interest and weight parameters, the corresponding parameter of one reference alternative should be fixed during model estimation. If γ ni and wnij Table 3.1: Matrix of mutual influences among alternatives. Alternative in choice set

1 ⋮ I ⋮ I

Target alternative 1



i



I

γ n1 ⋮ − γ n1 wn1i ⋮ − γ n1 wn1I

⋮ ⋮ ⋮ ⋮ ⋮

− γ ni wni1 ⋮ γ ni ⋮ − γ ni wniI

⋮ ⋮ ⋮ ⋮ ⋮

− γ nI wnI1 ⋮ − γ nI wnIi ⋮ γ nI

54

Junyi Zhang

are all the same across alternatives, then the r_MNL model based on Eq. (3.7) collapses into the conventional MNL model. 3.2.3.

Representing Quasi-Nested Choice Structure

When the number of alternatives in a choice set increases, one can expect that there is a higher possibility that correlations between alternatives matter. In such a case, grouping alternatives into several nests and adopting a nested logit model is a popular approach. However, when the nested choice structure is not homogeneous across population groups, the modelling task becomes difficult. One may apply, for example, a latent class modelling approach (Wu, Zhang, & Fujiwara, 2011); however, if there are more choice nests, the latent class modelling approach may not work well either. To represent this kind of heterogeneous choice structure flexibly, one can transform Eq. (3.7) into, X X Uni = γ ni wng ðvni − vnjg Þ þ eni ð3:10Þ j≠i

g

where wng is a weight parameter for group g rather than for a single alternative. Again, assuming that error term eni follows an independent and identical Weibull distribution results in the following choice model. P P exp ðγ ni j ≠ i wng g ðvni − vnjg ÞÞ P P ð3:11Þ Pni = P l exp ðγ nl j ≠ l wng g ðvnj − vnlg ÞÞ One can see that alternatives in the group g are, in fact, nested under alternative i with the help of wng This model is called a quasi-nested choice model (r_QNL model: Zhang & Fujiwara, 2004). 3.2.4.

Endogenous Modelling of Choice Set Generation

To date, endogenous modelling of choice set generation has been dominated by the two-stage procedure proposed by Manski (1977). The first stage treats the generation of a choice set and the second stage consists of choosing an alternative from the choice set. This approach seems logical at first glance. However, modelling practice suggests that the separate specification of choice sets provides no useful information for predicting choices beyond the information contained in the utility functions (Horowitz & Louviere, 1995). Furthermore, when the number of alternatives becomes large, the number of possible choice sets increases dramatically. Under these circumstances, model estimations become very difficult because compared to the large number of choice sets, the information available to estimate parameters, representing the different choice sets, is in fact very limited. Motivated by this shortcoming of the two-stage model, Zhang, Fujiwara, and Kusakabe (2005) applied the concept of relative utility to represent choice set

Relative Utility Modelling

55

generation endogenously within a unified modelling framework. First, transform Eq. (3.7) as, X Uni = ln Ψni þ γ~ ni wnij ðvni − vnj Þ þ eni ð3:12Þ j≠i

P P where 0 ≤ Ψni ≤ 1, − ∞ < γ~ ni < ∞ and γ ni = ðln Ψni þ γ~ ni j ≠ i wnij ðvni − vnj ÞÞ= j ≠ i wnij ðvni − vnj Þ. The validity of Eq. (3.12) is supported by the fact that relative interest parameters can take on any real value. Then, the choice probability can be re-written as, n P o Ψni ⋅exp γ~ ni j ≠ i wnij ðvni − vnj Þ n P o Pni = P l Ψnl ⋅exp γ~ nl k0 ≠ k wnkk0 ðvnk − vnk0 Þ

ð3:13Þ

If Ψni = 0, alternative i is excluded from the choice set. Because 0 ≤ Ψni ≤ 1, theoretically, one can use Ψni to represent the probability of an alternative being included in a choice set. Completely different from the Manski’s two-stage procedure, Eq. (3.13) (called r_GenMNL model) does not need any combination of alternatives. The utility of an alternative includes the information about not only the choice of the alternative but also whether the alternative is included in the choice set or not. Thus, choice set generation and choices of alternatives are represented simultaneously in a very flexible way. Especially, the new parameter γ~ ni still meets the condition γ~ ni ∈ð − ∞; ∞Þ, suggesting that it can still play the same role as the original interest parameter γ ni . In other words, one can apply Eq. (3.13) to represent choice set generation together with context dependencies. Further details are discussed in Section 3.5.

3.2.5.

Reflecting the Non-linearity of Context Dependency

All above model specifications assume a linear form to define relative utility. However, this does not mean that non-linear forms cannot be used. Redefine vni in Eq. (3.7) as, X X vni = δi þ π k xnik þ ηim gm ð3:14Þ m

k

where xnik is an alternative-specific attribute, gm is an alternative-generic attribute, δi is a constant term, and π k and ηim are unknown parameters of xnik and gnm , respectively. Substituting Eq. (3.14) into Eq. (3.7) results in the following Vni :

X Vni = γ ni ϕi þ Ψni þ μim gm m

ð3:15Þ

56

Junyi Zhang Ψni =

X

wnij

X

j≠i

π ðx − xnjk Þ k k nik



ð3:16Þ

where ϕi =

X j≠i

wnij ðδi − δj Þ = Δδi

X

wnij = Δδi ; Δδi = δi − δj

j≠i

Δθis = θis − θjs

μim =

X j≠i

wnij ðηim − ηjm Þ = Δηim

X

wnij = Δηim Δηim = ηim − ηjm

j≠i

It is obvious that Eq. (3.16) allows for attribute-based comparisons, as shown by xnik − xnjk . Distinguishing the sign of xnik − xnjk leads to three types of outcomes: positive outcome (also called gain or advantageous outcome), negative outcome (loss, regret or disadvantageous outcome) and indifferent outcomes. In all previous models, these outcomes are not distinguished. In other words, it is implicitly assumed that people show symmetric responses to advantageous and disadvantageous outcomes. However, behavioural economics suggests that this assumption may not always be true. To overcome the shortcoming of relative utility models described previously, one can integrate relative utility with the concept of prospect theory (see Zhang, 2013; Zhang, Wang, Timmermans, & Fujiwara, 2010) by re-specifying xnik − xnjk in the above Ψni as, þ − ðxnik − xnjk Þ ⇒ ðdnij;k Δ xnij;k Þ α − λð − dnij;k Δ xnij;k Þβ

ð3:17Þ

þ is equal to 1 if Here, two dummy variables are newly introduced: dnij;k − Δ xnij;k ð = xnik − xnjk Þ is non-negative and 0 otherwise; and dnij;k is equal to 1 if Δ xnij;k þ is negative and 0 otherwise. Thus, dnij;k Δ xnij;k represents the gain from a comparison − and − dnij;k Δ xnij;k indicates the loss. Parameters α and β (equal to or smaller than 1) determine the convexity/concavity of the utility function, and λ (equal to or larger than 1) describes the degree of loss aversion.

3.3.

Major Findings of Existing Studies

As seen above, the advantage/disadvantage of uni relative to unj , the relative importance parameter wnij in deriving the advantage/disadvantage of uni , and the relative importance parameter γ ni of each alternative are three key constructs to represent various context dependences. Especially, if the preference of alternative i

Relative Utility Modelling

57

is independent from some alternatives (i.e. weight wnij is zero), then the relative utility can be specified into two parts, namely the context-independent and context-dependent preferences. Thus, decomposing relative utility in different ways can derive choice models with more general context dependence structures. In this section, several empirical studies about relative utility models will be discussed and major findings for clarifying the models’ strong points and limitations will be highlighted.

3.3.1.

Choices of Destinations and Stop Patterns

The concept of relative utility was first applied to represent choices of destinations and stop patterns using data from a stated preference (SP) survey conducted in the Netherlands in 1997 (Zhang et al., 2004). The choice set in the SP survey contains five combinations of destinations and stop patterns, including a base alternative (i.e. none of combinations is preferred). For this analysis, 3013 valid SP responses from 335 respondents were used. Two types of relative utility models were built: an r_MNL model (Eq. (3.5)) and a nested logit model with relative utility (called r_NL model: Eqs. (3.18a)(3.18c)).

P exp ðγ nd d0 ≠ d ððvnd þ vndm Þ − ðvnd0 þ vnd0 m ÞÞÞ o PnðdjmÞ = P n P d0 exp ðγ nd 0 d″ ≠ d0 ððvnd 0 þ vnd 0 m Þ − ðvnd″ þ vnd″m ÞÞÞ

ð3:18aÞ

P exp ðμ⋅γ nm m0 ≠ m ððvnm þ v0 nm Þ − ðvnm0 þ v0 nm0 ÞÞÞ o Pnm = P n P 0 þ v0 nm0 Þ − ðvnm″ þ v0 nm″ ÞÞÞ exp ðμ⋅γ ððv 0 0 0 nm nm m m″ ≠ m

ð3:18bÞ

X X 0 þ vnd 0 m ÞÞÞ v0 nm = ln exp ðγ ððv þ v Þ − ðv and 0 < μ ≤ 1 nd ndm nd nd d d0 ≠ d

ð3:18cÞ

Here, d and m refer to either activity destination or stop, respectively, vnd , vnm and vndm are non-stochastic terms of utility, and μ is the parameter of logsum variable v0 nm . In this case study, relative utility parameters were directly estimated, that is Eq. (3.6) was not assumed. As a comparison, an MNL model and an NL model were also estimated. Tests of χ 2 statistics suggest that both the r_MNL model (adjusted McFadden’s rho-squared: 0.1150) and r_NL model (adjusted McFadden’s rho-squared: 0.1231) are superior to their competitors. The degree of model

58

Junyi Zhang

accuracy improvement by the r-NL model is lower than that of the r-MNL. This may be because the NL structure itself has partly incorporated alternative similarity, which is one type of context dependency. Furthermore, both models estimated that trip makers do not deal with different alternatives in the choice set equally (three are larger than that of the base alternative in the r_MNL model and only one is smaller than that of the base alternative in the r_NL model). Familiarity with alternatives and/or expectations about alternatives were mentioned as potential reasons causing the unequal interests on one hand and the influence of both observed and unobserved heterogeneity should be clarified in future on the other. Furthermore, 22 out of total 29 parameters in the r-MNL model are smaller than the ones in the MNL model and 27 out of total 29 parameters in the r-NL model are larger than the ones in the NL model (note that relative parameter values were compared, rather than absolute values).

3.3.2.

Dynamic Travel Mode Choice

Zhang, Sugie, Fujiwara, and Tamaki (2002)1 developed a dynamic combined SP/ RP model with relative utility and heterogeneous relative interest (dynamic r_SP/ RP model) using data on travel mode choice. Data were extracted from a fourwave SP panel survey implemented in Hiroshima, Japan, in 1987, 1990, 1993 and 1994. For this analysis, 904 valid SP responses were obtained with respect to choices of private car, bus, and a new transit system (NTS). In this case study, it is the first time to examine the influence of heterogeneity on relative interests (i.e. Eqs. (3.6) and (3.9a) were adopted). The dynamic r_SP/RP model can be expressed as follows: 8
< pil = 0 if pRli = 1 > : 0:5 if pRil = 0 ∧ pRli = 0

ð5:25Þ

That means that alternative i is chosen when it has enough rank advantage over l. In addition, it is assumed that a uniform random choice is applied when neither

The Heterogeneous Heuristic Modeling Framework

105

alternative is significantly better than the counterpart. Similarly, the expected choice probability considering latent preference structures turns out to be: pil =

K X

ð5:26Þ

pk piljk

k=1

where piljk is the probability of choosing alternative i over l based on discriminant threshold λRk . The same specification as in Eq. (5.16) can be used to model the selection of comparison heuristics, with extra modifications on the mental effort, risk perception and expected outcome. The stopping rule for attribute search under a heuristics depends on whether the two alternatives can or cannot be discriminated when subsequent possible value ranks are considered. Based on Eqs. (5.17)(5.21), expected effort is defined as: ! XX XX R R eh = e1 þ pia pla e2 Ya þ pia pla pib plb e3 Yab ð5:27Þ ia

( YaR =

R Yab =

la

0

ib

lb

if vRjiljabc < λR ∨vRjiljabc ≥ λR

∀b; ∀c

1 otherwise ( 0 if vRjiljabc < λR ∨vRjiljabc ≥ λR 1

∀c

ð5:28Þ

ð5:29Þ

otherwise

In Eq. (5.27), because by definition processing one attribute implies attributes of both alternatives are processed in this context, the major difference is that the probability beliefs of attribute states of both alternatives have to be included in order to form a joint probability. Identity function YaR is 0 when after x1 is processed, all subsequent possible value rank differences, vRjiljabc (|il| means regardless of compare sequences between alternatives), are all smaller than the discriminant threshold, implying that alternatives cannot be discriminated or are all no smaller than the discriminant threshold, meaning alternatives can be discriminated. The process can stop here. Otherwise, attribute search will continue and the effort for processing the R next attribute must be invested. The same definition logic applies to Yab . þ For the specification of risk perception, only rkh needs to be modified as: þ = rkh

K X K X s=1 t=1

pis plt YðjvRst j ≥ λR Þ

ð5:30Þ

where pis is the probability of alternative i having the overall value vs , and vRst is the rank difference between the overall value ranks of the two alternatives. The specification of the expected outcome changes accordingly, with o þ representing the situation that the two alternatives can be discriminated under λR , and o − representing the situation that they cannot be discriminated.

106

5.3.

Wei Zhu and Harry Timmermans

Applications

In this section, the proposed models are applied to modeling the decision-making of pedestrians in shopping streets. The satisficing model is applied to the decision of terminating the shopping trip, or the Go-Home decision; the comparative model is applied to the decision of walking direction choice. Both models are compared with the conventional MNL models. The data were collected in East Nanjing Road (ENR), a popular shopping street in Shanghai, China.

5.3.1.

The Go-Home Decision

5.3.1.1. Model specification Two kinds of real time are used in the models as explanatory factors: relative time (tR ) and absolute time (tA ), both in minutes. Relative time refers to the time elapsed since a pedestrian starts the shopping trip. It correlates with the progress of purchasing the planned items during the shopping trip, visiting schedules and how tired the pedestrian has become. Absolute time refers to the time difference between the current clock time and the 0:00 base. It correlates with available time budgets reflecting when the pedestrian must turn to other business. Under the framework of HHM, a pedestrian decides to go home, if X WX ΨðtX ≥ ΔX Þ ≥ λ X = R; A ð5:31Þ X

Here W is an N-element (more accurately, N is specific to each X) row vector of factor state values, ΔX = ½δX1 ; …; δXn ; …; δXN T is a column vector of factor threshold values and ΨðψÞ is an element-wise identity function being 1 for the true relationships ψ, being 0 for the false relationships. That means if the overall value of going-home is larger than the overall threshold λ, then the pedestrian will go home. Otherwise, he/she will keep shopping. To estimate the distribution of λ as depicted in Eq. (5.16), the value of each decision heuristic is calculated. The estimations of WX and ΔX provide the preference structure, from which the stopping conditions for each heuristic can be inferred. To complete the calculation of mental effort, the effort for processing each factor can be estimated as separate effort parameters. However, the results showed that estimating factor-specific effort parameters does not bring significant improvement to the model compared to only one effort parameter for all factors. Furthermore, this effort parameter cannot be separated from the weight parameter βe in Eq. (5.23). Thus, βe is assumed to be negative to represent some kind of cost. The last element required for calculating mental effort is the probability beliefs of factors being in certain states. Although people may have different belief distributions, which can be estimated for each factor state, the results showed that they add more complexity than goodness-of-fit to the model compared with uniform probability beliefs. The uniform probability means that the belief that a factor being in a X

= ½wX1 ; …; wXn ; …; wXN 

The Heterogeneous Heuristic Modeling Framework

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particular state is equally probable. With the probability beliefs, the risk perception of each heuristic can be calculated. Given the estimated distribution of preference structures, the expected probability of a pedestrian deciding to go home is estimated using the latent-class structure described in Eq. (5.14). In total, the parameters that are simultaneously estimated include, factor state weights, WX , and factor thresholds, ΔX . The number of their elements are not set a priori, but estimated through model selection. The parameter for mental effort, βe , as discussed before, is assumed to be negative; the parameter for risk perception, βr , is assumed to be positive because pedestrians, ceteris paribus, are assumed to prefer diverse decision outcomes to betting on very few highly probable outcomes; the sign of the parameter for expected outcome, βo , is not assumed, but determined empirically.

5.3.1.2. Results Table 5.1 shows the results of the two MNL models and the heterogeneous heuristic model (HHM). Comparing the two types of models, the best model in terms of CAIC is the MNL model with logged variables. The log-likelihood (LL) of HHM is the highest, but the complexity of the model is much higher. The optimal HHM turns out to have two thresholds for tR and three thresholds for tA . The pedestrians seem to have mentally represented tR into three states [ < 70 min, 70240 min, ≥240 min) and represented tA into four states [ < 14:30, 14:3016:00, 16:0020:00, ≥20:00). These segments are quite reasonable and conform with people’s habit of using typical clock hours as decision references. The positive weights mean that as time goes by, the value of going home increases, but not in a linear fashion. The negative βo suggests that decision strategies with strict judgment standards are preferred by the pedestrians in general. As a result, it is less likely that pedestrians decide to go home early during the trip so that they may have more opportunities to enjoy shopping. Three states for relative time and four states for absolute time mean that the number of preference structures is 13 = 3*4 + 1. Each preference structure implies two heuristics, one starting from processing tR and the other starting from processing tA . The probabilities of these 26 heuristics are estimated; the results are shown in Figure 5.1. In the figure, the larger the index for a preference structure, the higher the overall threshold. The general trend is that the probability increases as the standard becomes stricter, due to the negative βo , implying that simpler rules are preferred. The probabilities drop at Φ11 and Φ12 because they imply risky heuristics with high probabilities of rejection. However, although Φ13 implies one of the most risky strategies — unconditional rejection, its probability is still high because its processing effort is 0. The distributions of the two factor processing sequences differ very little before Φ6 , which means when the judgment standard is low (i.e. under which the pedestrians are more prone to go home), the factor processing sequence does not matter too much. While after this point, the importance of search sequence increases and tA becomes the first factor to process most of the time. Excluding the ‘no action’

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Table 5.1: Estimation results of the go-home models (ENR). MNL normal variables

MNL logged variables

Parameter

Estimate

Parameter

Estimate

βR βA βH NC NP LL CAIC

−0.005c −0.004c −6.006c 808 3 −415 854

βR βA βH NC NP LL CAIC

−0.869c −4.402c −35.424c 808 3 −410 843

HHM δR1 a

70 min

δA1

14:30

240 min 1.000c

δA2 δA3 wA1 wA2 wA3

16:00 20:00

δR2 (wR1 )b

0.766c

wR2 βe βr βo NC NP LL CAIC

−2.690c 4.526c −1.026c 808 8 −396 853

0.822c 0.710c 2.566c

a

Thresholds are not counted as free parameters as only their corresponding weights potentially have an effect. b Parameters in ( ) were set for the estimation. One value parameter is set to 1 because only the relative relationships between the values matter. c Parameters that are significant/effective. Only these parameters are counted for calculating CAIC.

All

AT→RT

RT→AT

Probability

0.2 0.15 0.1 0.05 0 1

2

3

4

5

6

7

8

9

10

11

12

Preference structure

Figure 5.1: Distribution of preference structures (go-home).

13

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heuristics implied by Φ1 and Φ13 , the probability of tA to be searched first is 62% in total, while tR has a probability of 18%. 5.3.2.

The Direction Choice Decision

5.3.2.1. Model specification If the pedestrian decides to continue shopping, the next decision assumed is to select a walking direction. For each direction, three factors are considered relevant for a pedestrian’s decision. The first factor is whether the direction is the same as the one that the pedestrian just came from, represented by a dummy variable dY (Y = E (East), S (South), W (West), N (North)). Because there is a natural tendency that pedestrians follow previous directions and try to minimize the number of back-turns, a positive influence is expected from this factor. The second factor is the total retail floorspace in the direction, qY . The variable represents the pedestrians’ estimate about the attractiveness of retail activities based on his/her perception of the environment. The third factor, lY , is the length of the part of the street that is pedestrianized in the direction, representing the amenity of walking. Tailored to the situation of ENR, the last factor is the location of The Bund. Because The Bund is a special landmark in this area and just located at the eastern end of ENR, it is common among pedestrians to use it for orientation, especially for tourists. It is represented by a dummy variable, bY . According to the HHM framework, the value function of each direction is defined as: vY = Wx ΨðxY ≥ Δx Þ þ wd d Y þ wb bY

x = q; l

ð5:32Þ

where Wx and Δx are vectors of state values and factor thresholds, similarly defined as in the go-home model; wd and wb are scalar state values for the two dummy variables. It is assumed that the pedestrian pairwise compares the values of alternative directions under a certain discriminant threshold, λR , and the direction with the highest value rank is selected. If no choice can be made on this basis, random choice is assumed. 5.3.2.2. Results Table 5.2 shows the estimation results. HHM has a significant improvement over the MNL models both in terms of LL and CAIC. It shows that only one threshold, 2005 m2, is used for representing retail floorspace. Directions with the more retail floorspace than this number are satisfactory on this factor. The length of the pedestrianized section is represented into three states [ < 110 m, 110341 m, ≥341 m). Maybe 100 m should be considered as the least length for a pedestrianized street to be constructed. The signs of wb and wd are both consistent with those of the MNL models. The positive βo suggests that smaller overall thresholds were preferred so that alternatives can be differentiated more easily.

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Table 5.2: Estimation results of the direction choice models (ENR). MNL normal variables

MNL logged variables

Parameter

Estimate

βq βl βb βd NC NP LL CAIC

3.466e − 6* 1.287e − 3* 0.638* −0.983* 2268 4 −1048 2131

HHM δq

2005 m2

wb

0.787*

(wq ) δl1

1.000* 110 m

wd βe

−6.936* −3.437*

δl2

341 m

βr

7.652*

wl1 wl2

7.452* 6.111*

βo

4.116*

NC NP LL CAIC

2268 8 −1002 2074

*

Parameter

Estimate

βq βl βb βd NC NP LL CAIC

0.432* 0.137* 0.509* −1.020* 2268 4 −1056 2147

Parameters that are significant/effective. Only these parameters are counted for calculating CAIC.

All

l 1st

d 1st

q 1st

b 1st

Probability

0.15 0.12 0.09 0.06 0.03 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Preference structure

Figure 5.2: Distribution of preference structures (direction choice). Nevertheless, Figure 5.2 shows that risk perception is the dominant force controlling the choice of heuristics. It indicates that the distribution of preference structures concentrates around Φ7 . From this point, when preference structure becomes smaller, the probability drops, suggesting that pedestrians tend to avoid using

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extremely low discriminant thresholds, which would make an alternative preferable to another even with trivial factor advantages, although the decision process tends to be quick. When the preference structure becomes larger, the probability also drops due to the fact that fewer alternatives can be differentiated under such high standards and random choices has to be made, which may give the pedestrians the feeling of losing control. The exception is that the probability of Φ24 is high, even though pedestrians make random choices all the time without considering any information, because this strategy is effortless. In general, pedestrians are risk averse and prefer information search in this particular decision problem. Their way of decision-making approximates rational mechanisms. The full factorial combination implies 24 factor search sequences, 6 for each factor. The figure also shows the probabilities of factors being searched first aggregated from all the 6 sequences starting from each factor. Under all preference structures, the length of pedestrianized section is always the most probable factor to be searched first. The second most probable first-to-search factor is previous direction. This relationship explains the fact that there are many pedestrians who stopped following their current walking direction into the non-pedestrianized section and turned back at the end of the pedestrianized section. The probabilities of searching floorspace and The Bund first are relatively low. Excluding Φ24 , the aggregate probabilities of factors being searched first are, l — 41%, d — 26%, q — 12% and b — 9%.

5.4.

Conclusion and Future Work

Increasing evidence suggests that choice behaviour in real-world settings may be more guided by principles of bounded rationality as opposed to typically assumed fully rational, utility-maximizing principle. Moreover, many choices and decisions in real-world settings are context-dependent. Thus, people may simplify the decision problem by considering information selectively, limiting choice sets, using simplifying decision heuristics, etc. Under such circumstances, conventional rational choice models do not validly mimic the decision processes. To model both outcomes and processes of decision-making, this chapter proposed a modeling approach which incorporates attribute thresholds as the basic mechanism to model attribute representation. It has been shown that such a discrete cognitive representation has the generality to be the source of several decision heuristics, including the typical conjunctive, disjunctive and lexicographic rules, assuming that the overall judgment threshold varies with individual and contextual differences. This approach allows modeling unobservable decision heterogeneity involved in a single decision, for example, in the form of a latent-class specification. Furthermore, the choice of decision strategies is modeled, taking into account mental effort, risk perception and expected outcome as explanatory factors. It should be emphasized that in this case the logit form represents the functional relationship between probabilistic choices and input variables and is not derived from assumptions of utility-maximizing behaviour.

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Two types of decision models are derived under the heterogeneous heuristic modeling framework: one models the satisficing decision; the other models the comparative decision. Both models are applied to a decision problem pertaining to pedestrian shopping behaviour and compared with conventional MNL. The results show that the proposed models may not be necessarily superior to conventional logit models in terms of model selection criteria due to the extra complexity related to modeling preference structure and heuristic selection. However, the new models suggest more interesting insights in the underlying decision processes. Inference can be made probabilistically whether a factor is used for the decision, the sequence of factor processing, the effort involved, the risk propensity and the outcome preference of the decision-maker, which together provide much richer information than what conventional utility-maximizing models can offer. Understanding decision processes additional to outcomes is a promising research direction. A more developed model should take into account more contextual and socio-demographic factors in the heuristic selection part, which then can provide possibilities to answer the question of who making decisions in what context and how. Of course, these assumptions of information processing must be subject to empirical tests to validate the model. Some tests need carefully designed experiments to test the ability of the model to reveal the specific decision style under control. Objective measures such as mental effort and risk propensity should be devised as criteria for validating the proposed elements in the heuristic choice part.

References Aran˜a, J. E., Leo´n, C. J., & Hanemann, M. W. (2008). Emotions and decision rules in discrete choice experiments for valuing health care programmes for the elderly. Journal of Health Economics, 27(3), 753769. Cantillo, V., & de Dios Ortu´zar, J. (2005). A semi-compensatory discrete choice model with explicit attribute thresholds of perception. Transportation Research Part B, 39(7), 641657. Cantillo, V., & de Dios Ortu´zar, J. (2006). Implications of thresholds in discrete choice modelling. Transport Reviews, 26(6), 667691. Dawes, R. M. (1964). Social selection based multidimensional criteria. Journal of Abnormal and Social Psychology, 68(1), 104109. Fishburn, P. C. (1974). Lexicographic orders, utilities and decision rules: A survey. Management Science, 20(11), 14421471. Gillbride, T., & Allenby, G. (2004). A choice model with conjunctive, disjunctive and compensatory screening rules. Marketing Science, 23(2), 391406. Hess, S., J. M. Rose, & Polak, J. (2007). Non-trading, lexicographic and inconsistent behaviour in sp choice data. In Proceedings of the 87th annual meeting of transportation research board, Washington, DC. Jedidi, K., & Kohli, R. (2008). Inferring latent class lexicographic rules from choice data. Journal of Mathematical Psychology, 52(4), 241249. Manski, C. F. (1977). The structure of random utility models. Theory and Decision, 8(3), 229254.

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McFadden, D. (1974). Conditional logit analysis of qualitative choice behavior. In P. Zarembka (Ed.), Frontiers in econometrics (pp. 105142). New York, NY: Academic Press. Parker, B. R., & Srinivasan, V. (1976). A consumer preference approach to the planning of rural primary health-care facilities. Operations Research, 24(5), 9911025. Payne, J. W., Bettman, J. R., & Johnson, E. J. (1988). Adaptive strategy selection in decision making. Journal of Experimental Psychology: Learning, Memory and Cognition, 14, 534552. Scott, A. (2002). Identifying and analysing dominant preferences in discrete choice experiments: An application in health care. Journal of Economic Psychology, 23(3), 383398. Simon, H. A. (1987). Bounded rationality. In J. Eatwell, M. Milgate, & P. Newman (Eds.), Bounded rationality (pp. 266268). London: Macmillan. Swait, J. A. (2001). Non-Compensatory choice model incorporating attribute cutoffs. Transportation Research Part B, 35(10), 903928. Swait, J., & Adamowicz, W. (2001). The influence of task complexity on consumer choice: A latent class model of decision strategy switching. Journal of Consumer Research, 28(4), 135148. Timmermans, H. J. P., Borgers, A. W. J., & Veldhuisen., K. J. (1986). A hybrid compensatory noncompensatory model of residential preference structures. Netherland Journal of Housing and Environmental Research, 1(3), 227233. Tversky, A. (1972). Elimination by aspects: A theory of choice. Psychological Review, 79(4), 281299.

Chapter 6

Investigating Situational Differences in Individuals’ Mental Representations of Activity-Travel Decisions: Progress and Empirical Illustration for the Impact of Online Alternatives Oliver Horeni, Theo Arentze, Benedict G. C. Dellaert and Harry Timmermans

Abstract Purpose  This chapter focuses on individuals’ mental representations of complex decision problems in transportation. An overview of approaches and techniques in this recent area of research is given as well as an illustration. The illustration concerns an application of CNET (causal network elicitation technique) to measure mental representations in a shopping activity scheduling task. The presence of an online shopping alternative is varied to investigate the influence of an online alternative on how individuals represent the choice problem. Theory  Mental-model and means-ends-chain theories are discussed. These theories state that individuals when faced with a decision problem construct a mental representation of the choice alternatives by activating relevant parts of their broader causal knowledge that allow them to evaluate consequences regarding their existing needs. Furthermore, these theories emphasise that situational and person dependence of this process can explain observed variability in preferences of travellers. Findings  The results indicate that considerable variation exists between individuals in terms of both the complexity, and the attributes and benefits that are activated in the mental representation of the choice problem. Presence of an online alternative has an influence on the benefits that individuals consider important. The impact is however small.

Bounded Rational Choice Behaviour: Applications in Transport Copyright r 2015 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISBN: 978-1-78441-072-8

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Originality and value  The chapter provides an overview of recent developments in the study of mental representations underlying choice behaviour. Traditionally, this has been the exclusive domain of qualitative research methods. The techniques reviewed enable larger samples and a formal representation of mental representations. Thus, the approach can help to better understand preference heterogeneity and incorporate this in (transport) choice models. Keywords: Mental representation; laddering; Bayesian belief network; benefits; heterogeneity

6.1.

Introduction

The issue of heterogeneity in urban transport demand analysis is of considerable importance. Conventional transport models assume all travellers of the same socioeconomic characteristics demonstrate the same type of behaviour. Models are based on averages and thus do not capture any behavioural differences in the structure of the model. In discrete choice modelling, heterogeneity is generally conceptualised as unobserved taste heterogeneity. Scholars are dealing with this issue in different ways (for a review, see Greene, Hensher, & Rose, 2006). One stream of research, focusing on mixed logit models, is estimating the parameters with some distribution for each parameter in order to reflect heterogeneity with regard to the effect of explanatory variables on the dependent variable of interest. Hence, such models still assume that the nature of the relationships between the explanatory variables and the dependent variable is the same for all respondents. A second approach identifies latent classes, each class having a different utility function; for example, depending on socio-demographic or context variables. Although both these approaches break down the choice problem into subproblems by segment respectively content, within each breakdown the assumption of homogeneous responses/behaviour is still needed. Psychological factors to explain taste heterogeneity have received attention in recent extensions of the standard discrete choice model (Ben-Akiva et al., 2002). This has led to so-called hybrid choice models which include attitudes and perceptions of individuals as latent factors causally related to preferences. Also in these hybrid frameworks, however, the assumption of stable, time-invariant preferences needs to be made. Thus, regardless of their sophistication and relative success, all these approaches are fundamentally limited in the sense that some degree of aggregation is still used. From a truly behavioural perspective, however, individuals and households face a different spacetime environment in which they need to cope with a different set of constraints in satisfying their needs and organising their activities. They have different experiences and hence will have different mental or cognitive maps of their built environment, the transportation system and the institutional context. They will vary in terms of their perception of the environment, which will be incomplete and partially incorrect. They hold different beliefs with regard to the most effective

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strategy of coping with constraints. Moreover, individual differences are not the only source of heterogeneity. Mental representations may also differ for one and the same individual depending on the specific situation that he or she faces. Different situational settings may activate different needs and hence different benefits and attributes in mental representations of choice alternatives. Arguably, modelling such individual variability and situational dependence could provide further insight in individuals’ decision-making and, therefore, is an important research goal to pursue. Especially in the domain of daily activity-travel choice behaviour the role of mental representations may be important given the complexity of such decisions. Travel is a derived demand from the activities individuals need or wish to perform in time and space and, consequently, travel decisions are an integral part of a broader activity scheduling problem. The decision process inherent to the realized activity-travel behaviour is thus a complex choice process with several interdependent decisions. Due to the complexity of such scheduling decisions, decision-makers’ mental representations of the decision are necessarily a simplification of reality (Beach & Mitchell, 1987; Johnson-Laird, 2001; Weber & Johnson, 2006). This allows them to evaluate their possible courses of action and to oversee the choice consequences. These mental representations are hence increasingly in the focus of transport research and policymaking as they are the key to understand and predict people’s travel and activity behaviour (Dellaert, Arentze, Chorus, Oppewal, & Wets, 2013; Dellaert, Arentze, & Horeni, 2014; Hannes et al., 2012). By knowing which attributes of choice alternatives people consider and which benefits they try to gain and how situational circumstances impact them in their decisions, better predictions about the effectiveness and consequences of policies on choice behaviour might be achieved. In this chapter, we first review recent progress in the modelling and measurement of mental representations involved in activity-travel choice of people (Section 6.2). Then we describe an application to illustrate the new approach (Section 6.3). In this application, the Causal Network Elicitation Technique (CNET) is used to collect data about mental representations in the context of an activity-travel scheduling task. To investigate how new alternatives may affect the activation of new benefits and thus lead to a shift in the consideration of spatial choice alternatives, the availability of online alternatives is varied in the experiment. The research method, sample and analytical results are presented next. The chapter closes with a section on conclusions and discussion (Section 6.4).

6.2.

Mental Representations of Complex Decision Problems

6.2.1.

Theory and Concepts

According to Johnson-Laird (1983) a mental representation (MR) is a temporary result of individual perception being stored in working memory for the moment of

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consideration. Building MRs depends on the individual’s experiences and long-term knowledge from which relevant information is retrieved, reordered or translated into other forms (Cox, 1999; Kearney & Kaplan, 1997). MRs consist of different components such as attributes and benefits (Myers, 1976) and also situational variables and causal relations between them. Depending on the point of view the decision itself can also be considered as a variable of the MR. While attributes relate to physically observable states of the considered choice options, benefits describe outcomes in terms of dimensions of more fundamental needs. Situational variables describe states of the system which cannot be influenced by the decision-maker or they result from a far-reaching decision in the past. Kusumastuti (2011) refers to attributes as instrumental and to situational variables as contextual aspects. Anyhow, as MRs represent causal knowledge of the environment, that is complex IF-THEN relations under different circumstances, they can be mapped as causal networks with nodes as variables and unidirectional arrows as causal links (Arentze, Dellaert, & Timmermans, 2008). This structure allows for an application as Bayesian Decision Network with additional parameters for conditional probabilities and utilities and facilitates the simulation of the decision process. Figure 6.1 shows an MR for an exemplary activity-travel task depicted as causal network. Introducing MRs for rational decision-making does not necessarily mean that individuals represent the real world genuinely. Due to the limited capacity of the working memory individuals will experience limitations on the amount of information that can be represented (Anderson, 1983). Consequently, MRs will generally involve a significant simplification of reality and are, thus, tailored to the specific task and contextual setting under concern (Johnson-Laird & Byrne, 1991). Limited cognitive capacity is not the only determinant for the construction of MRs. It was already stated above that the considered attributes and benefits depend

Figure 6.1: Arbitrary example of a mental representation for an activity-travel task.

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largely on respondents’ experience and information stored in long-term memory (Kearney & Kaplan, 1997). Taking just these two determinants into consideration MRs would barely differ between different contexts. Recently however, research has emphasised that depending on the context with which decision-makers are faced different aspects in the attribute-benefit chains may be more or less prominently activated (Dellaert, Arentze, & Timmermans, 2008; Ratneshwar, Warlop, Mick, & Seeger, 1997; Srivastava, Leone, & Shocker, 1981). In other words, contextual circumstances and the state of the person activate different needs which in turn cause the individual to define targets the choice alternatives should meet (Arentze, Dellaert, & Chorus, 2014). Nijland, Arentze, and Timmermans (2010) stated, for instance, that for daily recurring location and travel choices an individual’s needs for time saving, entertainment and convenience, etc. vary across situations depending on the individual’s state and contextual settings. External influences like advertisement and fashion on the one hand and internal psychological processes on the other hand are also considered as source of need activation (Ratneshwar et al., 1997). In terms of contextual settings of a decision task, different influences on the activation of needs and eventually on the construction of MRs have been discovered. The Construal Level Theory (Trope & Liberman, 2003) proposes that people’s mental representation of future events changes with temporal distance. The relative emphasis on benefits versus attributes would increase with the temporal distance of the events. This effect of temporal distance between the decision-maker and the considered task would also hold for other sorts of psychological distances like spatial or inter-personal distance (Liberman & Trope, 2008). Next to the time horizon for which the decision is considered also the importance of the (anticipated) consequences of the activity task affects the variation of the number of represented attributes and benefits and their causal interlinking (Dellaert et al., 2008). Trivial decisions are considered as less complex than decisions whose consequences have implications for a longer period of time or are perceived as severe or uncertain (Payne, Bettman, & Johnson, 1993). The latter type of decisions is hence likely to result in higher mental effort. Payne argues further that individuals are able to adjust the required mental effort according to the desired accuracy of MRs. Moreover, the choice set per se is likely to influence the size of MRs. Not only the number of choice alternatives might influence the number of considered attributes (Tversky, 1972), but also the (dis)similarity between the possible courses of action determines the mental effort which the decision-maker needs in order to find the optimal solution (Shocker, Ben-Akiva, Boccara, & Nedungadi, 1991). Thus, the quality and quantity of choice alternatives are assumed to influence the construction of MRs.

6.2.2.

Measuring Mental Representations

Many cognitive mapping and mind reading approaches exist ranging from verbal to non-verbal techniques and from recall to recognition techniques. In the context

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of the present chapter, only verbal techniques are relevant and the most prominent ones of this category will be presented here. Mental representations and hierarchical value maps from means-end-chain theory (Gutman, 1982) show structural similarities as both can be mapped as causal networks. In order to measure means-end-chains, several techniques proved to work and could hence also be adapted for measuring MRs. A widely used qualitative technique in this regard is laddering (Reynolds & Gutman, 1988). The original laddering technique is a structured face-to-face interview for understanding consumer’s values (ends) and how they are trying to attain them (means). The interviewer starts by asking respondents to name the most important attributes of some choice products which are subject to investigation. For each mentioned attribute they then are asked why they consider it. Ideally, the responses can be classified as consequences. Accordingly, the interviewer continues asking why these consequences are important for the respondent until a satisfying level of ends has been attained. The resulting ladder or means-end-chain does hence consist of more than two levels of abstractness. The exact number of levels depends on the interview depth and the desired precision determined by the interviewer. So, consequences can for instance also be grouped into the more concrete physical consequences and the next highest level of psychosocial consequences. A laddering interview performed in the described manner does not provide support in the memory retrieval process in terms of revealed attributes, consequences and values. Thus, laddering can be classified as recall-based technique. Since the emergence of a recognition-based variant of laddering (Botschen & Thelen, 1998) both versions are distinguished as soft (the recall-based version) and hard (the new version) laddering. Hard laddering presents predefined attributes, consequences and values from which respondents are asked to select the relevant ones. The sequence of the interview in the laddering format remains however the same. Next to soft laddering Russell, Busson, et al. (2004) and Russell, Flight, et al. (2004) applied a paper-and-pencil and a computerised version of hard laddering in an experiment on mothers’ opinions of the role of breakfast on their children’s physical and psychological well-being. In contrast to what was stated above about soft laddering Russell asked his respondents to select one to three important attributes from a list. Consequences and values were however elicited without auxiliaries by recall. The results showed that the hard laddering techniques yielded more ladders than soft laddering; a fact which is attributed to differences in participants’ cognitive processing (recall vs. recognition). While Russell, Busson, et al. (2004) recommend hard laddering if the focus of the research is on investigating strong links between certain predetermined elements, soft laddering would be more appropriate for gaining a fuller picture of participants’ cognitive structure. Yet, the drawbacks of a face-toface interview remain which make soft laddering not suitable for large-scale surveys. Ter Hofstede, Audenaert, Steenkamp, and Wedel (1998) suggested another measurement technique, called the association pattern technique (APT). Similar to the hard laddering variants respondents are faced with revealed attributes, consequences and values. The difference is only that the variables are not shown in list format and that the ladders are not elicited one-by-one. Rather, APT consists of two matrices

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(one for attributes and consequences and one for consequences and values) where respondents can indicate causal links by ticking off the corresponding cells. Hence, all ladders are elicited simultaneously which makes this technique quite difficult. The high complexity of the matrix format with which respondents might struggle can hardly be outweighed by the short interview duration. The advantage of APT is due to its simple analysis and the convenience it brings for the researcher. Thanks to the predefined labelling of variables no postprocessing of the responses is necessary, thus, making responses conveniently comparable. Yet, the downside of this convenience is that respondents are limited in their response freedom and possibly influenced by the revealed presentation of attributes, consequences and values. Furthermore, APT does not allow for a variation of the level of abstractness of the means-end-chains. To collect data on MRs specifically in the context of decision tasks, a semistructured interview protocol has been developed and tested in face-to-face sessions by Arentze et al. (2008) and Dellaert et al. (2008). The method assumes a choice task that may include multiple decision variables (e.g. the transport mode, destination and departure time for a trip). The so-called CNET starts by confronting the respondents with the decision variables in a random arrangement. They are asked to select them in the sequence in which they prefer to deal with them, assuming they were to make decisions. Next, the interview proceeds through the list of decision variables in the order indicated by the respondent and, for each variable, the respondent is informed about the decision alternatives and asked ‘What are your considerations when faced with these alternatives?’ From a list of predefined attributes and benefits, that is not visible to the respondent, those variables are identified that correspond to the response. If the response variable is not on the predefined list, the new attribute or benefit will be added. In any case, it is verified whether the respondent agrees with the classification and determined whether the attribute or benefit is causally linked to the decision variable. In case of doubts, these links are checked with the respondent. Having identified the variable, the next step depends on the variable type. If the variable is an attribute, the interview proceeds with the question ‘Why is this variable important in this case?’ This ‘why’ question generally results in the identification of an underlying benefit generated by the attribute, in which case no further ‘why’ questions are needed. If another attribute is mentioned, the ‘why’ question gets repeated until an underlying benefit emerges. When the originally mentioned variable is a benefit, the interview proceeds with the question ‘How is this variable influenced?’ and this ‘how’ question leads to the identification of other situational or alternative attributes. The causal links are also established and verified if in doubt. Further considerations are prompted by repeating this procedure until the respondent has no further considerations to mention. After the first decision variable is processed, the entire procedure is repeated for the next decision variable, and so on, until all decision variables are processed. Ultimately, this procedure leads to a completed representation of the attributes and benefits involved in respondents’ MR of the decision problem, as well as the causal links among these attributes and benefits and the action variables involved in the decision. Finally, after the MR is completed, the respondent is asked to select, for each decision

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variable, the alternative that he or she would choose in the given scenario. An example of an MR elicited in this way is shown in Figure 6.1. This protocol implies already that the interview is quite intensive and timeconsuming. Each variable is processed step-by-step so that all components of the MR are captured. However, the repeated prompts for consideration might possibly evoke too much deliberation on the respondent’s side so that he or she gives answers only in order to satisfy the interviewer. A somewhat tricky property of CNET is connected to its response freedom. Because respondents are not instructed in the labelling of the predefined variables the interpretation of their responses is subject to the interviewer. Still, a common set of variable labels is necessary to enable comparing MRs between individuals. The possibility to include even not predefined variables makes the MRs however strongly individually tailored. To eliminate possible interview bias and to allow application to large samples, an automated version of CNET has been developed and tested in Horeni, Arentze, Dellaert, and Timmermans (2014). The online tool is implemented in a PHP-based algorithm where a string recognition algorithm in cooperation with a predefined MySQL database takes over the job of the human interviewer. Originating from the semi-structured CNET protocol Kusumastuti, Hannes, Janssens, and Wets (2009) developed modifications in order to measure MRs underlying leisure-shopping trip decisions. Their first modification is called CNET card game as it works with revealed variables printed on cards. Instead of eliciting the components of the MR by recall the interviewee indicates thus the relevant variables from card stacks which he goes through one-by-one with the interviewer. The second modification is a computerised version of the card game (CB-CNET).

6.3.

Case Study

To illustrate the approach, in this section we consider an application of the online CNET tool. The application concerns the measurement and analysis of MRs in the context of an activity-travel choice task where the availability of an online shopping alternative is varied to investigate the impact on the way individuals represent the choice alternatives. In this section, we first describe the choice task and experimental design used in this application and next we discuss the characteristics of the sample and analysis results. 6.3.1.

Choice Task and Experimental Design

In order to collect data on mental representations with online CNET an activitytravel task has been chosen which was believed to be a familiar decision problem for respondents. It consisted of a scheduling task for working and grocery shopping on a usual workday. A fictive environment has been chosen in order to prevent that respondents apply their routine behaviour. When starting an online CNET session general information about the activity-travel task was provided and the alternatives

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for time moment of grocery shopping (during lunch break, after work, in the evening) were introduced. A map of the fictive city with images for home and work place gave visual support. Accordingly, the available shopping locations were presented. Again, visual images of the shopping locations on the map were given for a better orientation. Finally, the available transport modes (represented by images for car, bicycle and bus) with general information on travel times were introduced. To investigate the impact of choice-set composition on MRs, two variants of the choice task (scenarios) are considered: a scenario where an online shopping alternative is available and a scenario where such an alternative is not available. Hence, the variation of the choice set for the shopping alternatives led to the set up of a basic scenario with three shopping alternatives (week market, corner store and supermarket) and an e-commerce scenario (week market, corner store, supermarket, online shopping). All other elements of the setting remained unchanged. 6.3.2.

Data Collection and Sample

The study was split up into two waves of data collection with an aim to collect 200 respondents per wave. First, data for the basic scenario were collected in May 2010. Next the interviews with the e-commerce scenario took place in October 2010. Different respondents were approached for each wave. The selection of respondents was restricted to three criteria: possession of a driver’s license, aged between 18 and 60 years and having Dutch language skills. The former two characteristics were set up to ensure a real world reference for the experimental situations. The latter condition resulted from the interview language. A random sample of an existing national panel of 5000 Dutch households representative for the Netherlands was drawn (the panel originates from CentERdata, a research institute attached to Tilburg University and supported by the Dutch Organization for Scientific Research). In Table 6.1, sample characteristics are presented from all respondents who finished the online interview successfully, that is dropouts were excluded from further analysis. Statistical tests did not show significant differences between scenarios for any of the key socio-demographic variables.

6.3.3.

Analysis

The presence of an online shopping alternative may have an impact on both the complexity of mental representations and the contents of the representations. In this section we describe the results of analyses regarding these two aspects successively. 6.3.3.1. The complexity of respondents’ mental representations In a first step, the mental representations were analysed structurally in terms of number of recalled considerations, number of attributes, number of benefits, number of benefits per attribute and number of cognitive subsets. Table 6.2 shows the results of descriptive analyses.

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Table 6.1: Socio-demographics of the sample. Variable

Basic scenario

E-commerce scenario

185 42.8/11.4

290 43.5/10.9

43.8 56.2

47.2 52.8

13.5 31.4 49.7 4.3 1.1

13.1 27.2 53.8 5.5 0.3

2.7 21.1 11.9 33.0 22.7 8.6

3.5 20.1 12.1 27.3 27.7 9.3

N Age (years) (M/SD) Gender Men (%) Women (%) Status Single (%) Childless couple (%) Couple with child (%) Single parent (%) Other (%) Education Primary school (%) Practical professional training (%) Secondary education only (%) Higher level professional training (%) Bachelor’s degree (%) Master’s degree (%)

Table 6.2: Descriptive statistics of number of recalled considerations. N Mean

Basic scenario E-commerce scenario Basic scenario E-commerce scenario Basic scenario E-commerce scenario Basic scenario E-commerce scenario Basic scenario E-commerce scenario

SD

Std. error

95% Confidence interval for mean Lower bound

Upper bound

Min Max

185 290

4.12 4.71

2.33 2.65

.17 .16

3.78 4.41

4.46 5.02

0 0

12 19

185 290

3.77 4.25

2.20 2.45

.16 .14

3.45 3.97

4.09 4.53

0 0

11 17

185 290

5.78 6.99

3.52 3.59

.26 .21

5.27 6.57

6.29 7.40

1 1

18 17

182 285

1.79 1.67

1.35 1.33

0.10 0.08

1.59 1.51

1.99 1.82

0.50 0

10 9

185 10.05 9.80 290 11.74 10.46

.72 .61

8.63 10.53

11.47 12.95

0 0

46 68

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Number of recalled considerations. The variable number of recalled considerations (Table 6.2) counts in fact respondents’ spontaneous recalls in the open question phase no matter of which category. The data were however aggregated so that variables that were recalled twice or thrice for different decision variables appear only once. It can be seen that e-commerce respondents could recall on average about half a variable more than basic-scenario respondents. The significance of the difference is confirmed by an Independent Samples t-Test (t = −2.500, df = 473, p = .013). Minimum values of 0 are due to exclusions of non-interpretable inputs. Number of attributes and number of benefits. When looking at descriptive statistics for number of attributes (Table 6.2) it can be seen that MRs for the e-commerce scenario on average consist of 0.5 attributes more than MRs for the basic scenario. An Independent Samples t-Test (t = −2.144, df = 473, p = .033) confirmed the significance of this difference. The finding that Min equals 0 is partly caused by the fact that some respondents considered benefits without linking them to attributes and partly by the fact that some considerations could not be interpreted and were thus excluded from further analysis. As for number of benefits, the figures show that on average e-commerce respondents considered more than one benefit more than basic respondents. An Independent Samples t-Test (t = −3.601, df = 473, p < .001) confirmed the significance of this difference. Benefits per attribute. As a further measure of compactness of MRs the ratio of benefits per attribute has been computed for each respondent. It describes how many needs (represented by benefits) a decision-maker wants to satisfy by the consideration of a characteristic of the choice alternatives (conceptualised as attribute). Although this ratio is slightly bigger in the basic scenario (Table 6.2) an Independent Samples t-Test indicates that there is no significant effect of scenario (t = 0.996, df = 465, p = .320). Number of cognitive subsets. The fourth and final measure to describe and compare MRs is number of cognitive subsets. The term cognitive subset refers to a unique chain of linked variables. In other studies the terms (cognitive) subsets (Kusumastuti, 2011), hiesets (Farsari, 2006) or ladders (Reynolds & Gutman, 1988) are used to describe the same or similar constructs. In this analysis cognitive subsets can have two forms, namely ‘decision variable — attribute — benefit’ or ‘decision variable — benefit’. As situational variables are treated like attributes, cognitive subsets with situational variables fall within the first case. However, the link between decision variables and situational variables is not of causal nature. Rather, it stands for a mental association respondents have between these two. That explains the choice for the name of this measure. The basic scenario (10.05) resulted in less cognitive subsets per respondent than the e-commerce scenario (11.74) (Table 6.2). An Independent Samples t-Test (t = −1.759, df = 473, p = .079) indicates however that the difference is not significant. Minimum values of 0 are again a result of excluding non-usable inputs. In fact, all respondents indicated at least three cognitive subsets. Even more surprising

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are the high maximum values which means that at least one respondent indicated (almost) all of the revealed benefits. Conclusions on the complexity of respondents’ MRs. Three findings were significant: Number of recalled considerations; number of attributes and number of benefits were significantly higher in the e-commerce than in the basic scenario. These findings confirm indeed the expectation that the introduction of additional choice alternatives which was online shopping in that scenario leads to an activation of additional needs which in turn results in an increase of attribute considerations. With a variation of the choice set as it has been tested in the experiment MRs become hence structurally more extended. Whether the considered attributes and benefits between the scenarios differ with regard to their nature will be investigated in the content analysis later.

6.3.3.2. The content of respondents’ mental representations Besides on the structural variables mental representations may differ on their content components between the scenarios. Thus, the nature of attributes, benefits and cognitive subsets is examined on possible substantial shifts between scenarios. The first subsection will however report how respondents ranked the decision variables. Ranking of decision variables. At the beginning of the interview respondents were asked to rank the in random order shown decision variables time of shopping (TS), transport mode (TM) and shopping location (SL) according to the preferred sequence of decisions. In both scenarios respondents prefer to consider time of shopping before shopping location and transport mode. The mean ranks are 2.52 (TM choice), 1.89 (SL choice) and 1.59 (TS choice) in the base scenario and 2.23 (TM choice), 2.00 (SL choice) and 1.77 (TS choice) in the e-commerce scenario. Thus, respondents prefer to consider time of shopping before shopping location and transport mode. While Kusumastuti et al. (2009) report also a first order rank for time of (leisure) shopping, their findings for location and mode choice are reversed. Yet, according to Davidson et al. (2007) many activity-based models assume that the location choice is made before the mode choice which is herewith supported. A MANOVA confirms the significance of the rank order of decision variables across scenarios (F = 6262.1, df = 2, p < .001). As a second effect, the MANOVA also reveals a significant rank order difference between scenarios (F = 8.0, df = 2, p < .001). Although the rank order of the average scores does not change, respondents of the e-commerce scenario rank TS and SL choice more often lower and TM choice more often higher compared to basic respondents. Especially the latter finding is unexpected as the introduction of online shopping would not suggest an impact on transport mode choice on first sight. Possibly, respondents perceive more freedom by the increased number of shopping alternatives along with less travelrelated restrictive consequences which in turn allows them to decide on their transport mode before considering the other choices.

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The frequency of elicited attributes. Figure 6.2 shows all attributes that were elicited from at least 5% of respondents in any scenario. There is quite strong agreement between both scenarios in the sense that the available product assortment, distance from current location and number of bags to carry are most frequently considered attributes. While spatial accessibility attributes such as accessibility of the store and parking opportunities appear next frequent in the basic scenario, e-commerce respondents considered rather product-related attributes (the price level of the assortment) or personal/situational availability attributes such as available time to shop and leisure time more often. In general there is however only little variation between both scenarios with regard to the nature of attributes. For each of the top 10 attributes of either scenario cross tables have been set up with number of respondents who considered them and number of respondents who did not as rows and scenario as columns. In total, 11 Chi-Square tests have been performed. The results (not shown) indicate that there are no significant differences for the frequency of the 11 investigated attributes. The biggest difference was observed for Available product assortment Distance from current location Number of bags to carry Accessibility of the store Parking opportunities Price level of the assortment Non-interpretable/unclear Leisure time Available time to shop Simplicity of the travel route Recreation time during work Conflict with other arrangements Weather Product quality Availability of the TM Travel time Crowdedness in the store Capacity of the TM Working hours Required time to shop Durability of bought products Habituation to the TM Time pressure Possibility to store shoppings Necessity Combination with other activities Sort of bought products Familiarity with the SL 0%

5%

E-commerce scenario

10% 15% 20% 25% 30% 35% Basic scenario

Figure 6.2: Frequency of elicited attributes.

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accessibility of the store which failed however the level of significance slightly (X2 = 3.469, df = 1, p = .063). The frequency of elicited benefits. Figure 6.3 lists the frequencies of all elicited benefits for the scenarios. Ease of travelling, Time savings and Ease of shopping are the most frequently considered benefits in both scenarios while safety aspects, social acceptance and personal care receive little attention. Furthermore, it is remarkable that except for Health, Course of Fitness/Well-being and Safety in the shopping location all benefits yield higher frequencies in the e-commerce scenario. As for attributes, cross tables with scenario as column and number of considers versus number of non-considers as rows have been performed. It turned out that Time savings (X2 = 11.259, df = 1, p = .001), Ease of shopping (X2 = 10.196, df = 1, p = .001), Diversity in product choice (X2 = 7.682, df = 1, p = .006) and Travel comfort (X2 = 4.628, df = 1, p = .031) are considered significantly more often in the e-commerce scenario than in the basic scenario. The frequency of elicited cognitive subsets. A frequent item set analysis has also been performed to investigate the cognitive subsets. None in the top 10% of most Ease of travelling Time savings Ease of shopping Relaxation/recreation Shopping success Financial savings Mental ease Shopping pleasure Diversity in product choice Travel comfort Health Course of fitness/well-being Travel pleasure Shopping comfort Attractivity of the shopping environment Environmental protection Taste experience Safety in travelling Social acceptance Safety in the shopping location Personal care 0% 10% 20% 30% 40% 50% 60% 70% 80% E-commerce scenario

Basic scenario

Figure 6.3: Frequency of elicited benefits.

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frequent cognitive subsets included the decision for Time of shopping (TS) although most respondents started with this consideration, as discussed. As for attributes and benefits also the most frequent cognitive subsets were compared separately between scenarios by means of cross tables. Of the top eight cognitive subsets only TM — ease of travelling was considered significantly more often (X2 = 4.118, df = 1, p = .042) in the basic scenario. The subset SL — distance — time savings missed the level of significance slightly (X2 = 3.415, df = 1, p = .065). Centrality of variables. The previous three sections analysed MRs in the extent to which there is agreement among respondents in the considered attributes, benefits and cognitive subsets. While these three descriptors give insight in the salience of some components of MRs and how they differ between scenarios they say little about the role of these components within the causal network. For instance, it remains yet unclear whether the most frequent considered attribute is linked to one decision variable (DV) and benefit only or interlinked to several DVs and benefits. In the latter case, the attribute would have a central role for the decision-maker. Possibly, the role of some attributes undergoes shifts for different situational decision contexts. In order to determine the centrality of variables an implication matrix has been set up for each respondent or MR, respectively, where all variables which can be a parent node (DVs and attributes) are represented as rows and all variables which can serve as a child node (attributes and benefits) are represented as columns. All indicated causal links between them were coded as 1. All other cells were filled up with a 0. Adding then the row and column sum of a variable and dividing it by the matrix sum results in the centrality value of this variable which can take on values from the range between 0 and 1 (Knoke & Burt, 1982). In other words, the centrality c of a variable V represents the sum of its incoming (X, V) and outgoing (V, Y) links over the sum of occurring (X, Y) links in the MR of respondent j. In formula, the measure is defined as: P cVj =

P þ l ðV; YÞl m ðX; YÞm

VÞk k ðX;P

ð6:1Þ

Table 6.3 lists means for the top 10 central variables per scenario. DVs are highlighted in dark grey and benefits in light grey. Attributes are shown in normal style. Despite of the fact that DVs have no incoming links they score most central in both scenarios. This circumstance is not surprising when considering the fact that each cognitive subset has a DV as origin. Due to the limited number of decision variables their centrality values are this high. In both scenarios the benefits ease of shopping, time savings and ease of travelling belong to the top 10 central variables which speaks to the stability and general validity of these variables as underlying benefits. This holds also for attributes as available assortment, distance from current location and number of bags to carry do not differ considerably in their centrality value. The fact that the accessibility of the store fell out of the top 10 central variables in

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Table 6.3: Top 10 central variables in both scenarios. Variable SL decision TM decision TS decision Ease of travelling Available assortment Time savings Distance Accessibility of store Number of bags Ease of shopping

Basic scenario

Variable

E-commerce scenario

0.084 0.080 0.070 0.049 0.048 0.042 0.035 0.031 0.030 0.026

SL decision TM decision TS decision Available assortment Time savings Ease of travelling Distance Number of bags Ease of shopping Relaxation

0.088 0.078 0.070 0.052 0.049 0.042 0.031 0.029 0.029 0.022

the e-commerce scenario confirms the shift for this attribute as shown in Figure 6.2. What however is obvious is the fact that all shopping-related variables no matter of which category increased in their centrality values in the e-commerce scenario. All transport-related variables lost centrality in comparison to the basic scenario. A multivariate analysis of variance (MANOVA) has been performed with the top 11 central variables as dependent variables and scenario as factor which did not show a significant effect (F = 0.783, df = 11, p = .658). 6.3.4.

Conclusion

The analyses have shown that there is a structural shift in mental representations for the exemplary activity-travel task when an online shopping alternative is introduced. In this regard, the number of considered attributes and benefits increased significantly, that is MRs become more comprehensive. Yet, the ratio of benefits and attributes remains rather stable. Also the relative frequencies of cognitive subsets did not differ significantly between both scenarios. The substantial analysis showed a shift in the underlying benefits between the two scenarios: significantly more MRs in the e-commerce scenario included the benefits time savings, ease of shopping, diversity in product choice and travel comfort. This finding suggests that individuals perceive benefits of an online shopping alternative primarily in terms of a trade-off between convenience aspects (ease of shopping and travel comfort), time saving and product diversity. Since an online shopping alternative clearly stands out favourably on convenience and time-saving aspects, this finding suggests that product diversity is the crucial factor in the overall benefit this service can offer in the perceptions of the average shopper. No differences were however found among attributes. It is comprehensible that respondents’ mindset does not change qualitatively on the attribute level although the introduction of a new shopping alternative led to a stronger benefit activation

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and shift in benefit activation. A possible reason might be that individuals experience anchoring or order effects in updating their mental representations when new alternatives are introduced so that it takes a long time until they integrate the new attributes and customs connected with ICT options which could contribute to a willingness to adopt this new alternative (Hogarth & Einhorn, 1992). The reluctance of the elder generation in adapting to ICT-based communication and activity modes might be exemplary of this effect. The underlying attributes are hence quite stable for the decision task at hand. This is also evident in the frequency and centrality values for the MR components. Possibly, a more focused sample of individuals who currently make use of online (grocery) shopping services could lead to more insight in the differences between MRs of those who already adopted and those who have not adopted these services. A follow-up study with an extended sample could perhaps bring further clarity. From the analysis other interesting insights could also be gained. For instance, there is less variation in the ranking of the three decision variables in the e-commerce scenario. Probably, this is an effect of the unlimited availability of online shopping which caused a decrease in the importance of scheduling the time of shopping. In sum, a variation of the choice set in terms of additional ICT alternatives did not lead to a substantial shift in MRs in terms of attributes that are considered despite the fact that respondents consider quantitatively more and different collections of benefits for the experimental shopping trip. This finding suggests that the introduction of ICT services does not result in an important change of how individuals evaluate options for grocery shopping activities in time and space. Convenience and product diversity are the benefits that are specifically associated with the presence of an online alternative. In any case, our findings provide a further confirmation of earlier results that the introduction of online alternatives does not seem to fundamentally change individuals’ activity-travel patterns for shopping (Corpus & Peachman, 2003; Sim & Koi, 2002). Possibly, mental representations change over a longer period of time after ICT users got acquainted with the usage and the benefits of suchlike online offers.

6.4.

Conclusions and Discussion

In this chapter, we reviewed and illustrated a new approach to address the issue of personal variability and situational dependence of travellers preferences based on the notion of mental representations. When individuals need to explore and oversee their choice alternatives and the likely consequences of their behaviour, they conceptualise the causal interdependencies between the choice alternatives, the decision context, their inner needs and the task by means of simplified images of reality. These so-called mental representations are the core subject of this approach as they represent the individual and situation specific information being necessary to explain a potentially important source of heterogeneity in activity-travel preferences.

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Different ways of measuring mental representations have been briefly discussed focusing on techniques that work on a verbal level. According to the memory retrieval process stressed, one can group them into recall- and recognition-based techniques, respectively. For both categories a number of more or less sophisticated techniques exists among which laddering, face-to-face CNET and APT are the most prominent ones. Although all of them were applied successfully in small surveys for measuring mental representations, each of the techniques had specific drawbacks which prevented a large-based application on the investigation of individual variability in decision-making. While the structured recognition-based techniques are held insensitive for measuring individual and contextual shifts, the unstructured recall-based techniques were too time-consuming and their collected data difficult to analyse. A recently developed automated online version of this method largely overcomes this problem. The method was illustrated in an application to investigate the impact of online shopping alternatives on mental representation in shopping activity choice making. The results showed that the presence of an online shopping alternative has an impact on the complexity of the mental representations but, contrary to what could be expected, hardly any impact on the contents of the cognitive constructs. There is no doubt that the exploration of MRs is only about to start. The enormous amount of choice situations from all societal domains would provide many interesting approaches for deeper investigation of MRs. Inter-individual differences in benefit activation and its effect on decision-making might be a very interesting topic to study. Thus, do individuals from different age groups, education levels, cultures, man and woman, etc. differ in the way they image a decision problem? Besides these rather snapshot-like recordings of individuals’ MRs also the investigation of more dynamic effects due to learning and updating, effects of priming, habitualisation effects, etc. seems to be worth to be put high on the research agenda. Cross-technical comparisons might furthermore be of interest in order to solve the question which technique delivers the more genuine image respondents bear in mind. From a formal point of view, results of suchlike investigations could thus help to incorporate more heterogeneity in (transport) choice models. Another future project is the extension of discrete choice models to account for need activation and cognitive selectivity in choice processes. As such, Arentze et al. (2014) showed how both cognitive selectivity and choice of an alternative can be modelled in an integrated RUM framework and how the integrated model can be parameterized and estimated by loglikelihood methods based on observations of both MRs and choice outcomes. In this regard it is only a logical consequence to shed also light on the relevance of MRs for the prediction of choice behaviour. From a practical point of view, progress in measuring mental representations could also simply improve our knowledge about how individuals face a certain decision problem. These insights could be very worthwhile for policymakers, transport planners and marketing experts in order to tailor travel demand measures, transport options and consumer products according to the needs of (groups of) individuals.

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Srivastava, R. K., Leone, R. P., & Shocker, A. D. (1981). Market structure analysis: Hierarchical clustering of products based on substitution-in-use. Journal of Marketing, 45(3), 3848. Ter Hofstede, F., Audenaert, A., Steenkamp, J.-B. E. M., & Wedel, M. (1998). An investigation into association pattern technique as a quantitative approach to measuring meansend chains. International Journal of Research in Marketing, 15, 3750. Trope, Y., & Liberman, N. (2003). Temporal construal. Psychological Review, 110, 403421. Tversky, A. (1972). Elimination by aspects: A theory of choice. Psychological Review, 79(4), 281299. Weber, E. U., & Johnson, E. J. (2006). Constructing preferences from memories. In S. Lichtenstein & P. Slovic (Eds.), The construction of preference (pp. 397410). New York, NY: Cambridge University Press.

Chapter 7

Towards a Novel Classifier for the Representation of Bounded Rationality in Models of Travel Demand Davy Janssens and Geert Wets

Abstract Several activity-based transportation models are now becoming operational and are entering the stage of application for the modelling of travel demand. In our application, we will use decision rules to support the decision-making of the model instead of principles of utility maximization, which means our work can be interpreted as an application of the concept of bounded rationality in the transportation domain. In this chapter we explored a novel idea of combining decision trees and Bayesian networks to improve decision-making in order to maintain the potential advantages of both techniques. The results of this study suggest that integrated Bayesian networks and decision trees can be used for modelling the different choice facets of a travel demand model with better predictive power than CHAID decision trees. Another conclusion is that there are initial indications that the new way of integrating decision trees and Bayesian networks has produced a decision tree that is structurally more stable. Keywords: Bayesian networks; decision trees; BNT classifier; (un)supervised learning; rule complexity reduction

7.1.

Introduction to Decision Theory

In decision theory in general, the predominant paradigm is expected utility theory (EUT) founded in von-Neumann and Morgenstern’s utility theorem (McFadden, 2001). Here, a decision is considered to be a choice out of certain options, depending

Bounded Rational Choice Behaviour: Applications in Transport Copyright r 2015 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISBN: 978-1-78441-072-8

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on the probability of occurrence and a valuation of alternatives. Proponents of the EUT approach praise its univocal theoretical foundation and its concomitant, clear mathematical interpretation enabling advanced statistical elaborations. However, even at the source of this theory, that is in economics, critics adhering to behavioural perspectives argue that although the rigid homo economicus assumption is useful in normative or prescriptive applications (as a model of how people ought to choose), people’s everyday decision-making does not meet perfect rationality (Camerer, 1998). Indeed, the adoption of an EUT model implies a considerate, fully informed decision-maker, prone to a high degree of rationality; as opposed to approaches that account for bounded rationality (Simon, 1990), intuition (Plessner, Betsch, & Betsch, 2008) or uncertainty and lack of information of the decision-maker (Frederick, 2002; Tversky & Kahneman, 2002). This dichotomy (rational versus behavioural) in theoretical approaches of decisionmaking applies to the different types of decisions that characterize individual travel as well. On the one hand is the repetitive nature of trips (such as commuting, chauffeuring kids to school, grocery shopping) likely to render (once) conscious decisions into script-based or habitual behaviour (Ga¨rling & Axhausen, 2003). On the other hand is activity scheduling (including choices of destinations, travel modes and routes) likely to entail the coordination of competing goals and intentions (e.g. amongst household members) in a complex environment (e.g. traffic-jams, opening hours), similar to complex planning problems (Ga¨rling, Gillholm, Romanus, & Selart, 1997). A theoretical account of behavioural decision-making in travel behaviour is given in Svenson (1998). The actual decision-making mechanisms of daily activity and travel scheduling and execution are scrutinized further in this book chapter. To this end, we will demonstrate in this chapter a novel idea of combining decision trees and Bayesian networks to improve decision-making in travel behaviour and illustrate its application in The Albatross model (see Arentze & Timmermans, 2004) which is a rule-based model of activity-scheduling behaviour. The developed methodology can be seen as an example of bounded rationality in the sense that we assume with this approach that the decision-making of individuals is constrained by personal and scheduling obligations (e.g. bring/get children to school) as it is represented in the rule-based model of activity-scheduling behaviour that is used in our experiments. The remainder of this chapter is organised as follows. First we will introduce the concept of activity-based (AB) models, which is followed by a discussion about how decision theory fits/is used within these models. Next, we will move to a section where Bayesian networks are introduced. In addition to the general concepts of the technique; more technical algorithms related to parameter and structural learning as well as entering evidences will be illustrated by means of concrete examples. In the fourth section, we will identify a problem of inferring decision rules from a Bayesian network, which is a typical problem in Bayesian network classifiers. In order to solve the problem, we will develop a novel integrated classifier in the fifth section of this chapter, in which we propose the idea to derive a decision tree from a Bayesian network (that is built upon the original data) instead of immediately deriving the tree from the original data. The next section describes both the data and the design

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of the experiments, which is then followed by an empirical (results) section. The chapter concludes with a discussion of the results.

7.2.

Decision Theory in Activity-Based Models

7.2.1.

Introduction

AB demand models generate a sequential list of activities and trips connecting these activities for every person in the study area. Demand generation is embedded in a concept of daily activity demand from which the need for transport is derived (Kitamura, 1988). A major advantage of AB models is that the spatial and temporal consistency of travel behaviour can be ensured. This is superior to traditional demand generation where aggregate traffic quantities represent isolated trips. Generally speaking, an AB model of travel demand consists of various model components to streamline the process from population input to travel output. The main components of such a system are a population synthesizer, which is required to generate population input data sensitive to demographic evolutions. The scheduler uses this input and additional data such as transportation and land use characteristics from other model components to generate detailed activity and travel plans. Subsequently, various components to model route choice, to assign traffic to the road network and to account for interaction between demand and supply can be added to establish a full AB micro-simulation model. Clearly, the core of an AB model of travel demand is the scheduler since this model component produces a detailed calendar of activities and travel for each individual, indicating what to do, when, for how long, where and how to travel to that location. AB schedulers generally generate such a detailed activity calendar from scratch for each individual in the population based on their socio-demographic characteristics, although some schedulers use a predefined frame of activities depending on person characteristics. Roughly speaking, building the scheduler of an AB model involves a three phase process. Firstly, the actual model is learned and its parameters or rules are estimated based on observed activity and travel behaviour in diary data (the training set). In the next step, the model performance is tested. Therefore, current population characteristics are used to generate individual activity and travel calendars, and this output is compared to actual travel behaviour as observed in travel surveys (the test set). Once validated, the model can be applied to forecast the impact of policy measures, such as road pricing (e.g. Arentze & Timmermans, 2008), and general socio-demographic changes in so-called policy scenarios. 7.2.2.

Decision Theory in AB Models

Decision theory in AB travel demand modelling can be represented by three major lines of research as described in Rasouli and Timmermans (2014). The first line of

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research is represented by constraints-based models. Constraints-based models do not predict individual activity-travel patterns but check whether any given activity agenda is feasible to meet specific (spacetime) constraints. These models are not very strong from a behavioural point of view and the earliest examples of these models go back to PESASP (Lenntorp, 1976); CARLA (Jones, Dix, Clarke, & Heggie, 1983) and BSP (Huigen, 1986). In recent years several contributions by researchers active in spacetime geography have led to interesting contributions (see e.g. Neutens, Schwanen, Witlox, & De Maeyer, 2008; Soo, Zhang, Ottens, & Ettema, 2009) but these contributions are net yet integrated in full operational AB models. Secondly, the main research line of the representation of decision theory in AB models is represented by utility-maximizing models. These models are based on the premise that individual maximize utility in choosing between activity-travel pattern alternatives. Significant contributions in the field of AB travel demand modelling are, for example, the daily activity schedule model (Ben-Akiva, Bowman, & Gopinath, 1996); PCATS (Kitamura & Fujii, 1998) and CEMDEP (Bhat, Guo, Srinivasan, & Sivakumar, 2004), FAMOS (Pendyala, Kitamura, Kikuchi, Yamamoto, & Fujii, 2005) and SIMAGENT (Goulias et al., 2012). Finally, computational process models (CPMs) try to overcome the drawback of utility-based models, namely that travellers do not make ‘optimal’ decisions but rather contextdependent heuristic decisions. CPMs ‘... replace the utility-maximizing framework with behavioural principles of information acquisition, information representation, information processing, and decision making’ (Golledge, Kwan, & Ga¨rling, 1994). CPMs are basically also microsimulations due to their disaggregate nature, the sequential decision process and the use of heuristics. However, the heuristics employed by CPMs rather consist of ‘if-then’ rules than utility-maximizing decision criteria. Models in this line of research are SCHEDULER (Golledge et al., 1994), AMOS (Kitamura & Fujii, 1998; Pendyala, Kitamura, Reddy, & Chen, 1995), ALBATROSS (Arentze & Timmermans, 2004), FEATHERS (Bellemans, Janssens, Wets, Arentze, & Timmermans, 2010), Tasha (Miller & Roorda, 2003) and ADAPTS (Auld & Mohammadian, 2009). Within the context of CPMs, and more concretely within the Albatross and Feathers ABM’s, a series of Ph.D. dissertations and studies (Janssens, 2005; Moons, 2005; Sammour, 2013) have been conducted in the field of machine learning. The machine learning technique in question then typically produces an individual decision-making heuristic/rule set as output and as such using a typical -structure, is a nice representation of the concept of bounded rationality. Bounded rationality goes back to the work of Simon (1991), where the concept was introduced as having two interlocking components: the limitations of the human mind and the structure of the environments in which the mind operates. The first component means that models of human judgment and decision-making should be built on what we actually know about the mind’s capacities. In many real-world situations, like for instance in daily travel decisions, optimal strategies are unknown or unknowable (Simon, 1987). Because of the mind’s limitations, humans ‘must use approximate methods to handle most tasks’ (Simon, 1990, p. 6). These methods include recognition processes that largely obviate the need for further information

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search, heuristics that guide search and determine when it should end, and simple decision rules that make use of the information found. Machine learning techniques are very well suited to perform these tasks. The second component of Simon’s view of bounded rationality, environmental structure, is of crucial importance because it can explain when and why simple heuristics perform well: if the structure of the heuristic is adapted to that of the environment. Since we agree that there is no such thing as a uniform law of travel behaviour which explains daily activity-travel patterns all over the world; this is exactly what we do when we model travel behaviour. By incorporating a wide spectrum of geographical, contextual and land use characteristics in our models, our goal is to adapt the structure of the heuristic to that of the environment. The next section focuses on one particular approach (Bayesian networks) within the field of machine learning that is able to represent this concept of bounded rationality.

7.3.

Bayesian Networks

7.3.1.

General Concepts

A Bayesian network consists of two components (Pearl, 1988): first, a directed acyclic graph (DAG) in which nodes represent stochastic domain variables and directed arcs represent conditional dependencies between the variables (see Definitions 13) and second, a probability distribution for each node as represented by conditional dependencies captured with the directed acyclic graph (see Definition 4). Bayesian networks are powerful representation and visualization tools that enable users to conceptualize the association between variables. However, as explained later, Bayesian networks can also be used for making predictions. To formalize, the following definitions are relevant: Definition 1. A directed acyclic graph (DAG) is a directed graph that contains no directed cycles.’ Definition 2. A directed graph G can be defined as an ordered pair that consists of a finite set V of vertices or nodes and an adjacency relation E on V. The Graph G is denoted as (V, E). For each (a, b) ɛ E (a and b are nodes) there is a directed edge from node a to node b. In this representation, a is called a parent of b and b is called a child of a. In a graph, this is represented by an arrow which is drawn from node a to node b. For any a∈V, (a,a) ∉ E, which means that an arc cannot have a node as both its start and end point. Each node in a network corresponds to a particular variable of interest.’ Definition 3. Edges in a Bayesian network represent direct conditional dependencies between the variables. The absence of edges between variables denotes statements of

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independence. We say that variables B and C are independent given a set of variables A if P(c|b,a) = P(c|a) for all values a, b and c of variables A, B and C. Variables B and C are also said to be independent conditional on A.’ Definition 4. A Bayesian network also represents distributions, in addition to representing statements of independence. A distribution is represented by a set of conditional probability tables (CPT). Each node X has an associated CPT that describes the conditional distribution of X given different assignments of values for its parents.’ The definitions mentioned above are graphically illustrated in Figure 7.1 by means of a simple hypothetical example. First, this network introduced here clearly is acyclic and directed. Second, the variables ‘gender’, ‘driving license’ and ‘number of cars’ are parents of the mode choice variable. Finally, dependent and independent relationships, as well as examples of CPTs are shown in this figure. In the upper CPT for instance, the probability for mode choice being equal to bike, is 0.2, given that gender = male, driving license = yes and number of cars = 1. Learning Bayesian networks has traditionally been divided into two categories (Cheng, Bell, & Liu, 1997):

Gende r

Driving Number of cars License

Male Male Male Male Female Female Female Female

Yes Yes No No Yes Yes No No

1 >1 1 >1 1 >1 1 >1

CPT Mode Choice

0.75

P (mode choice = car)

0.2 0.6 0.7 0.4 0.4 0.8 0.1 0.3

0.8 0.4 0.3 0.6 0.6 0.2 0.9 0.7

Mode Choice

Gender

P (Gender = male)

P (mode choice = bike)

Driving License

P (Gender = P (Driving P (Driving female) License = Yes) License = No) 0.25

CPT Gender

0.6

0.4

CPT Driving License

Number of Cars

P (Number of cars = 1)

P (Number of cars > 1)

0.2

0.8

CPT Number of Cars

Figure 7.1: A small Bayesian network with its CPT.

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structural and parameter learning. Since these learning phases are relevant for the new integrated BNT classifier, the following sections elaborate on them into detail.

7.3.2.

Parameter Learning

Parameter learning determines the prior CPT of each node of the network, given the link structures and the data. It can therefore be used to examine quantitatively the strength of the identified effect. As mentioned above, a conditional probability table P(A|B1…Bn) has to be attached to each variable A with parents B1, …, Bn. Note that if A has no parents, the table reduces to unconditional probabilities P(A). According to this logic, for example Bayesian network depicted in Figure 7.1, the prior unconditional and conditional probabilities to specify are: P(Driving License); P(Gender); P(Number of cars); P(Mode Choice|Driving License, Gender, Number of cars). Since the variables ‘Number of cars’, ‘Gender’ and ‘Driving License’ are not conditionally dependent on other variables, calculating their prior frequency distribution is straightforward. Calculating the initial probabilities for the ‘Mode Choice’ variable is computationally more demanding. In order to calculate the prior probabilities for the ‘Mode choice’ variable, the conditional probability table for P(Mode Choice|Driving License, Gender, Number of cars) was set up in the first part of Table 7.1a. Again, this is straightforward Table 7.1a: Conditional and joint prior probability tables for the transport mode choice variable. Conditional prior probability table specifying P(Choice|Gender, Driving License, Ncar) Gender

Male

Driving license Number of cars Mode choice bike Mode choice car

Yes 1 0.2 0.8

Female No

>1 0.6 0.4

1 0.7 0.3

Yes >1 0.4 0.6

1 0.4 0.6

No

>1 0.8 0.2

1 0.1 0.9

>1 0.3 0.7

Joint prior probability table for P(Choice, Gender, Ncar, Driving License) Gender

Male

Driving license

Yes

Number of cars Mode choice bike Mode choice car

1 0.018 0.072

Female

No >1 0.216 0.144

1 0.042 0.018

Yes >1 0.096 0.144

1 0.012 0.018

No >1 0.096 0.024

1 0.002 0.018

>1 0.024 0.056

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mathematical calculus. In order to get the prior probabilities for the Mode Choice variable, we now first have to calculate the joint probability P(Choice, Gender, Number of cars, Driving License) and then marginalize ‘Number of cars’, ‘Driving License’ and ‘Gender’ out. This can be done by applying Bayes’ rule, which states that: PðChoice; Gender; Number of cars; Driving LicenseÞ = PðChoicejGender; Number of cars; Driving LicenseÞ PðGender; Number of cars; Driving LicenseÞ Since ‘Gender’, ‘Number of cars’ and ‘Driving License’ are independent, the equation can be simplified for this example as: PðChoice; Gender; Number of cars; Driving LicenseÞ = PðChoicejGender; Number of cars; Driving LicenseÞ PðGenderÞPðNumber of carsÞPðDriving LicenseÞ Note that P(Gender = male; Gender = female)=(0.75; 0.25), P(Driving License = yes; Driving License = no) = (0.6; 0.4) and P(Number of cars = 1; Number of cars > 1)=(0.2; 0.8), which are the prior frequency distributions for those three variables. By using this information, the joint probabilities were calculated in the second part of Table 7.1a. Marginalizing ‘Gender’, ‘Number of cars’ and ‘Driving License’ out of P(Choice, Gender, Number of cars, Driving License) yields P(Mode Choice = bike; Mode Choice = car) = (0.506; 0.494). These are the prior probabilities for the ‘Mode Choice’ variable. Of course, computations become more complex when ‘Gender’, ‘Number of cars’ and ‘Driving License’ are dependent. Fortunately, in these cases, probabilities can be calculated automatically by means of probabilistic inference algorithms that are implemented in Bayesian network-enabled software.

7.3.3.

Entering Evidences

In fact, Figure 7.1 only depicts the prior distributions for each variable. This is useful but not very innovative information. An important strength of Bayesian networks, however, is to compute posterior probability distributions of the variable under consideration, given the fact that values of some other variables are known. In this case, the known states of variables can be entered as evidence in the network. When evidence is entered, this is likely to change the states of other variables as well, since they are conditionally dependent. This is demonstrated by entering the evidence in the network that the ‘Mode choice’ variable is equal to ‘car’. In this case, evidence on ‘Mode choice’ now arrives in the form of P*(Mode Choice = bike; Mode Choice = car) = (0; 1), where P* indicates that we are calculating posterior probabilities (i.e. after entering evidences).

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Then; PðChoice; Gender; Number of cars; Driving LicenseÞ = PðNumber of cars; Gender; Driving LicensejMode ChoiceÞ PðMode ChoiceÞ = ðPðChoice; Gender; Number of cars; Driving LicenseÞ PðMode ChoiceÞÞ=PðMode ChoiceÞ This means that the joint probability table for ‘Choice’, ‘Number of cars’, ‘Driving License’ and ‘Gender’ is updated by multiplying by the new distributions and dividing by the old ones. The multiplication consists of annihilating all entries with ‘Choice’=‘bike’. The division by P(Mode Choice) only has an effect on entries with Mode Choice = ‘car’, so therefore the division is by P(Mode Choice = ‘car’). For this simple example, the calculations can be found in Table 7.1b. The distributions P*(Number of cars), P*(Gender) and P*(Driving License) are calculated through marginalization of P*(Choice, Gender, Number of cars, Driving License). This means that PðGender = male; Gender = femaleÞ = ð0:765; 0:235Þ P*(Number of cars = 1;Number of cars > 1) = (0.255;0.745); and P*(Driving License = yes; Driving License = no) = (0.522; 0.478), when evidence was entered that the ‘Mode choice’ variable equals car. Obviously, the calculation of this example is simple, however, in real-life situations it is likely that conditionally dependent relationships between the ‘choice’ variable and other variables exist as well, and as a result the evidence will propagate through the whole network. More information about efficient algorithms for propagation of evidence in Bayesian networks can be found in Pearl (1988) and in Jensen, Lauritzen, and Olesen (1990). 7.3.4.

Structural Learning

Structural learning determines the dependence and independence of variables and suggests a direction of causation (or association), in other words, the position of the Table 7.1b: Posterior probability table for the transport mode choice variable. The calculation of P*(Choice, Gender, Ncar, Driving License) = P(Choice, Gender, Ncar, Driving Licence|Mode Choice = car) Gender Driving license Number of cars Mode choice bike Mode choice car

Male Yes 1 0 0.146

>1 0 0.291

Female No

1 0 0.036

Yes >1 0 0.291

1 0 0.036

>1 0 0.049

No 1 0 0.036

>1 0 0.113

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links in the network. Experts can provide the structure of the network using domain knowledge. However, the structure can also be extracted from empirical data. Especially the latter option offers important and interesting opportunities for transportation travel demand modelling because it enables one to visually identify which variable or combination of variables influences the target variable of interest. It has to be noted however that algorithms which learn the structure of the network can sometimes have difficulties in capturing the correct (causal) relationships. Causality is extremely difficult to model, often involving human reasoning and it cannot be captured efficiently by any machine learning algorithm. Therefore, the intuitive interpretation of some directions of arrows might look strange. Therefore, it is better to consider the directed arc as an association rather than as a causality relationship per se. Structural learning can be divided into two categories: search & scoring methods and dependency analysis methods. Algorithms, belonging to the first category interpret the learning problem as a search for the structure that best fits the data. Different scoring criteria have been suggested to evaluate the structure, such as the Bayesian scoring method (Cooper & Herskovits, 1992; Heckerman, Geiger, & Chickering, 1995) and minimum description length (Lam & Bacchus, 1994). The underlying principle behind these scores is the well-known Ockham’s razor: the best model to describe a phenomenon is the one which best balances accuracy and complexity. The scores basically include two terms: one for accuracy and the other for complexity and the philosophy of the score is to find/select a model that rightly balances these terms. A Bayesian network is essentially a descriptive probabilistic graphical model that is potentially well suited for unsupervised learning. Unsupervised learning can be defined as the search for a useful structure without labelled classes, optimization criterion or any other information beyond the raw data. Unsupervised learning can help researchers to discover the whole set of probabilistic relationships existing within the data (association discovery) instead of only developing a learning function for one specific dependent variable (supervised learning). By tuning the technique, it also becomes suitable for the latter task (supervised (or classification) learning); just like other more traditional supervised learning algorithms like decision trees, neural networks or for instance support vector machines. A number of Bayesian network classifiers (e.g. Naı¨ ve Bayes, Tree augmented Naı¨ ve Bayes, General Bayesian network) have been developed for this purpose.

7.4.

Bayesian Network Classifiers: Problem Formulation

While Bayesian network classifiers have proven to give accurate and good results in a transportation context (Janssens et al., 2004; Torres & Huber, 2003), Achilles’ tendons obviously are the decision rules, which can be derived from the Bayesian network. As mentioned above, Bayesian networks link more variables in sometimes complex, direct and indirect ways, making interpretation more problematic. Second,

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each decision rule that is used for predicting a particular dependent variable within a network contains the same number of conditions resulting in potential suboptimal decision-making. To illustrate this, the procedure of transforming a Bayesian network into a decision table (i.e. rule-based form) is shown in Figure 7.2. We preferred to make this transition because decision rules and the corresponding decision table formalism have some advantageous properties. The first reason is that a decision table is exclusive, consistent and complete. This behaviour is not guaranteed by traditional production systems and it represents a clear advantage of decision tables for any modelling purpose. Secondly, the decision table provides a suitable formalism for representing various types of interactions between variables, such as conditional relevance and conceptual interaction. For an extensive review, we refer to Wets (1998). The left part of Figure 7.2 is an example of a pruned network. A pruned network is a network that is reduced in size such that the loss of accuracy on the dependent variable for the unseen test data is limited. In the left part of the figure, the different variables in the network are represented as boxes and each state in the network is shown with its belief level (probability) expressed as a percentage and as a bar chart. In the middle part of the figure, evidences are entered for every independent variable (see Section 3.3), resulting in a probability distribution of the target variable. This process is repeated for every possible combination of states (of independent variables). When an evidence is entered in the network, this is shown in the figure as a shaded box and as a 100% belief. As already introduced in Section 3.4, the direction of the arcs is preferably interpreted as an association rather than as a causality relationship. This means that not only child nodes but also parent nodes can influence the probability distribution of the dependent variable. For this reason, evidences need to be entered for every independent variable, regardless of whether these variables are child or parent nodes. There is one exception in this regard and that is the concept of d-separation. In this chapter, we will not elaborate into detail on this, but it means that in the case of d-separation, entering evidences for an independent variable will have no effect on the dependent variable. More information about this can be found in Pearl (1988) and in Geiger and Pearl (1988). As it can be seen from this figure, every rule contains the same number of condition variables for this particular network. For the example shown here, this number is equal to 4. Moreover, the number of rules that are derived from the network is fixed and can be determined in advance for a particular network (i.e. per dependent variable). This number is equal to every possible combination of states (values of the condition variables). Therefore, the total number of rules, which has to be derived from the network shown in Figure 7.2 is equal to 5*7*2*4 = 280, assuming that the with-whom attribute is taken as the class attribute. Especially when more nodes are incorporated, this number is likely to become extremely large. While this does not need to be a problem as such, it is obvious that a number of these decision rules will be redundant as they will never be ‘fired’. This flaw has no influence on the total accuracy of each Bayesian network classifier (see Janssens et al., 2004), but

Activity type 23.6 7.92 14.9 24.8 28.8 3.3 ± 1.5

Day 10.7 11.4 13.9 13.0 15.1 22.8 13.0 4.3 ± 1.9

N1 N2 N3 N4 N5

N1 N2 N3 N4 N5 N6 N7

Children 55.9 14.7 11.9 17.5 1.9 ± 1.2

N1 N2 N3 N4

 N1 N2 N3 N4 N5 N6 N7

N1 N2 N3 N4 N5

Day 100 0 0 0 0 0 0 1

Activity type 100 0 0 0 0 1

N1 N2 N3 N4

N0 N2

N1 N2 N3

Children 100 0 0 0 1

Availability 100 0 0

With Whom 90.6 .061 9.39 1.19 ± 0.58

Entering evidences (for every combination)

 N1

N1

N1

N1

N1

N0

N2

N1

N1

N1

N2

N1

… N4

… N1

… N7

… N5

9.39% 3.85% 10.8% 5.55% … 24.7%

0.06% 37.3% 0.03% 21.6% … 60.4%

90.6% 58.9% 89.2% 72.9% … 14.9%

N1

Children

With whom: N1 With whom: N2 With whom: N3

N0

N1

Day Availability

N1

Act. type

A decision table with probability distributions

Figure 7.2: Calculating probability distributions and entering them in a decision table.

Availability 16.2 83.8 0.84 ± 0.37

With Whom 41.7 36.7 21.6 1.8 ± 0.77

N0 N1

N1 N2 N3

The pruned network

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it is clearly a sub-optimal solution, not only because some of the rules will never be used, but also because this large number of conditions do not favour the interpretation. Clearly, decision trees do not suffer from this problem. In a decision tree, the ‘depth’ of the tree only determines the maximum number of conditions that is used in decision rules. This is a maximum, and non-fixed number. In Section 7.2, we will elaborate more into detail on this. For both reasons mentioned in this section, that is the possibility of combining the advantage of Bayesian networks (take into account the interdependencies among variables) and the advantage of decision trees (derive easy understandable and flexible (i.e. non-fixed) decision rules), and for the reason mentioned before, that is, deal with the variable masking problem in decision trees, the idea to integrate both techniques into a new classifier was conceived.

7.5.

Towards A New Integrated Classifier

In the integrated BNT classifier, the idea is proposed to derive a decision tree from a Bayesian network (that is built upon the original data) instead of immediately deriving the tree from the original data. By doing so, it is expected that the structure of the tree is more stable, especially because the variable correlations are already taken into account in the Bayesian network, which may reduce the variable masking problem. To the best of our knowledge, the idea to build decision trees in this way has not been explored before in previous studies. In order to select a particular decision node in the BNT classifier, the mutual information value that is calculated between two nodes in the Bayesian network is used. This mutual information value is to some extent equivalent with the entropy measure that C4.5 decision trees use. It is defined as the expected entropy reduction of one node due to a finding (observation) related to the other node. The dependent variable is called the query variable (denoted by the symbol Q), the independent variables are called findings variables (denoted by the symbol F). Therefore, the expected reduction in entropy (measured in bits) of Q due to a finding related to F can be calculated according the following equation (Pearl, 1988): IðQ; FÞ =

X X q

pðq; f Þlog f



pðq; f Þ pðqÞpðf Þ

 ð7:1Þ

where p(q, f) is the posterior probability that a particular state of Q(q) and a particular state of F(f) occur together; p(q) is the prior probability that a state q of Q will occur and p(f) is the prior probability that a state f of F will occur. The probabilities are summed across all states of Q and across all states of F. As a result of this calculation, where you in fact divide the posterior probability (that a particular state of Q(q) and a particular state of F(f) occur together) by the prior probabilities (of p(q) and p(f)), you can calculate which finding variable is the best in explaining the

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variability of the dependent variable. This means that this particular variable will be selected as the most important variable in our decision tree. To this end we calculated the expected reduction in entropy of the dependent variable for the various findings variables. The finding variable that obtains the highest reduction in entropy was selected as the root node in the tree. To better illustrate the idea of building a BNT classifier, we consider again the network that was shown in Figure 7.1 by means of example. In this case, the dependent variable is ‘Mode choice’ and the different finding variables are ‘Driving license’, ‘Gender’ and ‘Number of cars’. In a first step, we can for instance calculate the expected reduction in entropy between the ‘Mode choice’ and the ‘Gender’ variable. I = PðModebike ; Gendermale Þlog

PðModebike ; Gendermale Þ PðModebike ÞPðGendermale Þ

þ PðModecar ; Gendermale Þlog

PðModecar ; Gendermale Þ PðModecar ÞPðGendermale Þ

þ PðModebike ; Genderfemale Þlog þ PðModecar ; Genderfemale Þlog

PðModebike ; Genderfemale Þ PðModebike ÞPðGenderfemale Þ PðModecar ; Genderfemale Þ PðModecar ÞPðGenderfemale Þ

The calculation of the joint probabilities P(Modei, Genderj) for i = {bike, car} and j = {male,female} is the same as explained in Section 3.2. The calculation of the individual prior probabilities P(Modei) and P(Genderj) is straightforward as well (see Section 3.2). As a result, the expected result of formula (7.1) is: IðMode Choice; GenderÞ = 0:372log

0:372 0:378 þ 0:378log 0:5060:75 0:4940:75

þ 0:134log

0:134 0:116 þ 0:116log = 0:00087 0:5060:25 0:4940:25

In a similar way, I (Mode Choice, Driving License) = 0.01781 and I (Mode Choice, Number of cars) = 0.01346 can be calculated. Since I (Mode Choice, Driving License) > I (Mode Choice, Number of cars) > I (Mode Choice, Gender); the variable Driving License is selected as the root node of the tree (see Figure 7.3). Once the root node has been determined, the tree is split up into different branches according to the different states (values) of the root node. To this end, evidences can be entered for each state of the root node in the Bayesian network and the entropy value can be re-calculated for all other combinations between the findings nodes (except for the root node) and the query node. The node, which achieves the highest entropy reduction is taken as the node which is used for splitting up that particular branch of the root node. In our example, the root node ‘Driving License’ has two branches: Driving License = yes and Driving License = no. For the split in the first branch (Driving License = yes), only two

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Driving license Yes Number of cars >1

1 Gender

Car

Car

Female

Number of cars

Male Female Bike

Gender Male

Gender

Male Female

No

Bike

1 Bike

Number of cars >1 Car

1 Car

>1 Car

Figure 7.3: The final integrated BNT decision tree classifier (example).

variables have to be taken into account (since the root node is excluded): ‘Number of cars’ and ‘Gender’. The way in which the expected reduction in entropy is calculated is the same as shown above, expect for the fact that an evidence needs to be entered for the node ‘Driving License’, that is P(Driving License = Yes; Driving License = no)=(1; 0) (since we are in the first branch). The procedure for doing this was already described in Section 4.3. Again, I (Mode Choice, Gender) = 0.02282 and I (Mode Choice, Number of cars) = 0.07630. Since I(Mode Choice, Number of cars) > I((Mode Choice, Gender); the variable ‘Number of cars’ is selected as the next split in this first branch. Finally, the whole process then becomes recursive and needs to be repeated for all possible branches in the tree. A computer code has been established to automate the whole process. The final decision tree for this simple Bayesian network is shown in Figure 7.3.

7.6.

Data and Design of the Experiments

7.6.1.

Data

The activity diary data used in this study were collected in the municipalities of Hendrik-Ido-Ambacht and Zwijndrecht in the Netherlands (South Rotterdam region) to develop the Albatross model system (Arentze & Timmermans, 2004). The data involve a full activity diary, implying that both in-home and out-of-home activities were reported. The sample covered all seven days of the week, but individual respondents were requested to complete the diaries for two designated consecutive days. Respondents were asked, for each successive activity, to provide information about the nature of the activity, the day, start and end time, the location where

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the activity took place, the transport mode, the travel time, accompanying individuals and whether the activity was planned or not. A pre-coded scheme was used for activity reporting. After cleaning, a data set of a random sample of 1649 respondents was used in the experiments. There are some general variables that are used for each choice facet of the Albatross model (i.e. each oval box). These include (among others) household and person characteristics that might be relevant for the segmentation of the sample. Each dimension also has its own extensive list of more specific variables, which are not described here in detail.

7.6.2.

Design of the Experiments

The aim of this study is to examine both the predictive capabilities and the potential advantages of the BNT classifier. To this end, the predictive performance of this integration technique is compared with a decision tree learning algorithm (CHAID) and with original Bayesian network learning. For the CHAID decision tree approach, experiments were conducted for the full set of decision agents of the Albatross system. First, decision trees were therefore extracted from activity-travel diaries. Hereafter, these decision trees were converted into decision tables. Next, the decision tables were successively executed to predict the activity-travel patterns for the randomly selected sample of 1649 respondents. For the Bayesian network approach, a Bayesian network was constructed for every decision agent using a structural learning algorithm, developed by Cheng et al. (1997). This implies that the structure of the network was not imposed on the basis of a-priori domain knowledge, but was learned from the data. The structural learning algorithm was also enhanced by adding a pruning stage. This pruning stage aims at reducing the size of the network without resulting in a significant loss of relevant information or loss of accuracy on the unseen test data. Therefore, the aim is to find a favourable trade-off between the size of the network and the predicted accuracy, since significant ‘overpruning’ will obviously damage the final accuracy results. This means that nodes, which are not valuable for decision-making, are pruned away. In order to decide which nodes in the network are suitable for pruning, the reduction in entropy between two nodes was calculated using Eq. (1), shown in Section 5. Obviously, a huge entropy reduction indicates a potentially important and useful node in the network. An entropy reduction of less than 0.05 bits was used as a threshold to prune the network. Once the pruned network is constructed for every decision agent, the model can be used for prediction. To this end, probability distributions of all the variables in the networks have to be computed. A parameter learning algorithm developed by Lauritzen (1995) was used to calculate these probability distributions. The last step is to transform the predictive model to the decision table formalism. For building the BNT classifier, a decision tree is not derived directly from the original data, but from the Bayesian networks that are built in the previous step. The procedure for doing this was explained in Section 5. Once again, the decision tables are then sequentially executed to predict activity-travel patterns.

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In the next section, we report the results of detailed quantitative analyses that were conducted to evaluate the BNT classifier for every decision agent in the Albatross model. The results of the three alternative approaches are validated in terms of accuracy percentages. The techniques are compared at both the activity pattern level and the trip level.

7.7.

Results

7.7.1.

Model Comparison: Accuracy Results

To be able to test the validity of the presented models on a holdout sample, only a subset of the cases is used to build the models (i.e. ‘training set’). The decline in goodness-of-fit between this ‘training’ set and the validation set is taken as an indicator of the degree of overfitting. The purpose of the validation test is also to evaluate the predictive ability of the three techniques for a new set of cases. For each decision step, we used a random sample of 75% of the cases to build and optimize the models. The other subset of 25% of the cases was presented as ‘unseen’ data to the models; this part of the data was used as the validation set. The accuracy percentages that indicate the predictive performance of the three models on the training and test sets are presented in Table 7.2. It can be seen from this table that the accuracy percentages of the BN and BNT classifier outperform the accuracy results of CHAID decision trees. A paired t-test comparing the accuracies of the test data of the CHAID models with these of the Table 7.2: Comparison of accuracy percentages for CHAID decision trees, Bayesian networks and integrated BNT classifier. Decision agent

With-whom Duration Start time Trip chain Mode for work Mode other Location 1 Location 2 Average

Decision-making based on CHAID decision trees

Decision-making based on Bayesian networks

Decision-making based on integrated BNT classifier

Training set

Validation set

Training set

Validation set

Training set

Validation set

0.509 0.413 0.398 0.833 0.648

0.484 0.388 0.354 0.809 0.667

0.577 0.409 0.477 0.831 0.769

0.534 0.405 0.380 0.823 0.779

0.577 0.410 0.423 0.831 0.770

0.535 0.402 0.393 0.825 0.783

0.528 0.575 0.354 0.553

0.495 0.589 0.326 0.536

0.583 0.696 0.473 0.622

0.521 0.679 0.420 0.592

0.583 0.694 0.473 0.617

0.521 0.685 0.419 0.595

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BN and BNT models yields respective P-values of 0.006 (CHAID versus BN) and 0.0052 (CHAID vs. BNT). Therefore, the null hypothesis stating that there is no difference between the accuracies of the test sets of both models could not be accepted at a 5% significance level. On the other hand, the accuracy of the BNT classifier is not significant compared to Bayesian networks, as indicated by a P-value of 0.1643. This should not be surprising of course, because Bayesian networks were used as the underlying structure of the decision trees. More important is the observation that the newly proposed BNT classifier outperformed the CHAID decision trees for all nine decision agents of the Albatross model. This means that by using Bayesian networks as the underlying structure for building decision trees, better results can be obtained than using a traditional CHAID-based decision tree approach. In terms of validity of the three models, we can conclude that the degree of over-fitting (i.e. the difference between the training and the validation set) is low for all decision agents. Therefore, we conclude that the transferability of the models to a new set of cases is satisfactory.

7.7.2.

Model Comparison in Terms of Model and Individual Rule Complexity

Section 7.1 has illustrated that the idea of integrating Bayesian networks and decision trees holds out considerable promise in terms of predictive accuracy. However, since the accuracy of the BNT classifier was not significant compared to Bayesian networks, the exercise in this chapter only leads to an unambiguous added value if the interpretation of decision rules derived from BNT is superior to the interpretation of decision rules that are derived from Bayesian networks. On the one hand, model complexity can be approximated by the total number of decision rules that is derived from a model. For Bayesian networks, this total is equal to the product of the number of possible states per variable. For decision trees, the total number of rules equals the total number of leaves in the tree. However, while this may give an idea about the complexity of the full model, it does not give any indication about the ease of interpretation of a single decision rule. This latter form of individual rule complexity can be measured quite easily for Bayesian networks as it can be approximated by the number of independent variables which are present in the network, because every independent variable is used in every decision rule. For decision trees, the complexity of the derived decision rules can be approximated by the ‘depth’ of the decision tree. The depth of a decision tree is equal to the number of levels that occur in a decision tree. Indeed, if the structure of the decision tree is rather flat, the ease of understanding of an individual decision rule is easy, since the number of independent variables that is used for predicting the dependent variable is quite limited. As a result of this, the understanding of the joined impact of these variables on the dependent variable is facilitated. In Table 7.3 an indication is given about the model complexity in terms of number of rules and in terms of the complexity of every single rule. It can be seen from this table that the BNT classifier significantly improves the individual rule complexity in terms of the number of independent variables that is used in the decision rules.

With-whom Duration Start time Trip chain Mode for work Mode other Location 1 Location 2

Decision agent

4 4 11 11 8 5 9 10

Bayesian networks (number of independent variables in network) 3 2 5 6 5 4 7 6

BNT (‘depth’ of decision tree, in maximum number of levels)

Individual rule complexity

280 84 983,040 124,416 36,864 432 5760 9216

Bayesian networks (total number of rules)

70/16 6/4 210/58 384/175 187/64 108/45 253/215 131/124

BNT (number of leaves/number of nodes)

Model complexity

Table 7.3: Comparison for model complexity and individual rule complexity with respect to Bayesian networks and integrated BNT classifier.

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While it remains difficult to analyse the joined impact of several independent variables, it is still possible for relatively low numbers (let’s say at most 5 or 6), and it will become almost totally incomprehensible for higher numbers. A significant achievement is obtained in this respect for the facets ‘start time’, ‘trip chain’, ‘location2’ and ‘mode for work’. The reader should also note that the depth of the decision tree that is shown in the third column of Table 7.3 indicates the maximum number of levels in the tree. This means that for most branches in the tree, this maximum number will not be achieved and less independent variables will be used in the decision rules, making comprehension easier than in Bayesian networks. In addition to this, and as mentioned before in the chapter, this is not the case for Bayesian networks, since the number of independent variables in every decision rule is constant for every decision agent (see second column in Table 7.3). It would be possible however, to adopt some kind of pruning mechanism on the rules, by which the number of independent variables could be reduced. This was not done for two obvious reasons. First, the Bayesian networks that were used for prediction, as well as the BNT classifier, which uses the networks as its underlying structure for calculating the splits in the tree, already were pruned networks (see Section 6.2). Adding another post-pruning stage to the decision rules that are derived from these networks can potentially result in overpruned results, and it wipes out the original idea for using Bayesian networks, which is to analyse the joined impact of a large number of variables. Second, the rules that were derived from the BNT classifier were not pruned either, and this enables a fair comparison between both algorithms. The right part of Table 7.3 describes the model complexity in terms of the total number of rules that is used in each decision agent. As mentioned above, while the total number of rules has no impact on the total accuracy of each classifier, it creates an additional classification complexity and computational overhead. It has to be noted that a number of rules will be redundant as they will never be ‘fired’, because it is unlikely that every combination of variables in the network also occur in the real data. Based on Table 7.3, we have to conclude that the BNT classifier is a huge improvement over the Bayesian network approach and this for all decision agents. The next section examines the results of the models at pattern level.

7.7.3.

Activity Pattern Level Analysis

Obviously, the most important evaluation criterion for predicting individual choice facets is accuracy. In this respect, Section 7.1 has already illustrated that BNT and BN achieve comparable results and that they both perform better than CHAID. Section 7.2 has explained that BNT is better in terms of model and individual rule complexity, when compared to BN. Based on these two criteria, it is safe to conclude that the BNT classifier is the most convenient choice for the models under evaluation. However, this evaluation was only conducted at an individual choice facet level. The aim of this section is therefore to evaluate whether the same good results of BNT can be maintained at pattern level when compared with the original CHAID classifier. At the activity pattern level, sequence alignment methods (SAM)

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(Joh, Arentze, & Timmermans, 2001) were used to calculate the similarity between observed and generated activity schedules. One activity schedule is a sequence of activities, along with its start and end times, its location, with-whom and with which transport mode an activity is carried out during a 24-hour period. The activity schedule is generated by a sequential execution of decision tables, as described in the Albatross model (see Arentze and Timmermans, 2004). The SAM measure allows users to evaluate the goodness-of-fit. SAM originally stems from work in molecular biology to measure the biological distance between DNA and RNA strings. Later, it was used in time use research (Wilson, 1998). To account for differences in composition as well as sequential order of elements, SAM determines the minimum effort required to make two strings identical using insertion, deletion and substitution operations. The mean SAM distances between the observed and the predicted schedules are shown in Table 7.4. SAM distances were separately calculated for the qualitative activity pattern attributes (activity type, with-whom, location and mode). Also, both ‘UDSAM’ and ‘MDSAM’ measures were calculated. UDSAM represents a weighted sum of attribute SAM values, where activity type was given a weight of two units and the other attributes a weight of one unit. To account for the multidimensionality, which is incorporated in the Albatross model, the MDSAM measure (Joh, Arentze, Hofman, & Timmermans, 2001) was used. The lower the SAM measure, the higher the degree of similarity between observed and predicted activity sequences. The finding that was found in Section 7.2 was confirmed in Table 7.4, where it is shown that the decision tree using the Bayesian network as the underlying structure, performed better than the state-of-the-art CHAID decision tree approach at pattern level. 7.7.4.

Trip Matrix Level Analysis

The last measure to evaluate the predictive performance between CHAID and BNT is calculated at trip level. The origins and destinations of each trip, derived from the activity patterns, are used to build OD-matrices. The origin locations are Table 7.4: SAM distance measures for activity pattern level analysis. SAM distance measure SAM activity-type SAM with SAM location SAM mode UDSAM MDSAM

Decision-making based on CHAID decision trees

Decision-making based on integrated BNT classifier

2.86 3.225 3.181 4.599 16.725 8.457

2.151 2.6 2.144 3.784 12.83 6.25

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Table 7.5: Correlation coefficients between OD-matrices for trip level analysis.

None Mode Day Primary activity

Decision-making based on CHAID decision trees

Decision-making based on integrated BNT classifier

0.953814 0.876937 0.960293 0.889740

0.9542192 0.8495603 0.9504108 0.8281859

represented in the rows of the matrix and the destination locations in the columns. The number of trips from a certain origin to a certain location is used as a matrix entry. A third dimension was added to the matrix to break down the interactions according to some third variable. The third dimensions considered are day of the week, transport mode and primary activity. The bi-dimensional case (no third dimension) was considered as well. In order to determine the degree of correspondence between predicted and observed matrices, a correlation coefficient was calculated. To this end, cells of the matrix were rearranged into one array and the calculation of the correlation is based on comparing the corresponding elements of the predicted and the observed array. The results are presented in Table 7.5. It can be seen from Table 7.5 that the good performance of BNT at the activity pattern level could not be maintained at trip level. The correlation coefficient is especially low for the OD matrix, where the primary activity is taken as the third dimension. Although this should be the subject of additional and future research, it is believed that the integrated approach did predict less activities in the activity schedule than CHAID. This deficiency is less apparent at the pattern level than at the trip level.

7.8.

Conclusion and Discussion of the Results

Several AB models are nowadays operational and are entering the stage of application. Some of these models (like Albatross) rely on a set of decision rules that are derived from activity-travel diary data rather than on principles of utility maximization. While the use of rules may have some theoretical advantages, the performance of several rule induction algorithms in models of activity-travel behaviour is not well understood. This is unfortunate because there is some empirical evidence that decision tree induction algorithms are relatively sensitive to random fluctuations in the data. To add to the growing literature on the performance of alternate decisiontree induction algorithms, we proposed in this chapter a way of combining Bayesian networks and decision trees. This study was designed to examine whether a decision tree (which is implicitly always less complex) that uses the structure of a Bayesian network (referred to as Bayesian network augmented tree classifier, BNT) to select

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its decision nodes can achieve simultaneously accuracy results comparable to Bayesian networks and an easier and less complex model structure, comparable to traditional decision trees. In order to test the validity and the transferability to a new set of cases of the proposed approach, datasets were split up into training and validation sets. The predictive performance of the new approach was evaluated at three different levels. The test has shown that the BNT approach indeed achieved comparably good accuracy results than Bayesian networks and along with the Bayesian networks outperformed CHAID decision trees for all decision agents of the Albatross model. However, in terms of model understanding, the BNT approach outperformed Bayesian networks. Also at pattern level, the BNT classifier performed better than CHAID. Finally, at the third level of validation, the trip matrix level, correlation coefficients between observed and predicted origindestination matrices showed that CHAID outperformed BNT. Thus, the good results of the integrated decision tree were not maintained at the trip level. It is believed that the technique did predict fewer activities in the activity schedule. While the smaller number rules that are used may be responsible for this, this finding should be the subject of additional research. In summary then, this study has shown some interesting results. There are some initial indications that the new way of integrating decision trees and Bayesian networks may produce a decision tree that is structurally more stable and less vulnerable to the variable masking problem. Additionally, the results at the activity level and trip level suggest at least for the Albatross data, a trade-off between model accuracy and model complexity. When the main issue is the interpretation and the general understanding of the decision rules, the integrated BNT approach may be favoured above CHAID decision trees when decisions need to be made at pattern level. At a more detailed level, one may benefit from the use of the CHAID approach. However, when the main issue is model accuracy, Bayesian networks should be favoured. Finally, at this stage, the reader needs to adopt these techniques and particularly the BNT classifier in the context of other research domains with caution. At the current date, the performance of BNT was not examined in a broader context than the Albatross model. Therefore, additional and further research should examine the behaviour of the technique on more datasets and in more research domains.

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Chapter 8

Bounded Rationality in Dynamic Traffic Assignment WY Szeto, Yi Wang and Ke Han

Abstract Purpose  This chapter explores a descriptive theory of multidimensional travel behaviour, estimation of quantitative models and demonstration in an agent-based microsimulation. Theory  A descriptive theory on multidimensional travel behaviour is conceptualised. It theorizes multidimensional knowledge updating, search start/stopping criteria and search/decision heuristics. These components are formulated or empirically modelled and integrated in a unified and coherent approach. Findings  The theory is supported by empirical observations and the derived quantitative models are tested by an agent-based simulation on a demonstration network. Originality and value  Based on artificially intelligent agents, learning and search theory and bounded rationality, this chapter makes an effort to embed a sound theoretical foundation for the computational process approach and agent-based micro-simulations. A pertinent new theory is proposed with experimental observations and estimations to demonstrate agents with systematic deviations from the rationality paradigm. Procedural and multidimensional decision-making are modelled. The numerical experiment highlights the capabilities of the proposed theory in estimating rich behavioural dynamics. Keywords: Dynamic traffic assignment; Wardop’s principle; tolerance-based dynamic user optimal principle; nonlinear complementarity problem; departure time choice assignment

Bounded Rational Choice Behaviour: Applications in Transport Copyright r 2015 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISBN: 978-1-78441-072-8

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8.1.

Introduction

The aim of this chapter is to introduce boundedly rational dynamic user equilibrium traffic assignment to the beginners of this research area. The chapter has been prepared mainly based on the papers written by Szeto and Lo (2004, 2006), Szeto and Wong (2011) and Han, Szeto, and Friesz (2014). It firstly depicts dynamic traffic assignment (DTA) and its basic components, presents the literature of traffic assignment with bounded rationality, provides a problem formulation and discusses solution methods. The solution existence and uniqueness of the problem are also discussed. Some concluding remarks are finally given. The chapter can be served as a toolkit for the beginners.

8.2.

Dynamic Traffic Assignment

DTA is an important problem as its models have many applications in transportation planning, transportation policy evaluation and real-time traffic control and management. A comprehensive literature review on this problem area includes articles by Cascetta and Cantarella (1993), Peeta and Ziliaskopoulos (2001), Boyce, Lee, and Ran (2001), Szeto and Lo (2005a, 2005b), Jeihani (2007) and Szeto (2013). 8.2.1.

Problem Definition

DTA is a generalisation of static traffic assignment. In simple terms, static traffic assignment is a problem of determining the number of vehicles entering each highway in a specific area per hour (i.e. the vehicular traffic flow or flow pattern on each highway), given the vehicular demand for travel from each of the origins to each of the destinations in the area. In other words, the problem is to assign traffic to different highways according to certain behavioural rules. However, this problem cannot capture the realistic changes in the number of vehicles on the highways over time or the departure time choices of travelers. Therefore, DTA generalises static traffic assignment to determine the time-varying flow on each highway over a study period, given the overall demand for vehicular travel. DTA problems can be roughly divided into day-to-day adjustment (or dynamic) problems (e.g. Ben-Akiva, de Palma, & Kaysi, 1991, and Cascetta, 1989; Horowitz, 1984) and within-day DTA problems (e.g. Chang & Mahmassani, 1988; Friesz, Bernstein, Smith, Tobin, & Wie, 1993, 2013; Lam & Huang, 1995; Long, Huang, Gao, & Szeto, 2013; Ran & Boyce, 1996). Day-to-day adjustment problems are concerned with how the travel decisions of travelers change over days (or periods) and how their route or departure time choices on a particular day (or period) depend on their experience obtained in previous days (or periods). Within-day DTA problems include pure departure time choice problems, pure route choice problems, and simultaneous route and departure time choice problems, in the sense that the travel decision is considered in a typical day and there is no day-to-day adjustment.

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The solution of a within-day DTA problem can also be viewed as the final state solution of a day-to-day adjustment problem, where no traveler can make a better decision than their current one. For pure route choice problems, they can be further classified into the en route adjustment model or the reactive DTA problems (e.g. Ben-Akiva et al., 1991; Pel & Bliemer, 2009; Kuwahara & Akamatsu, 1997) and no en route adjustment problems or the predictive DTA problems (e.g. Friesz, Luque, Tobin, & Wie, 1989, 1993; Lo & Szeto, 2002a; Sumalee, Zhong, Pan, & Szeto, 2011; Szeto, Jiang, & Sumalee, 2011; Szeto & Lo, 2006). The en route adjustment problems allow drivers to switch their routes during their trips in response to having more updated traffic information. For example, a driver will switch to another route if he/she realizes that there is a heavy traffic jam ahead on his/her originally planned route. This adjustment problems contrast with the no en route adjustment problem, which assume that choices do not change during trips and that travelers select routes based on pre-trip information and predicted travel times. The focused problem in this chapter is actually within-day DTA problems with no en route adjustment consideration, but the literature review also includes those related to day-to-day DTA problems and en-route adjustment problems. Unless specified otherwise, DTA in the rest of this chapter, except the section of literature review (i.e. Section 8.3), refers to within-day DTA problems without en-route adjustment consideration.

8.2.2.

Example

A simple example of (within-day) DTA is as follows. Figure 8.1 depicts a road network with two nodes and two links. Node A represents the origin and node B represents the destination. The links represent the highways connecting the origin and the destination. Any driver can go from A to B by car via one of the two routes, that is via either Link 1 or Link 2. However, the minimum travel time via Link 2 is 30 minutes less than that via Link 1. If the arrival rate of vehicles at the bottleneck in the middle of Link 2 at any instant is not greater than the capacity (i.e. the maximum number of vehicles that can pass through the bottleneck per hour) of 2000 vehicles per hour, all vehicles can pass through the bottleneck without delay and their travel time is 30 minutes. Otherwise, a queue is formed behind the bottleneck and the travel time via Link 2 is increased. The longer the queue, the higher the travel time. There is also a bottleneck in Link 1 with a capacity of 4000 vehicles per hour and the minimum travel time via Link 1 is 1 hour. It is known that between 6:00 am and 10:00 am, a total of 8000 drivers travel to B from A along both routes. All of these drivers must reach B on or before 9:00 am. These drivers have a choice of departure time in addition to link (or route). They can select a departure time so that they arrive at B at 9:00 pm sharp but they waste a lot of time in queuing. They can also depart early to have less queuing time (or waiting time in queue) and arrive at B early. However, arriving at B too early is not desirable as the time between the arrival time and 9:00 am is wasted at B, because the time can be reserved for other

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8000 drivers A

B

Link 1

Bottleneck with capacity of 4000 veh/h

Figure 8.1: Example network. activities. Given that the demand from A to B during that period is 8000 vehicles, the problem is to determine the numbers of vehicles using Links 1 and 2 over the study period. In other words, the problem is to find out the time-varying demand splits. Note that the splits depend on the travel times on both links and the travel times also depend on the splits. Figure 8.2 represents a solution for this simple example. The cumulative arrival curves represent the total number of drivers entering the links over time, whereas the cumulative departure curves represent the total number of drivers leaving the links over time. The vertical distance between the cumulative arrival and departure curves at a particular time gives the number of vehicles on the link at that time. The horizontal distance between two curves gives the travel time of a particular driver. As the minimum travel time on Link 2 is initially less than that on Link 1, drivers initially select Link 2. As the arrival rate of Link 2 is greater than the capacity of the bottleneck, the travel time on this link increases until it is equal to the minimum travel time of Link 1 of 60 minutes. Then, both links are chosen by drivers and all drivers can reach B before 9:00 am. As can be seen in Figure 8.2, the vertical distance between the cumulative arrival and departure curves is changing over time, meaning that the numbers of vehicles on the two links are changing over time. This is because the queuing time changes over time, which affects the departure time and route choice of drivers. The end result is that the minimum travel time from A to B is 30 minutes and the maximum travel time is 90 minutes. Some drivers depart earlier to have less travel time and queuing time. The first driver to leave A can travel to B without facing congestion and arrives at B at 7:00 am, whereas the last driver leaves A at 7:30 am and requires a travel time of 90 minutes to arrive at B sharply at 9:00 am. Furthermore, the number of drivers using each of the two links (which is the height of the curve) is equal to 4000 and the sum is equal to the demand of 8000 vehicles. However, in general, the usage of each link may not be the same. This example also illustrates that DTA consists of two main components, namely travel choice and traffic flow. The travel choice component determines the traffic flow level on each road at each instant of time, given the road network performance in terms of the time-varying travel times on each road. The traffic flow component depicts how vehicular traffic propagates inside a road network, given the demand split to each route over time, and governs the performance of the road network, in the sense that more traffic on a link results in a higher travel time. The output of

Bounded Rationality in Dynamic Traffic Assignment Cumulative number of drivers

Cumulative arrivals at link 1

167

Cumulative departures from link 1

4000 8000 Time Cumulative number of drivers Cumulative arrivals at link 2

6:00

6:30

2000

4000

7:00

7:30

8:00

Cumulative departures from link 2

9:00

Time

Figure 8.2: Cumulative arrivals and departures on Links 1 and 2 (Szeto & Wong, 2011). the travel choice component is the input of the traffic flow component while the output of the traffic flow component is the input of the travel choice component. DTA is then used to determine the flow pattern that satisfies the two components simultaneously. In this example, the demand of 8000 vehicles is required to propagate on either one of the two links. The travel choice component determines the split of the demand (or the number of vehicles entering each link) over time, based on the timevarying travel time of each link. Then, the traffic flow component propagates each vehicle on the respective links and determines the travel time. The travel time on each link over time must be the same as that used to determine the time-varying demand splits if the splits are optimal. Otherwise, a new set of splits based on the output of the traffic flow component should be used to determine the traffic flow on each link over time.

8.2.3.

Travel Flow Component

The traffic flow component depicts how traffic propagates within the transport network and governs the network performance in terms of travel time given route flows or link flows. This component can be represented by two approaches: point queue (e.g. Friesz et al., 1989; Han, Friesz, & Yao, 2013a, 2013b; Ran & Boyce, 1996) and physical queue (e.g. Kuwahara & Akamatsu, 1997; Szeto, 2008). The point-queue1

1. In the literature, point queues usually refer to deterministic queues. We use a more general definition here.

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(including but not limiting to deterministic-queue) representation treats vehicles as points without physical lengths, whereas the physical-queue representation considers the vehicle lengths. Because of the difference in their assumption, only the physicalqueue representation can capture junction blockage and queue spillback. Both the point-queue and physical-queue representations of traffic dynamics can be modelled as side constraints in the solution set or a unique mapping of route (link) flows yielding route (link) travel times. Representing traffic dynamics as side constraints explicitly is cumbersome and makes solutions difficult to be obtained efficiently (Lo & Szeto, 2002a). Modelling the traffic-flow component as a unique mapping of route (link) flows opens up a new way to analyse DTA problems (e.g. Bressan & Han, 2013; Han, Friesz, & Yao, 2013c; Lo & Szeto, 2002a, 2012b, 2004, 2005; Szeto & Lo, 2004). The outputs of this mapping are route (link) travel times. This means that in this approach, route (link) travel times are modelled as a function of route (link) flows. For the route-based approach, the unique mapping can be expressed as: n = ΦðfÞ

ð8:1Þ

where f is the route flows; n is route travel times and ΦðfÞ is a unique travel time mapping from route flows through a point-queue or physical-queue traffic flow model. There are two advantages of this approach. First, it can automatically ensure the consistency between link travel times and link exit flows in DTA because link travel times are uniquely derived from exit link flows. Second, this approach allows us to determine the existence and uniqueness of solutions of DTA problems directly by simply checking whether the unique mapping is continuous and strictly monotonic respectively. For more discussion on this component, please see Mun (2007) and Szeto (2003).

8.2.4.

Wardrop’s First Principle and its Dynamic Extensions

Traditionally, the travel choice component of DTA is developed based on Wardrop’s (1952) first principle of static traffic assignment. Wardrop’s first principle or the user equilibrium principle states that the journey times on all routes actually used are equal and are not greater than those which would be experienced by a single vehicle on any unused route. In other words, the travel times of all used routes between the same origin-destination (OD) pair are equal and minimal. This principle assumes that each traveler is identical, non-cooperative and rational in selecting the shortest route, and knows the exact travel time he/she will encounter. If all travelers select routes according to this principle the road network will be at equilibrium, such that no one can reduce their travel times by unilaterally choosing another route of the same OD pair. This principle has been extended to consider generalised travel cost instead of travel time, where generalised travel cost can include the monetary cost of in-vehicle travel time, tolls, parking charges and fuel consumption costs.

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User equilibrium can also be explained from the economic concept of utility maximisation (Oppenheim, 1995), where utility measures the degree of satisfaction travelers derive from their choices. In the simplest case, a traveler’s utility equals his/her budget or income minus the travel time. In this sense, the assumption of Wardrop’s first principle can be viewed as travelers selecting routes to maximise their individual utility and, at equilibrium, no traveler can change his/her route to obtain a higher utility. The following equilibrium principles for the travel choice component of DTA can be considered as simple dynamic extensions of the travel choice principles adopted in static traffic assignment: • the dynamic user equilibrium (DUE) or dynamic user optimal (DUO) route choice principle (Friesz et al., 1989); • the DUE departure time choice principle (Vickrey, 1969) and • the DUE route/departure time choice principle (Mahmassani & Herman, 1984). The dynamic user equilibrium (DUE) route choice principle, which is the simplest dynamic extension of Wardrop’s (1952) first principle, states that for each origindestination pair, any routes used by travelers departing at the same time must have equal and minimal travel time. This principle is used when the demand at each departure time is known. That is, the principle is often used in the pure dynamic route choice problem. The DUE departure time choice principle considers departure time choice instead of route choice. This principle requires that the generalised travel costs for travelers between the same OD pair departing at any time are equal and minimal. The principle is adopted when the route choice is pre-determined or there is no route choice for travelers (i.e. the pure departure time choice problem) and the generalised travel cost normally includes the penalty cost due to early and/or late arrivals in addition to the travel time cost. The DUE route/departure time choice principle considers departure time choice in addition to route choice, and considers generalised travel cost instead of travel time. This principle states that for each OD pair, the generalised travel costs incurred by travelers departing at any time using any route are equal and minimal. This principle is essentially a generalisation of the DUE route choice principle and the DUE departure time choice principle, and can be used in the simultaneous route and departure time choice problem. In fact, the example discussed in the introduction is constructed using this principle. Assuming that 1 minute of travel time is equivalent to 1 dollar and the cost of early arrival is 0.5 dollar per minute, the minimum generalised travel cost is equal to 90 dollars for all drivers. The travel time cost for the last driver is 90 dollars as the travel time is 90 minutes. The penalty cost for the last driver is 0 dollar, as he/she arrived at B at 9:00 am sharp. The travel time cost for the first driver is 30 dollars and the penalty cost is 60 dollars (i.e. 120 minutes multiplied by 0.5 dollar per minute). If this driver chose Link 1 instead, he/she would arrive at B at 7:30 am, and his/her travel time cost and penalty cost would become 60 dollars and 45 dollars, respectively, leading to a general travel

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cost of 105 dollars. Therefore, the choice of departing at 6:30 am and going to B via Link 1 is not optimal and no driver picks this choice at equilibrium. To sum up, this section briefly reviews the common types of travel choice principles used in DTA. For a more detailed review on this component, please refer to Szeto and Wong (2011).

8.3.

Bounded Rationality in Traffic Assignment

As a relaxation of the otherwise restrictive Wardropian assumption, the notion of bounded rationality was proposed by Simon (1957, 1990, 1991) and introduced to traffic modelling by Mahmassani and Chang (1987). In prose, the common notion of bounded rationality postulates a range of acceptable travel costs that, when achieved, do not incentivize travelers to change their departure times or route choices. Such a range is phrased by Mahmassani and Chang (1987) as ‘indifference band’. The width of such band, usually denoted by ɛmax , is either derived through a behavioural study of road users (for example, by surveys) or calibrated from empirical observation through inverse modelling techniques. It could be defined differently for each traveler (e.g. Mahmassani & Chang, 1987), on the traffic flow level, and on the origin-destination pair (e.g. Ge & Zhou, 2012). Bounded rationality has gradually become a major field of inquiry especially in static traffic assignment, with an incomplete list of papers including Karakostas, Kim, Viglas, and Xia (2011), Zhang (2011), Di, Liu, Pang, and Ban (2013), Di, He, Guo, and Liu (2014) and Zhao and Huang (2014a, 2014b). Some studies further considered boundedly rational user equilibrium in road toll applications (e.g. Lou, Yin, & Lawphongpanich, 2010) and planning and policy applications (e.g. Arslan & Khisty, 2005; Gifford & Checherita, 2007; Marsden, Frick, May, & Deakin, 2012). Bounded rationality has been examined and used in both within-day and day-today DTA frameworks. It was investigated via simulation-based approaches in the venue of within-day dynamic modelling (Gao, Frejinger, & Ben-Akiva, 2011; Hu & Mahmassani, 1997; Jayakrishnan, Mahmassani, & Hu, 1994; Jou, Lam, Liu, & Chen, 2005; Mahmassani & Jayakrishnan, 1991; Mahmassani & Liu, 1999; Mahmassani, Zhou, & Lu, 2005; Nakayama, Kitamura, & Fujii, 2001; Srinivasan & Mahmassani, 1999). It was also used in within-day DTA models (e.g. Ge, Sun, Zhang, Szeto, & Zhou, 2014; Ge & Zhou, 2012; Han et al., 2014; Szeto, 2003; Szeto & Lo, 2006). These models do not consider en-route adjustment. Ridwan (2004) applied the theory of fuzzy systems to the study of bounded rationality with the consideration of en-route adjustment. Fonzone and Bell (2010) proposed an en-route adjustment model for passenger traffic assignment. Others considered boundedly rational user equilibrium in day-to-day dynamic problems (e.g. Di, Liu, Ban, & Yu, 2014; Guo & Liu, 2011; Han & Timmermans, 2006; Wu et al., 2013; Yang & Jayakrishnan, 2013) and their toll applications (e.g. Guo, 2013). The notion of bounded rationality was used rather imprecisely (in terms mathematics) during the early days of within-day DTA research and defined by various

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ways. In particular, bounded rationality was studied in a so-called laboratory setting by Mahmassani and Chang (1987), but they did not provide a meaningful mathematical articulation of bounded rationality for within-day DTA. They defined ɛ max differently for each traveler and named the resultant equilibrium ‘boundedly rational user equilibrium’. Bounded rationality was used in a similarly ad hoc fashion for simulations by Mahmassani and Jayakrishnan (1991), Peeta and Mahmassani (1995), Mahmassani and Liu (1999), and Chiu and Mahmassani (2002), in which efforts were again limited by the lack of a complete mathematical model of bounded rationality for within-day DTA. Recognising the lack of a theory of traffic assignment that directly incorporates bounded rationality, Ridwan (2004) applied the theory of fuzzy systems to the study of bounded rationality. Bogers, Viti, and Hoogendoorn (2005), again driven by the lack of a suitable theory, conducted more laboratory studies about bounded rationality on route choice. For analytical within-day DTA models, Szeto (2003) and Szeto and Lo (2006) proposed a mathematical model for the route-choice boundedly rational dynamic user equilibrium (BR-DUE) traffic assignment problem, although the authors did not take into account drivers’ departure time choices. ɛmax was defined on the traffic flow level, irrespective of OD pairs, and the resultant equilibrium principle was referred to as tolerance-based DUO principle. The route-choice BR-DUE traffic assignment problem was formulated as a discrete-time nonlinear complementarity problem in the study of Szeto and Lo (2006), where a heuristic routeswapping algorithm adapted from Huang and Lam (2002) was proposed to solve the problem. Ge and Zhou (2012) considered the route-choice BR-DUE traffic assignment problem with signals and endogenously determined tolerances by allowing the width of the indifference band ɛmax to depend on OD pair, departure time and the actual path flows. Ge et al. (2014) extended the concept of DUO with variable tolerances to more scenarios, including the discontinuity of path travel times or costs, too high demand levels, capacity shortage or sharp change, etc. Therefore, the applications of this concept are not limited to signalised road networks any more. Ge et al. (2014) focused on a comparison of the concept of DUO with variable tolerances to DUE and tolerance-based DUO, including the differences between the three alternative definitions of DUO, their relationships and the existence conditions of these DUO states. However, no solution method was proposed in these two papers. Contributions by Szeto (2003), Szeto and Lo (2006), Ge and Zhou (2012) and Ge et al. (2014) achieved enhanced (yet partial) integration of bounded rationality and DUE but did not establish and analyse a complete theory, where by ‘complete’ we mean a mathematical formulation consistent with known empirical results and surmised behaviours, qualitative properties and a computational approach that is demonstrably effective. To the best of our knowledge, there has not been any analytical treatment of the simultaneous route and departure time choice BR-DUE traffic assignment problem with exogenous tolerances or with variable (endogenous) tolerance in the literature, in terms of formulation, qualitative properties and computation until Han et al. (2014) which successfully bridged such a gap in the literature by providing the first

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complete analytical framework capable of formulating BR-DUE problems into canonical mathematical forms and continuous time, analysing their qualitative properties and computing solutions with convergent algorithms. As a final remark in this section, although Mahmassani and Chang (1987) defined ɛ max at the individual level and Szeto and Lo (2006) defined it at the flow level, their terminologies ‘boundedly rational user equilibrium’ and ‘tolerance-based user equilibrium’ were used interchangeably in the traffic assignment literature.

8.4.

Boundedly Rational Dynamic User Equilibrium Route Choice Assignment

Boundedly rational dynamic user equilibrium route choice assignment can be considered a special case of (within-day) DTA, in which the travel choice principle or route choice principle is defined by the tolerance-based dynamic user optimal principle.

8.4.1.

Tolerance-Based Dynamic User Optimal Principle

The traditional DUO principle (e.g. Ran & Boyce, 1996) requires that all used paths between the same OD pair have equal and minimum travel time. As a relaxation, the tolerance-based DUO principle only requires the travel times of all used routes between the same OD pair to be similar, or within an acceptable tolerance ɛmax from the minimum OD route travel time, where the tolerance level is purely a function of the behaviour of the network users. The relaxation recognises the important fact that it is physically impossible to always fulfil the requirement that all used routes on the same OD pair have exactly the same travel time, as is demonstrated in a study with the more realistic physical queue representation (Szeto, 2003). This relaxation, adapted from the boundedrationality behavioural notion, can be expressed as: If fprs ðtÞ > 0; then ηrs p ðtÞ − πðtÞ ≤ ɛ max rs ηrs p ðtÞ − π ðtÞ ≥ 0;

∀ rs; p; t

ð8:2Þ ð8:3Þ

where fprs ðtÞ and ηrs p ðtÞ are respectively the flow between OD pair rs entering route p at time t and the corresponding route travel time; π rs p ðtÞ is the shortest OD travel time between OD pair rs for flows departing at time t; ɛmax is the acceptable tolerance, a non-negative parameter obtained through travel behaviour surveys and experiments. In Eqs. (8.2) and (8.3), t is an instant of time; in this chapter, t is a time-slice index, as we consider a discrete-time DTA formulation. Condition (8.3) is included in this principle to ensure π rs ðtÞ to be the shortest OD

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travel time among all the possible routes between OD pair rs for flows departing at time t. By employing the following transformation function:  0 if 0 ≤ y ≤ u ɛðy; uÞ = ð8:4Þ y if y > u where u and y are independent non-negative variables, the tolerance-based DUO principle can be alternatively formulated as: rs fprs ðtÞ⋅ɛðηrs p ðtÞ − π ðtÞ; ɛ max Þ = 0;

rs ɛðηrs p ðtÞ − π ðtÞ; ɛ max Þ ≥ 0;

∀ rs; p; t ∀ rs; p; t

ð8:5Þ ð8:6Þ

Condition (8.6) implies condition (8.3) due to the requirement of the non-negative independent variable to the transformation function. Condition (8.5) implies (8.2), meaning that the travel time of a used route is greater than the minimum route travel time by not more than an acceptable level ɛ max . According to (8.5), if route p carries a positive flow at time t (i.e. fprs ðtÞ > 0), the transformation function rs rs rs ɛðηrs p ðtÞ − π ðtÞ; ɛ max Þ must be equal to zero, implying 0 ≤ ηp ðtÞ − π ðtÞ ≤ ɛ max due to the first condition of Eq. (8.4). In other words, if route p carries a positive flow at time t, the travel time of route p is greater than the minimum route travel time by not more than an acceptable level ɛmax . Note that if route p carries zero flow at rs time t (i.e. fprs ðtÞ = 0), ɛðηrs p ðtÞ − π ðtÞ; ɛ max Þ must be nonnegative due to Eq. (8.6) and rs hence the route travel time ηp ðtÞ must be greater than or equal to the minimum route travel time π rs ðtÞ. As a special case, if ɛmax equals zero, conditions (8.5)(8.6) can be reduced to the following: rs fprs ðtÞ⋅ɛðηrs p ðtÞ − π ðtÞ; 0Þ = 0;

rs ɛðηrs p ðtÞ − π ðtÞ; 0Þ ≥ 0;

∀ rs; p; t ∀ rs; p; t

ð8:7Þ ð8:8Þ

According to Eq. (8.4), we have  rs ɛðηrs p ðtÞ − π ðtÞ; 0

=

rs 0 0 ≤ ηrs p ðtÞ − π ðtÞ ≤ 0 rs = ηrs rs rs rs rs p ðtÞ − π ðtÞ ηp ðtÞ − π ðtÞ if ηp ðtÞ − π ðtÞ > 0

Therefore, Eqs. (8.7)(8.8) are simplified to rs fprs ðtÞ½ηrs p ðtÞ − π ðtÞ = 0;

∀ rs; p; t

ð8:9Þ

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∀ rs; p; t

ð8:10Þ

or If fprs ðtÞ > 0;

rs then ηrs p ðtÞ − π ðtÞ = 0

rs ηrs p ðtÞ − π ðtÞ ≥ 0;

∀ rs; p; t

ð8:11Þ ð8:12Þ

Conditions (8.9)(8.10) are the ideal DUO conditions. That is, if ɛmax equals zero, Eqs. (8.5)(8.6) can be reduced to the ideal DUO conditions. According to this result, the tolerance-based principle is a generalisation of the traditional DUO principle.

8.4.2.

Nonlinear Complementarity Problem (NCP) Formulation

For fixed demands, the DTA problem with the tolerance-based DUO route choice principle is to find a route flow vector f  such that: rs fprs ðtÞ⋅ɛðηrs p ðtÞ − π ðtÞ; ɛ max Þ = 0; rs ɛðηrs p ðtÞ − π ðtÞ; ɛ max Þ ≥ 0;

X

fprs ðtÞ = qrs ðtÞ;

∀ rs; p; t ∀ rs; p; t

ð8:13Þ ð8:14Þ

∀ rs; t

ð8:15Þ

∀ rs; p; t

ð8:16Þ

p rs ηrs p ðtÞ = Φp;t ðfÞ;

f ≥0

ð8:17Þ

where qrs ðtÞ is the demand of OD pair rs at time t; Φrs p;t ð⋅Þ is a unique mapping yielding route travel times for a given route flow vector f. Eqs. (8.13)(8.14) express the tolerance-based DUO principle. Eqs. (8.15)(8.17) are the flow conservation and non-negativity conditions. Eq. (8.16) considers the traffic flow component as a unique functional mapping yielding route travel times for given route flows. The tolerance-based DUO route choice problem (8.13)(8.17) can be reformulated as an NCP by introducing three more conditions. By attaching π rs ðtÞ to the flow conservation condition (8.15), we obtain " # X rs rs rs π ðtÞ fp ðtÞ − q ðtÞ = 0; ∀ rs; t ð8:18Þ p

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P rs As π rs ðtÞ, the shortest OD travel time, must be greater than zero, p fp ðtÞ − qrs ðtÞ = 0 must hold at optimality. Adding this to the problem (8.13)(8.17) will not alter the optimality condition. For mathematical completeness, we also introduce two more conditions to the original problem: π rs ðtÞ ≥ 0; X

∀ rs; t

fprs ðtÞ − qrs ðtÞ ≥ 0;

ð8:19Þ ∀ rs; t

ð8:20Þ

p

P They include π rs ðtÞ > 0 and p fprs ðtÞ − qrs ðtÞ = 0 as special cases and therefore do not change the optimality condition of the original problem ((8.13)(8.17)). The DUO route choice problem with fixed demands can then be considered as the problem of finding a route flow vector f  to satisfy Eqs. (8.13)(8.20). By putting Eq. (8.16) into Eqs. (8.13) and (8.14), the DUO route choice problem with fixed demands (8.13)(8.20) can be written as an NCP: to find x such that   f x= ≥0 ð8:21Þ π xT ⋅Fðx ; Φðf  ÞÞ = 0

ð8:22Þ

Fðx ; Φðf  ÞÞ ≥ 0

ð8:23Þ

and

where

ðf rs ðtÞ; ∀ rs; p; tÞ x = rsp π ðtÞ; ∀ rs; tÞ

 ð8:24Þ

"

rs ∀ rs; p; tÞ ðɛðΦrs p;t ðfÞ − π ðtÞ; ɛ max Þ; Fðx; ΦðfÞÞ = P rs rs ∀ rs; t p fp ðtÞ − q ðtÞ;

# ð8:25Þ

and ΦðfÞ is a vector of ðΦrs p;t ðfÞ; ∀ rs; p; tÞ; π is a vector of lowest OD travel times, rs rs rs and; fprs ðtÞ, ηrs t ðtÞ, π ðtÞ, ɛ max , q ðtÞ, and Φp;t ð⋅Þ follow earlier definitions. 8.4.3.

Solution Existence and Uniqueness

For ease of explanation, we first define a non-negative gap function HðfÞ: rs rs HðfÞ = max½δrs p ðtÞ⋅ðηp ðtÞ − π ðtÞÞ;

∀ rs; p; tÞ

ð8:26Þ

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where the indicator variable δrs p ðtÞ equals 1 if the flow on route p between OD pair rs at time t is nonzero, and equals zero otherwise. This gap is the largest difference between the travel time of all used routes and the corresponding shortest OD travel times under the route flow pattern f, and measures how far the current solution is from the traditional DUO solution. To analyse the existence and uniqueness of solutions, we define the theoretical gap (TG), which is the minimum of the largest difference between the travel time of all used routes and the corresponding shortest OD travel times of all feasible flow patterns. In other words, TG is the smallest gap of all the feasible solutions. Mathematically, the theoretical gap (TG) is expressed as follows: TG = min HðfÞ ≥ 0 f

ð8:27Þ

The value of the theoretical gap depends on the network and the demand pattern. In DTA formulations with the more realistic physical-queue representation, the route travel time functions may not be always continuous with respect to the route flows, rendering the existence of solutions not always possible (Szeto, 2003), or equivalently their theoretical gap never reaches zero. For the tolerance-based DUO route choice problem with physical queues, in a similar manner, solution existence is not guaranteed. A solution exists to the problem if and only if the theoretical gap is less than or equal to ɛ max : TG = min HðfÞ ≤ ɛmax f

ð8:28Þ

According to Eq. (8.28), the existence of solutions to the physical-queue tolerance-based DUO problem depends on the theoretical gap (which is related to the network topology and demand pattern) and the parameter ɛmax (which is related to the behaviour of the network users). In general, as ɛmax and hence constraint (8.28) are gradually relaxed, solution existence gradually becomes easier. However, we emphasise the importance of specifying the tolerance level from a behaviour perspective rather than as a numerical means for obtaining equilibrium solutions. If the tolerance level is specified at a level higher than the actual behaviour, then the solution will stop at a premature ‘equilibrium’, even though in reality, travelers are still swapping routes in the search of better ones. For point queue DTA models, the route travel time is always continuous with respect to route flows. The solution set is convex. Once a continuous transformation function (Han et al., 2014) is used to reformulate the problem, a solution always exists to the problem regardless of the value of ɛmax . In summary, solution existence is both a function of the network topology and travel demand, the behavioural tolerance of the users on route swapping, and choice of travel flow models. On the issue of uniqueness of solution, as one can see, when ɛmax approaches infinity, all feasible gap values satisfy (8.28) and hence the corresponding flows are solutions, implying that multiple solutions are possible in this problem. In fact, even if

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the tolerance is zero, multiple solutions can still be possible, similar to the case of the traditional DUO problem (Lo & Szeto, 2002a; Szeto, 2003).

8.4.4.

Solution Method

Three approaches can be employed to obtain tolerance-based DUO solutions. The first one is to solve the formulation (8.21)(8.25) directly. Since the transformation function is not continuous and differentiable, it is difficult to develop convergent algorithms for the problem. One may transform the NCP formulation into a mathematical programme through a gap function, and solving the resultant mathematical programme through global optimisation algorithms such as genetic algorithm (e.g. Lo & Szeto, 2002b) and simulated annealing algorithm. Solution efficiency of this approach is an issue to be addressed. The second approach is that the NCP formulation is to reformulate the NCP to a variational inequality or fixed point problem using Theorem 1.3 and Proposition 1.4 in Nagurney (1993) and then using existing algorithms developed for these problems such as projection algorithm (e.g. Han & Lo, 2002) to solve the tolerance based problem. The third approach is to employ existing solution algorithms that maintain feasibility of the traditional DUO route choice problem while using the new stopping rule as: Stop if HðfÞ ≤ ɛmax That is, the solution procedure of the traditional DUO route choice formulation is repeated until a tolerance-based DUO solution is found. This approach is based on the fact that the traditional DUO solution (if exists) is also a solution to the tolerance-based DUO route choice problem. For this purpose, we employ a search direction developed within the context of traditional DUO problems to solve the tolerance-based DUO problem. The algorithm stops when the gap reaches the tolerance level. The heuristic algorithm depicted in this chapter was proposed by Szeto and Lo (2006) and is modified from the route-swapping algorithm in Huang and Lam (2002). The iterative algorithm is analogous to the day-to-day route-swapping process (e.g. Smith & Wisten, 1995) and the route choice adjustment process based on projected dynamical systems (e.g. Nagurney & Zhang, 1997a, 1997b). The advantage of this algorithm is that no matter the tolerance-based or traditional DUO solutions exist or not, the algorithm can simulate day-to-day flow patterns in addition to the within-day flow patterns. That is, the transition from disequilibrium to equilibrium or one state of disequilibrium to another can be described. This allows us to study the daily variations in network performance. The basic idea of the algorithm is that a portion of travelers on the unacceptable routes swaps to the acceptable routes. The specific swaps are

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proportional to three factors: the flows on the current route, the difference in travel time between the current route and the shortest route, and the swapping rate ρ, which is a parameter representing travelers’ sensitivity to the travel time differences. Higher swapping rates indicate travelers’ higher sensitivities to the travel time differences, or higher eagerness to swap routes. Moreover, travelers farther from the acceptable travel time and on routes with larger flows are more inclined to change their routes than otherwise. Due to flow non-negativity, the flows swap out of each rs rs unacceptable route are the minimum of ρfprs ðtÞ × ½ηrs p ðtÞ − π ðtÞ and fp ðtÞ. The detailed algorithmic steps are the following: Step 1: Set the iteration counter τ = 1. Choose the initial route flow fprs ðtÞτ . Step 2: Determine the route travel time ηrs p ðtÞτ through the procedures of network loading and route travel time extraction. Find π rs ðtÞτ = min½ηrs p ðtÞτ ; ∀ p. Step 3: Update the route flow as below: rs fprs ðtÞτ þ 1 = max½0; fprs ðtÞτ − ρfprs ðtÞτ ½ηrs p ðtÞτ − π ðtÞτ ;

fprs ðtÞτ þ 1 = fprs ðtÞτ þ

ψ rs ðtÞ ; jPrs t;τ j

p ∈ Prs =Prs t;τ

p ∈ Prs t;τ

where ψ rs ðtÞτ =

X p∈Prs =Prs t;τ

½fprs ðtÞτ − fprs ðtÞτ þ 1 

rs rs Prs t;τ = fp : ηp ðtÞτ − π ðtÞτ ≤ ɛ max g

Prs is the set of all feasible path between OD pair rs. Step 4: Stop if HðfÞτ ≤ ɛmax or τ = τmax , rs rs where HðfÞτ = max½δrs ∀ rs; p; t and p ðtÞτ ⋅ðηp ðtÞτ − π ðtÞτ Þ; τmax is the maximum number of iterations; Otherwise, set τ = τ þ 1 and return to Step 2. In Step 1, the initial solution can be obtained by many methods. As an example, one can determine the initial solutions based on either an all-or-nothing or average loading assignment procedure. All-or-nothing assignment loads all demands on the shortest time routes, assuming that the travel time is not flow-dependent. Average loading assignment ignores the travel time and sets the flow on a route between an OD pair to be the demand divided by the number of routes between that OD pair. Step 2 does not restrict the choice of traffic flow models to determine the route travel times. As an example, one can employ CTM as the dynamic traffic flow model. To determine the route travel times from route flows, one can refer to Lo (1999), Lo and Szeto (2002a, 2002b) and Long, Gao, and Szeto (2011) for the details of encapsulating CTM in DTA and/or the travel time determination procedure.

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In Step 3, the algorithm first deducts flows from unacceptable routes based on the basic idea described before, and then distributes (induces) these flows equally to rs the acceptable routes in Prs t;τ . The sum of deduction ψ ðtÞτ should however be equal to the sum of induction to ensure the feasibility of the solution during the adjustment process. Step 4 checks whether the stopping condition is reached. The algorithm terminates when the current gap is not larger than the tolerance or the maximum number of iterations is reached. Otherwise, the algorithm increases the iteration counter by one and repeats the procedure.

8.5.

Boundedly Rational Dynamic user Equilibrium Route and Departure Time Choice Assignment

The formulation in Section 8.4 can be easily extended to consider both route and departure time choices. Instead of using route travel times to define tolerance based equilibrium, we use generalised route travel costs, which includes travel time and other costs. As in the literature (e.g. Yang & Meng, 1998), travelers going to the same destination s are assumed have similar desired arrival times, expressed as the arrival time window ½t~s − Δs ; t~s þ Δs , where t~s is the desired arrival time and Δs is the interval of arrival time flexibility. Travelers acquire no schedule delay cost cs ðtÞ if they arrive within the desired arrival time window. Otherwise, they incur schedule delay costs for both early and late arrivals outside of this arrival time window. Various schedule delay functions can be used to model this. In this chapter, we model it by a piecewise linear function: 8 ρ ½ðt~ − Δs Þ − ðt þ ηrs > p ðtÞÞ > < s s cs ðtÞ = 0 > > : ψ ½ðt þ ηrs ðtÞÞ − ðt~ þ Δ Þ s

p

s

s

if t~s − Δs > t þ ηrs p ðtÞ ~ if t~s − Δs ≤ t þ ηrs p ðtÞ ≤ ts þ Δs if

ð8:29Þ

t~s þ Δs < t þ ηrs p ðtÞ

whereρs and ψ s correspond to the unit costs of early and late arrival for travelers heading for destination s. Figure 8.3 shows the schedule delay cost as a function of arrival time. Note that αs , ρs , ψ s , Δs , and t~s are independent of origin r but only dependent on destination s, meaning that travelers heading to the same destination have the same desired arrival time window and the same schedule delay cost function. This assumption, as proposed by Yang and Meng (1998), is reasonable for morning commute traffic. Commuters heading for the same office have similar desired arrival times, regardless of their resident locations. This assumption, nevertheless, can be relaxed by inserting additional variables to segregate travelers by classes, with each class having different desired arrival time windows, different schedule delay functions, or different origins. This relaxation can be accommodated without major conceptual difficulty.

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Punctual arrival

Late arrival

Schedule delay cost

Early arrival

ψs 1

ρs 1 ts − Δ s

ts

ts + Δ s

Arrival time

Figure 8.3: The schedule delay cost as a function of arrival time (Szeto & Lo, 2004).

The total generalised travel costs of each traveler make up of two parts: (i) the travel time cost and (ii) the schedule delay cost, expressed as: rs ϕrs p ðtÞ = αs ηp ðtÞτ þ cs ðtÞ;

∀ rs; p; t

ð8:30Þ

where ϕrs p ðtÞ and αs are, respectively, the generalised travel cost for travelers between OD pair rs departing at t entering route p and the cost of unit travel time for travelers heading for destination s. In general, different parameters and shapes of the cost functions were used in previous studies (Bernstein, Friesz, Tobin, & Wie, 1993; Ran & Boyce, 1996; Vythoulkas, 1990; Wie et al., 1995). Small (1982) found the following the empirical results: ψ s > αs > ρs > 0

ð8:31Þ

That is, the unit cost of late arrival (ψ s ) is higher than the cost of unit travel time (αs ), which is in turn higher than the unit cost of early arrival (ρs ). For fixed demands, the tolerance-based DUO route and departure choice DTA problem is to find a route flow vector f  to satisfy Eqs. (8.15)(8.17), (8.29)(8.31) and the following: rs 0 fprs ðtÞ⋅ɛðϕrs p ðtÞ − ϕ ; ɛ max Þ = 0;

rs 0 ɛðϕrs p ðtÞ − ϕ ; ɛ max Þ ≥ 0;

X

qrs ðtÞ = Qrs ;

∀ rs; p; t ∀ rs; p; t

∀ rs

ð8:32Þ ð8:33Þ

ð8:34Þ

t

where Qrs is the total demand of OD pair rs during the modelling period. ϕrs is the minimum generalised travel cost for travelers between OD pair rs. Note that this

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cost is independent of departure time. ɛ0 max is the acceptable tolerance in terms of travel cost. Following the technique in Section 8.4, the DUO route and departure time choice problem with fixed demands can be written as an NCP: to find x such that   f y= ð8:35Þ μ yT ⋅Gðy ; Φðf  ÞÞ = 0

ð8:36Þ

Gðy ; Φðf  ÞÞ ≥ 0

ð8:37Þ

and

where

y=

ðfprs ðtÞ; ∀ rs; p; tÞ ϕrs ; ∀ rs



3 rs 0 ∀ rs; p; tÞ ðɛðϕrs p ðtÞ − ϕ ; ɛ max Þ; 6 7 Gðy; Φðf  ÞÞ = 4 X X f rs ðtÞ − Qrs = 0; ∀ rs 5 p

ð8:38Þ

2

t

ð8:39Þ

p

μ is a vector of minimum generalised OD travel cost, that is μ = ðϕrs ; ∀ rsÞ, and ϕrs p ðtÞ is defined by Eqs. (8.16), (8.29)(8.31). As we can see, the problem structure is similar to that for the route choice case. Hence, the discussion on the existence and uniqueness of solutions and solution methods are similar to those for the route choice case and are omitted here. Moreover, when each OD pair has only one route, the simultaneous route and departure time choice problem reduces to a pure departure time choice problem case. Hence, the formulation for the pure route choice problem is also excluded in this chapter.

8.6.

Concluding Remarks

This chapter introduces the tolerance based principles for DTA and boundedly rational dynamic user equilibrium traffic assignment problems. The problems are formulated as NCPs. The existence and uniqueness of solutions are discussed. Some solution methods are mentioned. A brief literature about bounded rationality in traffic assignment and an introduction to DTA and its components are also given. Although the problems can be formulated as NCPs in a discrete time setting, they can be formulated into variational inequality or fixed point problems in either

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a continuous or discrete time setting. In fact, Han et al. (2014) reformulated the simultaneous route and departure time choice problem into a fixed point problem in a continuous time setting. This allows developing a convergent fixed point algorithm to solve their problem. Nevertheless, a sufficient condition for the convergence of the fixed point algorithm is strong monotonicity together with Lipschitz continuity of ‘transformed route cost’, which is too tight and may not be satisfied in most cases. Developing a convergent algorithm using a looser convergence requirement can be a future research direction. Another future research direction can be developed novel formulations for the problems so that more efficient or convergent algorithms can be developed.

Acknowledgements This work is jointly supported by a grant from the National Natural Science Foundation of China (71271183), a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (HKU 716312E) and a grant from the University Research Committee of the University of Hong Kong (201211159009).

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Simon, H. A. (1991). Bounded rationality and organizational learning. Organization Science, 2(1), 125134. Small, K. A. (1982). The scheduling of consumer activities: Work trips. American Economic Review, 72, 467479. Smith, M. J., & Wisten, M. B. (1995). A continuous day-to-day traffic assignment model and the existence of a continuous dynamic user equilibrium. Annals of Operations Research, 60, 5979. Srinivasan, K. K., & Mahmassani, H. S. (1999). Role of congestion and information in tripmakers’ dynamic decision processes: Experimental investigation. Transportation Research Record, 1676(1), 4452. Sumalee, A., Zhong, R. X., Pan, T. L., & Szeto, W. Y. (2011). Stochastic cell transmission model (SCTM): A stochastic dynamic traffic model for traffic state surveillance and assignment. Transportation Research Part B: Methodological, 45, 507533. Szeto, W. Y. (2003). Dynamic traffic assignment: Formulations, properties, and extensions. Doctoral thesis, The Hong Kong University of Science and Technology, Hong Kong. Szeto, W. Y. (2008). The enhanced lagged cell transmission model for dynamic traffic assignment. Transportation Research Record: Journal of the Transportation Research Board, 2085, 7685. Szeto, W. Y. (2013). Cell-based dynamic equilibrium models. In K. Ozbay and S. Ukkusuri (Eds.), Advances in dynamic network modeling in complex transportation systems (pp. 163192). New York, NY: Springer. Szeto, W. Y., Jiang, Y., & Sumalee, A. (2011). A cell-based model for multi-class doubly stochastic dynamic traffic assignment. Computer-Aided Civil and Infrastructure Engineering, 26(8), 595611. Szeto, W. Y., & Lo, H. K. (2004). A cell-based simultaneous route and departure time choice model with elastic demand. Transportation Research Part B: Methodological, 38(7), 593612. Szeto, W. Y., & Lo, H. K. (2005a). Dynamic traffic assignment: Review and future research directions. Journal of Transportation Systems Engineering and Information Technology, 5, 85100. Szeto, W. Y., & Lo, H. K. (2005b). Properties of dynamic traffic assignment with physical queues. Journal of the Eastern Asia Society for Transportation Studies, 6, 21082123. Szeto, W. Y., & Lo, H. K. (2006). Dynamic traffic assignment: Properties and extensions. Transportmetrica, 2(1), 3152. Szeto, W. Y., & Wong, S. C. (2011). Dynamic traffic assignment: Model classifications and recent advances in travel choice principles. Central European Journal of Engineering, 2, 118. Vickrey, W. S. (1969). Congestion theory and transport investment. The American Economic Review, 59, 251261. Vythoulkas, P. C. (1990). Two models for predicting dynamic stochastic equilibria in urban transportation networks. In M. Koshi (Ed.), Transportation and traffic theory (pp. 253272). New York, NY: Elsevier. Wardrop, J. (1952). Some theoretical aspects of road traffic research. In ICE Proceedings: Part II, Engineering Divisions (Vol. 1, No. 3, pp. 325362). Thomas Telford. Wie, B. W., Tobin, R. L., Friesz, T. L., & Bernstein, D. (1995). A discrete time, nested cost operator approach to the dynamic network user equilibrium problem. Transportation Science, 29(1), 7992.

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Wu, J., Sun, H., Wang, D. Z. W., Zhong, M., Han, L., & Gao, Z. (2013). Boundedrationality based day-to-day evolution model for travel behavior analysis of urban railway network. Transportation Research Part C: Emerging Technologies, 31, 7382. Yang, H., & Meng, Q. (1998). Departure time, route choice and congestion toll in a queuing network with elastic demand. Transportation Research Part B: Methodological, 32(4), 247260. Yang, I., & Jayakrishnan, R. (2013). Modeling framework to analyze effect of multiple traffic information service providers on traffic network performance. Transportation Research Record, 2333, 5565. Zhang, L. (2011). Behavioral foundation of route choice and traffic assignment: Comparison of principles of user equilibrium traffic assignment under different behavioral assumptions. Transportation Research Record, 2254, 110. Zhao, C.-L., & Huang, H.-J. (2014a). Modeling bounded rationality in congestion games with the quantal response equilibrium. Procedia-Social and Behavioral Sciences, 138, 641648. Zhao, C.-L., & Huang, H.-J. (2014b). Modeling the equilibrium bus line choice behavior and transit system design with oblivious users. Discrete Dynamics in Nature and Society, 2014, 5.

Chapter 9

Incorporating Bounded Rationality in a Model of Endogenous Dynamics of Activity-Travel Behaviour Ifigenia Psarra, Theo Arentze and Harry Timmermans

Abstract Purpose  This chapter discusses the formulation of an agent-based model to simulate day-to-day dynamics in activity-travel patterns, based on short and long-term adaptations to exogenous and exogenous changes. Theory  The model is based on theoretical considerations of bounded rationality. Agents are able to explore the area, adapt their aspirations and develop habitual behaviour. If they experience dissatisfaction, stress emerges and this may lead to short or long-term adaptations of an agent’s activity-travel patterns. Both cognitive and affective responses are taken into account, when agents evaluate available options. Moreover, memory-activation and forgetting processes play a significant role in the development of habitual behaviour. Findings  Results of numerical simulations show the effect of memoryactivation and emotion-related parameters on habit formation, on the decisionmaking process and on overall model behaviour. Effects of specific aspects of bounded rationality on the evolution of dynamics in the activity-travel patterns of an individual are illustrated. Effects seem realistic, behaviourally rich and, therefore, more sensitive to a larger spectrum of policies. Originality and value  The model is unique in its kind. It is one of the first attempts to formulate a dynamic model of activity-travel behaviour, based on principle of bounded rationality, which includes both cognitive and affective mechanism of adaptation. Keywords: Dynamic activity-travel pattern; memory decay; aspiration level; learning; habits

Bounded Rational Choice Behaviour: Applications in Transport Copyright r 2015 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISBN: 978-1-78441-072-8

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Introduction

The models of activity-travel behaviour play a significant role in the policy-analysis and policy-making process, as they help to predict both direct and secondary effects of interventions in the spatial, transportation or land-use system of an area. Most operational activity-based models of travel behaviour that have been developed in many countries are static models, which are based on cross-sectional data and can simulate a typical day (Rasouli & Timmermans, 2014). Therefore, the next challenge is to incorporate the notion of dynamics in those models, in order to make them more sensitive and realistic. In the context of the current study, it is assumed that individuals tend to develop habits and subsequently that their activity-travel behaviour has a propensity to reach an equilibrium state (Ga¨rling & Axhausen, 2003). Specifically, people’s behaviour is characterised by inertia, as they avoid to perpetually search for the best alternative within their choice set. However, as the environment where they act is highly stochastic and non-stationary, many factors may lead to dissatisfaction with the current habitual choices. This deviation between the aspirations of an individual and the performance of this habitual state can be regarded as stress (Habib, Elgar, & Miller, 2006). Stress tends to accumulate and can result in the distortion of habitual behaviour and the evolution of dynamics in the activity-travel behaviour of an individual. Specifically, reacting to stress can lead either to short-term or even to longterm adaptations of activity-travel behaviour. The difference between short and long-term change is that the former occurs within a specific context of resources or restrictions (e.g. following a different route during the commuting trip, or going for shopping by bike instead of going by car), while the latter changes this context of long-term decisions which characterises an individual’s life in the mobility-accessibility domain (e.g. deciding to buy a car, or deciding to move to another residential location) (Miller, 2005). Moreover, both a bottom-up and a top-down chain of influence can be noticed between those two time perspectives. Specifically, the long-term decisions influence the possibilities and the constraints within which a short-term decision can be made (top-down process), while the short-term decisions explore and feedback whether this context of long-term decisions is satisfactory enough or whether a long-term change should be decided in order to improve it. Most prior research on dynamics of activity-travel behaviour has either focused on the dynamics that would be triggered by a specific external factor (Klo¨kner, 2004; Timmermans et al., 2014; Verhoeven, Arentze, Timmermans, & van der Waerden, 2005) (e.g. the implementation of a specific type of policy) or on the topdown process of influence (Beige & Axhausen, 2006; Prillwitz & Lanzendorf, 2006) (e.g. how the residential relocation affects the activity-travel patterns of an individual). The model described in this chapter focuses on the bottom-up process of influence and specifically on the evolution of endogenous dynamics, by hierarchically

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linking short and long-term adaptation responses to stress. Specifically, it is modelled how an individual decides to implement short or long-term changes and how they develop habitual behaviour again. In order to reach that goal, several mechanisms of the choice-making process are included in this model. Specifically, both cognitive and emotional responses are considered, the formation of context-specific dynamic choice sets is modelled, learning and forgetting processes are taken into account, as well as the adaptation of beliefs and aspirations, based on previous experiences. Therefore, bounded rationality of individuals is taken into account in various aspects of this model. Actually, the above mentioned bounded rationality mechanisms, in combination with the stochasticity of the system (e.g. generated delay) trigger the dynamics of the model. It should be stated that a detailed description of the model, as well as some basic case numerical simulation results can be found in Psarra, Liao, Arentze, and Timmermans (2014). Moreover, in Psarra, Arentze, and Timmermans (2014a, 2014b, 2014c) there is a focus on specific aspects of the model, such as on the exploration process, on tolerance to short and long-term stress and on choice set formation, respectively. Finally, in Psarra, Arentze, and Timmermans (2014d) there are some first numerical simulation results, depicting the effect of affective responses on the model behaviour. This chapter focuses on two specific aspects of bounded rationality: memory limitations and emotional responses during the decision-making process. Specifically, the effect of specific memory-activation parameters on habit formation process, as well as on the overall behaviour of the model, will be illustrated with some numerical simulation results. Additionally, adding on Psarra et al. (2014d), the effect of specific emotion-related parameters on the evaluation of the various alternatives during the decision-making process, as well as on the overall model behaviour, will also be depicted in some simulation graphs. The remainder of this chapter is organised as follows. First, the model structure is presented, followed by a description of the numerical simulation process and some simulation results. Finally, the chapter is completed with conclusions and discussion about future research.

9.2.

The Model

In this section, we will give a short description of the main parts of the suggested model. However, it was considered important to include the main points in this chapter, in order to introduce the reader to the overall context of endogenous dynamics and the key role of emotional responses and memory-activation mechanisms. In this way, the simulation results that follow in the next section will become clearer. Furthermore, at this point, it should also be stated that the structure of the described model is based to a great extent on the work of Han et al. on the

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modelling of dynamic choice sets of shopping locations (Han, Arentze, & Timmermans, 2007; Han, Arentze, Timmermans, Janssens, & Wets, 2009). The model extends this work, as it does not focus only on the shopping activity type; it models the choice set formation of activity profiles, which constitute combinations of several activity attributes, and models apart from the short-term dynamics, the point of time that a long-term adaptation is considered.

9.2.1.

Activity Profiles and Universal Choice Set

In the context of this model, it is assumed that when an individual travels in order to reach an activity location, (s)he actually chooses from his/her choice set an activity profile and implements it. An activity profile consists of a combination of states of various activity attributes (Arentze, Ettema, & Timmermans, 2007). For instance, two examples of activity profiles that could be included in the choice set of an individual follow: • Shopping, supermarket, going by car, starting-time of trip at 17:00, origin of the trip: office, via route A • Shopping, grocery shop, going by bike, starting-time of trip at 18:00, origin of the trip: home, via route B When an activity profile is habitually followed by an individual, it is considered to be a script that is repeatedly implemented when an activity needs to be conducted. A universal choice set includes all the feasible activity profiles that an individual can select. Therefore, for every activity type that is included in an individual’s agenda and under every specific context-condition (e.g. weekday and rush-hour), there is a universal choice set of possible activity profiles. Undoubtedly, this universal choice set is directly influenced by the long-term decisions of that individual, such as the work and residential location, the number of available cars, etc. However, when an individual is engaged in the decision-making process and needs to choose one specific activity-profile, usually (s)he does not consider all the activity profiles that are included in the universal choice set. Instead, (s)he considers a subset of it, which constitutes the choice set of that individual at this specific point of time, under the specific context-condition and for the specific activity type. In the following section the modelling of this dynamic process of choice set formation is explained.

9.2.2.

Decision-Making Process

Every time that an individual needs to travel in order to conduct an activity, (s)he needs to select and experience a specific activity profile. This choice-making process can take place consciously or sub-consciously, while both cognitive and emotional considerations can influence the final decision.

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9.2.2.1. Cognitive responses In this model it is assumed that depending on attributes’ base utilities and an individual’s preferences, a cognitive expected utility value corresponds to every activity profile in the universal choice set of that individual. In case that an activity profile was experienced before, an emotional value is also assigned to it. Thus, some cognitive and some emotional responses correspond to that activity profile and when they are combined, they constitute the overall expected utility value of that activity profile, for this individual, at a specific point of time and under a specific contextcondition. In order to calculate the cognitive part of this overall expected utility value, it is assumed that every activity profile, belonging to the universal choice set of an individual, consists of some dynamic and some static activity attributes. In the context of this model, dynamic attributes are those attributes whose value varies among the various context-conditions. For instance, travel time is considered to be dynamic, for the reason that the state of travel time in an activity profile depends on whether it is a rush or a non-rush hour. In order to calculate the expected utility (from the perspective of the individual) of an activity profile, the partial expected utilities of both the static and the dynamic activity attributes are taken into account. Specifically, the expected utility of an activity profile ik of activity type k is: t EUtik = EUstatic þ EUdynamic; ik ik

EUstatic = ik

XX

ð9:2Þ

n

j

EUdynamic;t = ik

βstatic jn xjn Iik j ðxjn Þ

ð9:1Þ

XX j

βjn xjn Ptik j ðxjn jct Þ

ð9:3Þ

n

where EUstatic is the expected partial utility of an activity profile ik for static activity ik attributes j under state n, Xjstatic = fxj1 ; xj2 ; ⋯; xjN g are the static attributes, βstatic is the jn individual’s preference regarding state n of attribute j and Iik j ðxjn Þ equals to 1 if state n of the attribute j is contained in the activity profile ik, otherwise equals to 0. EUdynamic;t is the expected partial utility of activity profile ik, with respect to the ik dynamic attributes under states xjn under context ct and time t, Xjdynamic are the dynamic attributes, βjn is the individual’s preference regarding dynamic attribute j with state n, Ptik j ðxjn jct Þ is the conditional, time-varying probability distribution across various states of dynamic attribute Xjdynamic at time t. Individuals’ beliefs Ptik j ðxjn jct Þ regarding the state of the dynamic attributes, under specific context c, are updated with Bayesian principles and the decision tree induction method (Arentze & Timmermans, 2003). Specifically, individuals do not make decisions based on the actual states of dynamic activity attributes, but rather on their current beliefs about these activity attributes.

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9.2.2.2. Emotional responses When experiencing an activity profile, specific emotional responses emerge, which will be taken into account the next time that this activity profile will be evaluated during the decision-making process. When implementing an activity profile, individuals experience the actual state of the activity attributes. The experienced utility of an activity profile ik at time t equals: XX

AUTtik =

j

βjn xjn I t ðxjn Þ þ ɛtik

ð9:4Þ

n

where I t ðxjn Þ is 1, if state n of the attribute j was experienced at time t, otherwise equals to 0. ɛ tik is the surprise that an individual experiences at time t. If the cognitive expected utility deviates from the experienced utility, negative or positive emotions regarding this experience event emerge. The emotional value of an experience event of an activity profile ik at time t equals to: Rtik = AUTtik − EUtik

ð9:5Þ

If an activity profile is experienced several times, the emotional values of the experience events will be accumulated and result in a positive or negative overall affective value of that activity profile. The emotional value of an activity profile ik at time t, under context c, is:  ð1 − a1 ÞEitk− 1 ðcÞ þ a1 Rtik− 1 ; if Iitk− 1 = 1 and Ict − 1 = 1 Eitk ðcÞ = ð9:6Þ ð1 − a1 ÞEitk− 1 ðcÞ; otherwise where 0 ≤ α1 ≤ 1 is the trade-off between accumulated past emotional values and the most recent ones. This emotional value is incorporated in the overall expected utility of the activity profile ik at time t, under context c (where both a cognitive and an emotional component are included): EUEti ðcÞ = EUti þ a2 Eit ðct Þ k

k

ð9:7Þ

k

where 0 ≤ α2 ≤ 1 is the trade-off between rational (based on expected utility) and affective behaviour (based on emotional value). When α2 increases, emotionally driven behaviour emerges, whereas when α2 decreases, affective responses are taken less into account during the evaluation process of the activity profiles. 9.2.2.3. Choice set formation During the decision-making process, an individual evaluates a number of activity profiles and arrives at a single selection, which is actually the choice of that individual at that time and under a specific context. However, due to memory restrictions, or due to limited information, usually (s)he is not aware of all the activity profiles that are included in the universal choice set. Instead, only those activity profiles that (s)he is aware of, will be considered during the decision-making process. The extent

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to which an individual is aware of an activity profile is contingent on memory and follows the processes of memory decay and reinforcement. The awareness level of an activity profile ik at time t, under context c, equals to:  t − 1 8 t−1 < Sik ðcÞ þ R ik ; Stik ðcÞ = Stik− 1 ðcÞ þ θRtik− 1 ; : λ1 Stik− 1 ðcÞ;

if Iitk− 1 = 1 and Ict − 1 = 1 if Iitk− 1 = 1 and Ict − 1 = 0 if Iitk− 1 = 0

ð9:8Þ

where 0 ≤ λ1 ≤ 1 is the awareness retention rate and 0 ≤ θ ≤ 1 is the context-wise awareness reinforcement parameter. Iitk− 1 equals to 1 if the activity profile ik was experienced at time t − 1, otherwise it is 0. Ict − 1 equals to 1 if the context c was experienced at time t − 1, otherwise equals to 0. Thus, the stronger the emotional impact of the event experience, the longer the activity profile stays in memory. At time t, a choice set consists of those activity profiles whose awareness level exceeds a threshold, reflecting limited human memory retrieval ability. Thus, the choice set of an individual for activity type k, under context c, at time t, is: Φtk ðcÞ = fik ðcÞjStik ðcÞ ≥ ωg

ð9:9Þ

where ω is the minimum awareness level for event memory retrieval ability. 9.2.2.4. Aspirations and stress One of the major theoretical underpinnings of the current research study is the satisficing theory of Herbert Simon (Simon, 1957, 1959). In contrast with utility maximisation theory, Simon stated that individuals cannot continuously search for the best alternative in their choice set and perpetually maximise the utility they get from their choices. Instead, they tend to develop habits, which actually are satisfactory activity profiles. Therefore, every individual holds specific aspiration values, which serve as thresholds of satisfaction. In case that the performance of an activity profile is below that threshold, then it is considered to be a dissatisfactory one, as it does not meet the aspirations of the individual. In the context of this model, it is assumed that an individual holds for every activity type both a conditional and an unconditional aspiration value. Both of these values are dynamic and context-specific. The difference is that a conditional aspiration value ðUk ðcÞjhÞ takes into account the context of the current long-term decisions and the restrictions that they may imply. Therefore, it is more realistic and tailored to the present situation of that individual. On the other hand, an unconditional aspiration value ðUk ðcÞÞ represents the general expectations of an individual, without taking into account the context of the current long-term decisions that characterise his/her life. Specifically: Uk ðcÞjh =

X j

Ukj ðcÞjh

ð9:10Þ

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where Ukj ðcÞjh is the conditional aspiration level of an attribute j of activity type k, under context c, while: X Uk ðcÞ = Ukj ðcÞ ð9:11Þ j

where Ukj ðcÞ is the unconditional aspiration level of an attribute j of activity type k, under context c. Stress is defined as the discrepancy between the overall expected utility value of the habitual activity profile and the conditional aspiration value of the corresponding activity type. It is also assumed that every individual tends to develop some inertia to his/her habitual behaviour and therefore, depending on his aversion to change and explore for other alternatives, (s)he demonstrates some tolerance to stress. Thus, there is a tolerance to stress value (σ1) which reflects the extent to which an individual’s behaviour is characterised by inertia and reluctance to change. When stress exceeds this tolerance to stress value, a short-term change occurs in an individual’s activity-travel patterns. 9.2.2.5. Habit formation Depending on the extent to which an individual is aware of an activity profile at a specific point of time and under a specific context-condition, this profile will be included in the choice set of that individual or not. However, depending on previous experiences, those activity profiles which have been implemented recently are firstly considered, during the decision-making process. Specifically, a memory-activation level corresponds to each activity profile included in the choice set. The activation level is a habit indicator, as the profile with the maximum activation level in the choice set is the habitual option for that individual, for the specific activity type, under a specific context-condition and at a specific point of time. Specifically the updated activation 0 level Witk ðcÞ of activity profile ik ðcÞ ∈ Φtk ðcÞ, at time t0 and context c equals to: 0

Witk ðcÞ = log ðWitk ðcÞ þ 1Þ

ð9:12Þ

where Witk ðcÞ =

 t−1 Wik ðcÞ þ γ λ2 Witk− 1 ðcÞ

if Iitk− 1 = 1 and Ict − 1 = 1 ; where ik ðcÞ ∈ Φtk ðcÞ otherwise

ð9:13Þ

and γ > 1 is the recency weight and 0 ≤ λ2 ≤ 1 is the retention rate. In Eq. (9.12), the logarithmic transformation is used because it is assumed that when an activity profile is newly experienced, its activation level rapidly increases until reaching a saturation point, where the activation level increase slows down. On the other hand, when an activity profile is no longer experienced, its activation level is dramatically lowered. The habitual activity profile can be denoted as ik ðcÞ ∈ Φtk ðcÞ, which is the alter0 native with argmax Wit ðcÞ in the current choice set. Thus, habitual behaviour means

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that the activity profile with maximum activation level is selected and it is continually chosen as long as stress (if any) is below the stress threshold. That is, if: Uk ðcÞjh − EUEti ðcÞ < σ 1

ð9:14Þ

k

then the habitual alternative ik ðcÞ is chosen at time t. 9.2.2.6. Short-term dynamics In case that stress exceeds the tolerance level of an individual, then (s)he acts in conscious mode and short-term dynamics are triggered. Exploitation choice mode. It is assumed that if the difference between the overall expected utility of the habitual option and the conditional aspiration of an individual exceeds stress tolerance threshold, then the individual will search for the best option within his/her current choice set, which means for the activity profile with the maximum overall expected utility. If: Uk ðcÞjh − EUEti ðcÞ < σ 1

ð9:15Þ

k

then the i k ðcÞ exploitation activity profile is selected at time t. Exploration choice mode. In case that the overall expected utility of the exploitation option and the conditional aspiration of an individual exceeds stress tolerance threshold, then the individual will search for another activity profile, beyond his/her current choice set. The process of exploration is not random, but goal-directed, in the sense that is guided by those activity attributes that were not satisfied by the exploitation activity profile. Therefore, the probability of finding another activity profile of a certain activity type and deciding to follow it is proportional to the Vitk ðJ 0 jcÞ utility of that activity profile, which is calculated based on the expected utility of the attributes of the exploitation activity profile that caused dissatisfaction ðeui j ðcÞÞ. If the individual had perfect information, the probability of finding the k best activity profile for a certain activity type would be one. In reality, the degree that an individual is well-informed varies from situation to situation. From the perspective of the analyst, the probability that an individual in exploration mode finds and implements a new activity profile is: .X Pt ðik ðcÞjJ 0 Þ = exp ðVitk ðJ 0 jcÞ=τÞ exp ðVitk ðJ 0 jcÞ=τÞ

ð9:16Þ

Ik

where τ is the degree of information lack in the study area, J 0 are the dissatisfactory activity attributes of the exploitation activity profile J 0 = fjjUkj ðcÞjh − eui j ðcÞ > σ 1 g k

∀j

ð9:17Þ

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and Vitk ðJ 0 jcÞ is the utility of the activity profiles that are not included in the choice set, based on the dissatisfactory activity attributes J 0 : X Vitk ðJ 0 jcÞ = eui ð9:18Þ j ðcÞ k j∈J 0

9.2.2.7. Long-term dynamics In some cases, it is possible that short-term changes do not seem to be effective enough in alleviating stress. Then, a long-term change may be considered. One of the main goals of the current study is to link those two time horizons (short and long-term) in one model of dynamic activity-travel behaviour. Lowering aspirations. It is assumed that an individual will keep a record ðNkt − 1 Þ of how many consecutive times (s)he conducted an exploration for an activity type, under the same context c, without finding something satisfactory. As the exploration effort is built up, the probability that the individual will lower the conditional aspirations on the level of the dissatisfactory activity attributes increases. Lowering aspirations means that an individual lowers the conditional aspiration values of the attributes that caused dissatisfaction to the corresponding attribute values of the exploitation option. Specifically, the probability that an individual lowers the conditional aspiration level of an activity attribute j ∈ J 0 of an activity type k, is: Pt ðloweringUkj ðcÞjhÞ = exp ðμ þ νðNkt − N0 ÞÞ=½1 þ exp ðμ þ νðNkt − N0 ÞÞ

ð9:19Þ

where μ, ν and N0 are parameters of the logistic function. Becoming ‘awake’. After several decreases of the conditional aspiration values of an individual, it is possible that (s)he realises that (s)he needs to extent the possibilities and the universal choice set within which (s)he acts. In that case, the individual becomes ‘awake’ and realises that a long-term change needs to be considered. In particular, the conditional aspiration value of an individual for activity type k equals to: X X Uk jh = freqc Uk ðcÞjh= freqc ð9:20Þ c

c

where freqc is the frequency of experiencing context c. In a similar way, the unconditional aspiration value of an individual for activity type k ðUk Þ is calculated, as well. The mean value of the conditional aspiration values of the K total number of activity types included in an individual’s agenda equals to the actually experienced utility of the current long-term decisions of that individual: Uh =

X k

Uk jh=K

ð9:21Þ

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Similarly, the mean value of the unconditional aspirations of the K activity types equals to the aspiration of an individual, regarding his/her context of long-term decisions ðUr Þ. Let σ2 be the stress tolerance threshold for the long-term level. We assume that if: Ur − Uh < σ 2

ð9:22Þ

an individual does not consider making a long-term change, as (s)he does not suffer by stress on the long-term level. However, if this difference exceeds the long-term stress tolerance threshold σ2, (s)he becomes ‘awake’ and considers conducting a long-term change. Predicting the point of time and the situation under which an individual becomes ‘awake’ is the boundary of this model. The process which led to a long-term change, the specific activity-type and context-condition which generated stress, and the effort of an individual to reduce it, can be predicted by the model. In this way, it gives a clear image of how the sequence of short to long-term adaptation efforts took place. Finally, the role that emotional responses played in this process, as well as the fluctuation of various conditional and unconditional aspiration values can also be traced. 9.2.3.

Updating Phase

The decision-making process, which was described in the previous section, is followed by the implementation of the selected activity profile. Based on this experience, an individual updates the emotional value of the selected activity profiles, his/her beliefs and his/her aspiration values (as it was already explained). Moreover, the awareness and the activation level of the activity profiles are updated based on a learning/forgetting process (according to Eqs. (9.8) and (9.13)).

9.3.

Numerical Simulations

Numerical simulations were conducted in order to check the face validity of the suggested model. The aim of those simulations was threefold: to check the behaviour of the model under a basic case, to monitor the behaviour of the model under specific scenarios and to illustrate the effect of every parameter on the overall behaviour of the model. In the current chapter, a few basic case results are going to be presented in order to help the reader to get a clear image of how the model works. However, the focus of this chapter is on the effect of the γ and λ2 memory-activation parameters on the model behaviour and especially on the process of habit formation, as well as on the effect of the α1 and α2 emotion-related parameters on the decision-making process. 9.3.1.

Simulation Settings

Due to simplification reasons the conducted simulation focused on three activity types (work, shopping and leisure), while there are five agents acting. When

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an agent becomes ‘awake’, it stops acting. The time needed for processing one agent is 1.29 seconds. On simulation day t = 600, the simulation run is stopped by the user. The spatial setting of the simulation is based on the Eindhoven area. A residential and a work location, as well as a supermarket close to their neighbourhood, a social club and a friends’ home location correspond to each agent. The schedule of the three activities is predefined and same for all the agents. There are four possible context-conditions c (weekday/rush-hour, weekday/non-rush-hour, weekend/rushhour, weekend/non-rush-hour). A Monte Carlo simulation, based on predefined probabilities for each context c, determines which c is experienced by an agent, on simulation day t. The input universal choice set, including the possible activity profiles, is similar for the five agents. Travel time and cost for every activity profile were calculated via maps.google.com. Due to simplification reasons, travel cost is regarded as static; while travel time as dynamic (three levels of delay are considered). The model starts with the agents not being aware of the simulation setting. Two tables are considered, one representing the agents’ beliefs regarding the probability of delay for every context and one representing the system probabilities. Based on the first one (which is updated after every experience), agents’ beliefs are formulated, while based on the second one, a Monte Carlo simulation generates the actual delay that is experienced by each agent, on simulation day t. The surprise term for experienced utility is generated based on a normal distribution with mean 0 and standard deviation 0.25. When an activity needs to be conducted, each agent selects an activity profile in order to reach the location of that activity. Based on this experience, the emotional value, the awareness and the activation level of the experienced activity profile are updated. On the other hand, the same indicators of the non-selected profiles decrease, based on a forgetting process. The basic settings for these mechanisms are (1) awareness threshold ω = 0.01, (2) awareness retention rate λ1 = 0.99, (3) parameter for updating activation levels γ = 1.5, (4) activation level retention rate λ2 = 0.9, (5) parameters for the probability of lowering aspirations μ = 0.1, v = 1 and N0 = 3, (6) stress threshold for short-term horizon σ1 = 1 and for long-term horizon σ2 = 2 and (7) uncertainty parameter for exploration τ = 20.

9.3.2.

Basic Case Results

In Figure 9.1, there is an example of the first 10 consecutive choices that the fourth agent made in order to travel for a leisure activity, when it was weekend and nonrush hour. Both the overall expected utility of the chosen activity profiles and the conditional aspiration values of that activity type, under this context, are depicted in the graph. Specifically, after four unsuccessful exploration efforts, the agent lowers its conditional aspiration value, so that it becomes more realistic. Then the exploitation option becomes satisfactory enough to be selected on simulation day

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Figure 9.2: Fluctuation of aspiration values of the second agent over time.

t = 63 and to constitute the habitual option of that agent for this activity type and under this context. In Figure 9.2, the aspiration values, that the second agent holds, are depicted. Specifically, the conditional aspiration values for work, shopping and leisure activity type, as well as the aspiration for the long-term decisions and the actually experienced utility of the long-term decisions, are included. Initially, those aspiration values are adjusted according to the experiences of the agent and after some point of time, they are stabilised. The deviation between the aspiration for longterm decisions and the experienced utility of the long-term decisions represents stress at the long-term time horizon and if it exceeds σ2 value, the agent becomes ‘awake’.

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202 9.3.3.

Effect of Memory-Activation Parameters

9.3.3.1. Effect of λ2 parameter Parameter λ2 is the retention rate, included in the forgetting part of the activation level update equation (Eq. (9.13)). As λ2 decreases, memory activation fades out very fast and therefore the agents ‘stick’ less to their previous experiences. Thus, while λ2 decreases, the agents develop habits with bigger difficulty, resulting in a decreasing frequency of habitual behaviour (Figure 9.3a). However, as λ2 decreases, more exploitation choices take place (Figures 9.3a and b). This is because memory activation decreases very fast and the agents need to exploit better their choice-set, as they don’t have a strong habit indicator. On the other hand, as λ2 increases, the agents develop strong habits and do not exploit the rest of their choice-set enough. This implies that even if there were better options than their habits, they are discarded from memory very fast. However, if at some point of time these strong habits are not satisfactory any more, the choice set hardly contains any other alternatives and the agents have to explore the area in order to find a better solution. Thus, when parameter λ2 increases, exploratory behaviour increases, as well (Figure 9.3b). Finally, choice-set size is also affected, as it tends to increase, due to the increasing number of explorations.

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Figure 9.3a and b: Effect of λ2 parameter on frequency of choice modes (habit, exploitation, exploration) and lowering aspiration incidents.

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As λ2 decreases, the agents cannot develop easily habits and they tend to better exploit their choice set. This results in a higher mean overall expected utility of the habitual and exploitation choice modes (Figure 9.4). Consequently, the overall expected utility of their choice set also improves (Figure 9.5). Specifically, when parameter λ2 decreases, the agents better exploit their choice-set and the alternatives that perform well have a lower chance to be discarded from memory. Finally, aspiration values are also affected. Specifically, they also tend to increase, when parameter λ2 decreases, because of the increased overall expected utilities of the habit and exploitation choice modes.

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Figure 9.4: Effect of λ2 parameter on overall expected utility of habit, exploitation and exploration choice mode.

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Figure 9.5: Effect of λ2 parameter on choice set overall expected utility.

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Thus, when the retention rate of memory activation is too high, habitual behaviour becomes too strong and the agents do not manage to exploit well their choice set. It also appears that when activity-travel behaviour is driven by strong habits, there is a decline of welfare over time and an increase of exploration effort. 9.3.3.2. Effect of γ parameter Parameter γ is the recency weight included in the equation describing the update of the activation level of the experienced activity profiles (Eq. (9.13)). In this equation, the logarithmic form is used to account for saturation. Specifically, when an activity profile is newly experienced, its activation level rapidly increases, until reaching a saturation point, where the activation level increase slows down. Thus, the higher the value of γ, the higher the impact of the experience and the faster the increase of the activation level of this activity profile. As γ increases, the activation values become stronger and memory fading takes more time, which actually implies that the agents ‘stick’ more to their previous choices. On the other hand, as γ decreases, the activation value decreases faster (during the interval time of experiencing again the same combination of activity type and context-condition), resulting in a greater flexibility and more exploitations of the choice set (Figure 9.6). Regarding the overall expected utilities of the choice modes (Figure 9.7), the habitual and exploitation choice mode seem to be slightly affected (as γ increases, there is a tendency to ‘stick’ more to previous choices, which seems to result in a slight decrease of the overall expected utility of the habitual and exploitation choices). Similarly, the mean overall expected utility of the choice set also seems to slightly decrease, as γ increases. This is plausible because the agents exploit the choice set better when γ is lower and, therefore, the alternatives that perform well have a lower chance to be discarded from memory. The size of the choice set, as Activation level recency weight (γ) impact Frequency of choice modes 80

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Figure 9.6: Effect of γ parameter on frequency of choice-modes (exploitation, exploration) and lowering aspirations incidents.

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Figure 9.7: Effect of γ parameter on overall expected utility of choice-modes (habit, exploitation and exploration).

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Figure 9.8: Effect of γ parameter on aspiration values. well as the aspiration values are not affected by γ (Figure 9.8), as this parameter only influences the impact of an experience event on activation memory. 9.3.4.

Effect of Emotion-Related Parameters

9.3.4.1. Effect of α2 parameter Parameter α2 represents the trade-off between rational and emotional behaviour and it is included in Eq. (9.7). Higher value of α2 parameter implies a more emotionally driven behaviour. As it is explained in Psarra et al. (2014), higher influence of

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affective responses results in spending more effort in exploring new alternatives. This is because after a negative experience there is a high chance that in next choice an agent will try to explore another alternative. Additionally, even when positive emotions are experienced; the activity profile may not perform structurally well, which also increases the possibility that in next choice an agent will need to find another solution. The increased number of explored activity profiles leads to an increase of the average choice-set size, as it is indicated by the trend line in Figure 9.9. Moreover, as α2 increases, aspiration values tend to decrease. This is because emotionally driven behaviour leads to more exploration efforts, which, in turn, result in more lowering aspiration incidents. Consequently, when an agent focuses more on emotional responses when making a decision, it is getting easily dissatisfied

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Figure 9.9: Effect of α2 parameter on choice-set size. Rational-affective trade-off (a2) impact Aspirations 3

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Figure 9.10: Effect of α2 parameter on aspiration values.

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and its aspiration values tend to decrease (Figure 9.10), in order to become more realistic and allow the development of habitual behaviour. Finally, it should be mentioned that as the aspiration values get lower, the probability that a long-term change is decided increases. Thus, emotionally driven behaviour brings more short and long-term dynamics into the system. 9.3.4.2. Effect of α1 parameter Parameter α1 represents the trade-off between accumulated past emotional values and the most recent ones. When α1 approaches 1, the emotional value of an activity profile is totally influenced by the most recent emotional experience event. Accordingly, as α1 approaches 0, past emotional experiences are taken into account to a greater extent. Undoubtedly, the salience of the effect of parameter α1 on model behaviour depends on the value of parameter α2. For instance, in case that α2 is too low, emotions are not taken into account during the decision-making process and, therefore, the influence of α1 on the model behaviour will be very small. On the other hand, if α2 approaches 1, affective responses influence to a great extent the evaluation of alternatives and the trade-off between recent and accumulated past emotional experiences can play a significant role on the decision-making process and the overall model behaviour. As it is depicted in Figure 9.11, when the value of α1 increases, the frequency of exploration behaviour also increases. This implies that when the choice-making process is based only on the most recent affective responses, more effort is spent in exploring new alternatives, while habitual behaviour decreases. Finally, the intensive exploration behaviour results in an increased frequency of the lowering aspiration incidents. When the value of parameter α1 increases, the overall expected utility of the exploitation and exploration choice mode tends to decrease (Figure 9.12). Past-recent emotional values trade-off (a1) impact Frequency of choice modes 90 80 Number of choices

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Figure 9.11: Effect of α1 parameter on frequency of choice modes (exploitation, exploration) and lowering aspiration incidents.

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Specifically, when the decision-making process is based to a greater extent on the most recent emotional experience events, the agents tend to get more easily disappointed and conduct more exploration effort. Consequently, much more activity profiles are explored, many of which do not perform very well. This results in a decrease of the mean overall expected utility of the exploration choice mode. Similarly, the overall expected utility of the exploitation choice mode tends to decrease, as well. This is explained by the fact that more lowering aspiration incidents take place, as α1 increases. If the value of parameter α1 increases, the mean overall expected utility of the agents’ choice sets decreases (Figure 9.13). This implies that when taking into

Past-recent emotional values trade-off (a1) impact Expected utility of choice modes 2.5

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Figure 9.12: Effect of α1 parameter on overall expected utility of choice modes (habit, exploitation and exploration).

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Figure 9.13: Effect of α1 parameter on choice-set overall expected utility.

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Past-recent emotional values trade-off (a1) impact Aspirations 3

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Figure 9.14: Effect of α1 parameter on aspiration values. account only the most recent emotional experiences and not the accumulated past emotional experience, the evaluation of the alternatives and the decision-making process leads to more explorations and ultimately to a decrease of welfare over time. This result is plausible and it is also in line with the fact that the overall expected utilities of the exploitation and exploration choice modes are also in decreasing trend. Finally, the decreasing trend line of the aspiration values in Figure 9.14 is explained by the increased frequency of the lowering aspiration incidents. This result is also reasonable, as when the decision-making process is only based on the most recent emotional response, the agents get easily disappointed and become engaged in a more intensive exploration behaviour. Ultimately, this results in a decrease of their aspirations (in order to manage to develop habitual behaviour at some point of time), which, in turn, increases the probability that long-term dynamics emerge.

9.4.

Conclusions and Discussion

This chapter focused on a model of short and long-term dynamics of activity-travel behaviour, where specific bounded rationality mechanisms are incorporated. Specifically, the decision-making process is not based on utility maximisation and full cognition, as memory activation and emotion responses are taken into account. The behaviour of this agent-based system is illustrated with some numerical simulation results. The agents are able to learn the area where they act, adjust their aspirations and develop habits. In case that these habits are not satisficing any more, some short or long-term adaptations may occur, until habits emerge again. Specific dynamic, experience-based learning and forgetting processes determine which option is the one that first comes to an agent’s memory when it is about to decide how it will travel in order to conduct an activity. This is the habitual option,

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which has the highest memory-activation level in the current context-specific choice set. As long as the overall expected utility of this option is within the stress threshold of the agent, it will be selected and implemented, while the memory-activation level of this habitual activity profile will be further increased. In order to clarify to what extent this process is affected by specific memoryactivation parameters, a series of numerical simulations, where the values of these parameters systematically vary, was conducted. The results of these simulations are plausible and logical. Specifically, they indicate that low retention rates lead to difficulty in developing habitual behaviour and engagement in intensive exploration behaviour. Usually, this results in a decrease of their aspirations, and lower expected utility of their choice-sets. Apart from the memory-activation process and the strength of habit, emotional responses can also affect the activity-travel behaviour of an agent. Numerical simulations illustrate the effect of the tradeoffs between past and recent emotional experiences, as well as between cognitive and affective responses. They indicate that higher influence of emotional responses in choice-making process results in more exploration effort and in decreasing aspiration values. Moreover, when only the recent emotional experiences are taken into account and the accumulated past experiences are ignored; there is a similar effect on the model behaviour. Specifically, the agents tend to get more easily disappointed and conduct more explorations. Summing up, emotionally driven behaviour brings more short and long-term dynamics into the system. Future research needs to focus on analysing empirical data, in order to estimate the parameters and validate the suggested model. These data are going to be collected with a stated adaptation experiment (Psarra, Arentze, & Timmermans, 2013). This data collection method enables the acquisition of information regarding the current habits and aspirations of the respondents, as well as the formulation of personalised, realistic scenarios. Based on the response to these scenarios, tolerance to stress in short and long-term time horizon can be detected. Nevertheless, the validation of specific parameters, such as the trade-off between cognition and emotions, would need specific structured experiments, which are beyond the scope of the current research. The calibration of these parameters will be based on numerical simulation results of the same type as the ones presented in this chapter. Lastly, it should be mentioned that the suggested model is considered to be scalable, in the sense that it can be applied to study areas of relatively large size (e.g. city region), as this would influence only the time needed for a simulation run (approximately N*1.29 seconds, for simulating a population of N size). Furthermore, the overall system can help in getting a better insight into the dynamics of activitytravel behaviour and specifically into various aspects of it, such as the formation and distortion of habits and the effect of emotional responses in decision-making. Finally, the suggested model can be regarded as part of a more comprehensive model of activity-travel behaviour, currently under development, where both exogenous and endogenous dynamics are taken into account (Timmermans et al., 2010).

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Acknowledgements This research received funding from the EU Research Council under the European Community’s 7th Framework Programme (FP7/20072013)/ERC grant agreement n° 230517 (U4IA project). The opinions in this publication represent those of the authors only. The ERC and European Community are not liable for any use of this information.

References Arentze, T. A., Ettema, D. F., & Timmermans, H. J. P. (2007). Micro-simulation of individual space-time behavior in urban environments: A new model and first experience. In Proc. 10th International Conference on Computers in Urban Planning and Urban Management (CUPUM), Iguassu Falls, Brazil. Arentze, T. A., & Timmermans, H. J. P. (2003). Modeling learning and adaptation processes in activity-travel choice: A framework and numerical experiments. Transportation Journal, 30, 3762. Beige, S., & Axhausen, K. W. (2006). Long-term mobility decisions during the life course: Experiences with a retrospective survey. In Proc. 11th International Conference on Travel Behavior Research, Kyoto, Japan. Ga¨rling, T., & Axhausen, K. W. (2003). Habitual travel choice. Transportation Journal, 30, 111. Habib, K. M. N., Elgar, I., & Miller, E. J. (2006). Stress triggered household decision to change dwelling: A comprehensive and dynamic approach. In Proc. 11th International Conference on Travel Behavior Research, Kyoto, Japan. Han, Q., Arentze, T. A., & Timmermans, H. J. P. (2007). Modelling the dynamic formation of activity location choice sets. In Proc. 11th World Conference on Transportation Research (WCTR), Berkeley. Han, Q., Arentze, T. A., Timmermans, H. J. P., Janssens, D., & Wets, D. (2009). Developing dynamic models of activity-travel behavior: Principles, mechanisms, challenges in data collection and methodological issues. In Compendium of Papers CD-ROM, the 88th Annual Meeting of the Transportation Research Board, Washington, DC. Klo¨kner, C. (2004). How single events change travel mode choice a life-span perspective. In Proc. 3rd International Conference of Traffic and Transport Psychology, Nottingham, England. Miller, E. J. (2005). An integrated framework for modeling short- and long-run household decision-making. In Proc. Progress in Activity-Based Models, Maastricht, the Netherlands. Prillwitz, J., & Lanzendorf, M. (2006). Impact of life course events on car ownership. In Compendium of Papers CD-ROM, the 85th Annual Meeting of the Transportation Research Board, Washington, DC. Psarra, I., Arentze, T. A., & Timmermans, H. J. P. (2013). Capturing short and long term dynamics of activity travel behavior: Design of a stated adaptation experiment. In Proc. 13th World Conference on Transportation Research (WCTR), Rio de Janeiro, Brazil. Psarra, I., Arentze, T. A., & Timmermans, H. J. P. (2014a). Simulating exploration processes and aspiration adjustment in a model of endogenous dynamics of activity-travel behavior. Submitted for publication.

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Psarra, I., Arentze, T. A., & Timmermans, H. J. P. (2014b). The effect of stress tolerance on dynamics of activity-travel behavior: Numerical simulation results. Submitted for publication. Psarra, I., Arentze, T. A., & Timmermans, H. J. P. (2014c). Simulating choice set formation processes in a model of endogenous dynamics of activity travel behavior: The effect of awareness parameters. Submitted for publication. Psarra, I., Arentze, T. A., & Timmermans, H. J. P. (2014d). The effect of emotional responses on endogenous dynamics of activity-travel behavior: Numerical simulation results. Procedia Computer Science, 32, 750755. Psarra, I., Liao, F., Arentze, T. A., & Timmermans, H. J. P. (2014). Modeling contextsensitive, dynamic activity-travel behavior, by linking short and long-term responses to accumulated stress: results of numerical simulations. Transportation Research Record, 2412(1), 2840. Rasouli, S., & Timmermans, H. J. P. (2014). Activity-based models of travel demand: Promises, progress and prospects. International Journal of Urban Sciences, 18(1), 3160. Simon, H. A. (1957). Models of man: Social and rational. New York, NY: Wiley. Simon, H. A. (1959). Theories of decision-making in economics and behavioral science. The American Economic Review, 49, 253283. Timmermans, H. J. P., Arentze, T. A., Cenani, S., Ma, H., Pontes de Aquino, A., Sharmeen, F., & Yang, D. (2010). U4IA: Emerging urban futures and opportune repertoires of individual adaptation. Presented at the 10th International Conference on Design & Decision Support Systems in Architecture and Urban Planning, Eindhoven, the Netherlands. Timmermans, H. J. P., Khademi, E., Parvaneh, Z., Psarra, I., Rasouli, S., Sharmeen, F., & Yang, D. (2014). Dynamics in activity-travel behavior: Framework and selected empirical evidence. Asian Transport Studies, 3, 124. Verhoeven, M., Arentze, T. A., Timmermans, H. J. P., & van der Waerden, P. J. H. J. (2005). Modeling impact of key events on long-term transport mode choice decisions: Decision network approach using event history data. In Compendium of papers CD-ROM, the 84th Annual Meeting of the Transportation Research Board, Washington, DC.

Chapter 10

Multidimensional Travel Decision-Making: Descriptive Behavioural Theory and Agent-Based Models Chenfeng Xiong, Xiqun Chen and Lei Zhang

Abstract Purpose  This chapter explores a descriptive theory of multidimensional travel behaviour, estimation of quantitative models, and demonstration in an agentbased microsimulation. Theory  A descriptive theory on multidimensional travel behaviour is conceptualised. It theorizes multidimensional knowledge updating, search start/ stopping criteria, and search/decision heuristics. These components are formulated or empirically modelled and integrated in a unified and coherent approach. Findings  The theory is supported by empirical observations and the derived quantitative models are tested by an agent-based simulation on a demonstration network. Originality and value  Based on artificially intelligent agents, learning and search theory, and bounded rationality, this chapter makes an effort to embed a sound theoretical foundation for the computational process approach and agent-based microsimulations. A pertinent new theory is proposed with experimental observations and estimations to demonstrate agents with systematic deviations from the rationality paradigm. Procedural and multidimensional decision-making are modelled. The numerical experiment highlights the capabilities of the proposed theory in estimating rich behavioural dynamics. Keywords: Learning; search; imperfect knowledge; decision rules; hidden Markov chain

Bounded Rational Choice Behaviour: Applications in Transport Copyright r 2015 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISBN: 978-1-78441-072-8

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Background

The study of travel demand estimation, forecasting and adjustment has long been a vital topic in the field of transportation planning. Being an induced demand, travel demand is often regarded as the product of other activities. Individuals commute to work, drop-off family members, travel for leisure, fly to customers/suppliers, visit relatives/friends and so forth. While these activities are often differentiated by locations and time, how these spatial/temporal details can be accounted for becomes an essential question for transportation planners and researchers. Moreover, these activities encompass interrelated travel decisions including destination, mode, departure time and route. Therefore, the complexity arising from the mutual effects of these multidimensional decisions upon each other and from their decision timing needs to be represented. Traditional travel demand modelling structure distinguishes four decision dimensions: deciding the frequency of travel, choosing a destination, selecting a travel mode and travelling via a route. These decision dimensions are assumed to follow a predefined sequential manner of trip generation, trip distribution, modal split and traffic assignment, as known as the ‘Four-Step’ method. Travel behaviour research gradually moved from aggregate demand models to more disaggregate individuallevel and activity-based models (Bhat & Koppelman, 1999; Bowman & Ben-Akiva, 2001; Vovsha, Donnelly, & Gupta, 2008; Xiong & Zhang, 2013b; Zhang, Southworth, Xiong, & Sonnenberg, 2012). While the majority of interest focuses on advancing single-dimensional (single-facet) choices and more advanced representation of activity pattern such as scheduling (Bowman & Ben-Akiva, 2001; Golledge, Kwan, & Garling, 1994), land use influence (Salvini & Miller, 2005) and location choices (Bhat & Guo, 2007), the linkages among different travel behavioural dimensions are largely ignored (Pinjari, Pendyala, Bhat, & Waddell, 2011) and individuals’ embedded behavioural processes that influence them to change certain dimension(s) of their travel behaviour remain unexploited. Besides the rigid sequential assumption, travel demand models also rely on other simple and sometimes unrealistic behavioural assumptions in order to keep themselves analytically tractable. Perfect rationality theory is one of the well-known assumptions that individuals are fully rational, have perfect information and always maximise utility (Savage, 1954; von Neumann & Morgenstern, 1947). Being an approach with rich results, mathematical rigour and interesting applications, perfect rationality and utility maximisation allow structural insights and explain similarities and differences in travel behaviour. However, if using this theory to calculate how certain variations in the situation are predicted to affect travel behaviour, ‘these calculations obviously do not reflect or usefully model the adaptive process by which subjects have themselves arrived at the decision rules they use’ (Lucas, 1986). The opposite holds true for the computational process models, a group of new methods that depart from rationality assumptions and implement learning, adaptations, information acquisition and decision-making efficiently by taking the advantages of computer power. These models are microsimulations relying on heuristic

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arguments and imitation of human behaviour. A large number of real-world or benchmark problems can be analysed by applying these models to simulate numerical results in different set-ups. Examples on the rapidly growing list include Pendyala, Kitamura, and Prasuna Reddy (1998), Arentze and Timmermans (2004), Wahba and Shalaby (2011), and Auld and Mohammadian (2012). On the one hand, these models introduce more complex learning, adaptation and behavioural rules instead of utility maximisation. But on the other hand, multiagent simulation cannot prove but only suggests a certain feature of travel pattern and still assumes sequential decision process. Thus it requires additional theories to conceptualise the more rigorous behavioural foundation and better explain behaviour adjustments along multiple choice dimensions (see Arentze, Hofman, & Timmermans, 2004; Pinjari et al., 2011). Urged by the above-mentioned theoretical and modelling issues, this chapter describes an alternative framework to modelling multidimensional aspects of travel behaviour. Descriptive theory and models are built upon economics and travel behaviour research on learning (Arentze & Timmermans, 2005; Golledge et al., 1994), search theory (Stigler, 1961) and bounded rationality (Mahmassani & Chang, 1987; Simon, 1955). The theory recognises that there are inconveniences and risks associated with each behaviour adjustment dimension, which is conceptualised as a search cost unique to each individual and each behaviour dimension. On the other hand, an individual, based on his/her spatial knowledge, personal travel experiences and beliefs, forms subjective expectations on potential gains (search gain) from behavioural adjustments along each behavioural dimension. It is the interplay of these search gains and search costs along all feasible behavioural adjustment dimensions that collectively determine when individuals start seeking behaviour changes, how they initially change behaviour, how they switch behaviour adjustment dimensions, and when they are satisfied and stop changing behaviour. The theorisation of multidimensional knowledge updating, search model and behaviour process becomes a unified and coherent approach that models the activity and travel decision-making with a consistent behavioural foundation and increased rigour. The theory is supported by empirical observations and the derived quantitative models are tested by agent-based simulation on a demonstration network. For improved readability, emphasis is given to modelling knowledge and multidimensional search. Certain details about search rules and decision rules are excluded. Readers interested in these topics are referred to appropriate references.

10.2.

Theory and Models

The multidimensional travel decision-making theory is conceptualised in Figure 10.1. The theory starts with the definition of artificially intelligent agents and their characteristics. Each agent i is treated differently with socio-demographic

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Experience Eit Past travel experience Supplement Information: ATIS, media, internet, map, etc.

Subjective Beliefs Pitd Multidimensional knowledge Beliefs and expectations

Memory Mitd Learning attributes of places, modes, paths, times, etc. Forgetting outdated/unrepresentative information Memorizing experiences subjectively

t=t+1

Multidimensional Search Gain and Search Cost gdt | d ∈{mode, departure time, route, etc.}

Searching Dimension - d Prioritizing the behavioural dimensions Search cost exceeds gain for all d

Search Scope Agents find alternatives by heuristics

Decision Rules Agents select one alternative by heuristics

Travel Experience Habitual behaviour

Figure 10.1: Conceptualization of multidimensional travel decision-making theory. attributes, personal experience, knowledge and subjective beliefs. At any given time, an agent has a certain level of knowledge about places, activities and transport networks in an urban area. This spatial/temporal knowledge can be employed to solve various spatial/temporal decision tasks such as choosing destination, departure time and routes. This problem-solving process consists of several procedural steps in the true behavioural sense. Firstly, each agent i at a given time period t possesses experiences, denoted as Eit. Agents acquire Eit through past searches or through information sources such as internet, media and advanced traffic information system (ATIS). Eit is time-variant as the agent searches and accumulates a-priori experiences in the urban transportation network day-by-day. Travel experiences with similar payoffs that occur routinely may reinforce the agent’s memory, while the travel experiences that are not representative may be easily forgotten (Arentze & Timmermans, 2003). Moreover, agents are assumed to be able to search information about one behavioural adjustment dimension at a time, for example agents may search for an alternative route or search for an alternative travel mode. Thus each past experience can be mapped into one single dimension d and form a multidimensional memory space Md.

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The memory space keeps updating, alters the aspiration level and changes subjective beliefs Pdit . An agent thus determines the expected gain gdt from a search for alternatives in each behavioural dimension d based on his/her subjective beliefs. Information acquisition and other mental efforts are explicitly modelled as perceived search cost scdi when agents are searching for alternatives for each behavioural dimension. These search cost variables are recognised in this theory as inconveniences and risks associated with each behaviour adjustment dimension. It is the interplay of these subjective search gains and costs that jointly determines when a search for alternatives in dimension d is initiated or stopped in time period t. Although the subjective search gain is defined by individual’s beliefs and therefore can be quantitatively derived, it is much more difficult to theoretically determine the magnitude of perceived search cost which should be individually different. Once the multidimensional behavioural adjustment evidences can be observed, the perceived search cost and its relations with other variables can be empirically derived. If an agent decides not to search in a dimension, habitual behaviour in that dimension is executed. Otherwise, the agent will employ a set of search rules to search from her/his knowledge and identify a new alternative. After identifying an alternative, she/he needs to determine whether or not to switch to that alternative. The decision rules constitute a mapping from spatial/temporal knowledge (especially experienced travel conditions corresponding to different alternatives) to a binary decision: switch to the alternative or retain habit. Both the search rules and the decision rules should be empirically estimated from observed search processes.

10.2.1. Modelling Imperfect Knowledge Search, learning, and knowledge play a crucial role in making a decision. A rational person will choose the best alternative from the set of feasible alternatives. The term ‘rationality’ would also require that this rational person holds the knowledge that is derived from coherent inferences. In contrast, more realistic models are intended to allow modellers to construct agents who systematically do not possess perfect knowledge and do not make correct inferences but make biased ones. An agent explores decision opportunities by searching her/his feasible environment and learns knowledge about various payoffs related to the search and decisions. Here the spatial/temporal knowledge is generalised as multidimensional vectors with each vector corresponding to a particular dimension. Assume that each agent i at any given time period t possesses a list of past experiences, Eit. Each experience is characterised by a generalised cost: CE =

X

λn ψ n

ð10:1Þ

n

where n denotes the index of different related attributes such as travel time, cost, schedule delays and mode comfort. ψ denotes the vector of attributes; λ denotes the

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coefficient to translate values into monetary costs (e.g. value of time). This generalised cost is adopted to measure the outcome of each experience and to set an anchoring point for the search model. Assuming that in each behavioural dimension d, an individual’s perceptual capabilities allow the separation of generalised cost into a number of categories. If CE that falls into the generalised-cost category j has been observed mj times in prior experiences, the memory this individual has about the generalised cost in dimension d is fully described by a vector Md = (m1, …, mj, …, mJ). Individuals update memory space through learning and forgetting processes. Bayesian learning relies on the premise of some prior knowledge. When new information from various sources becomes available, learning occurs and obeys the Bayes’ rule. Forgetting relies on the cognitive weighting of each past experience, which can be measured as a function of the recentness and representativeness of the experience (Arentze & Timmermans, 2003). Once the weight is lower than a certain threshold parameter, the experience will be eliminated from Eit. ~ When Bayesian learning theory relies on the premise of some prior memory ðMÞ. new evidence (e) from various information sources is available, learning occurs and follows Bayes theorem. That is: the posterior memory is updated using conditional ~ = PðejMÞ ~ ⋅ PðMÞ=PðeÞ ~ probabilities: PðMjeÞ (this equation can also be expressed as posterior = likelihood prior/evidence). When a new alternative in this dimension is experienced and the associated generalised cost falls into category j, the updated memory becomes Md = (m1, …, mj + 1, …, mJ). Let the vector Pd = (p1, …, pj, …, pJ) represent an individual’s subjective beliefs, where pj is the subjective probability that an additional search in dimension d would lead to an alternative with jth level of generalised cost. In order to quantitatively link Md and Pd, we assume that individuals’ prior beliefs and memory follow a Dirichlet distribution, which is a J-parameter distribution. Therefore the posterior beliefs will also be Dirichlet distributed since the Dirichlet is the conjugate prior of the multinomial distribution (Rothschild, 1974). The probability density function is defined as: P=

ΓðNÞ

J

mj − 1

∏ pj

∏Jj= 1 Γðmj Þ j = 1

ð10:2Þ

where N denotes the total number of Md observations and Gamma function Γðmj Þ = ðmj − 1Þ! According to the law of large numbers, as sample size N grows, this assumption asymptotically converges to: Eðpj Þ =

mj N

ð10:3Þ

Bayesian learning is capable of describing updates of spatial knowledge about the attributes of spatial objects, and relations between spatial objectives when repeated observations are available. Travel time on a roadway section, waiting time at a transit station, level of congestion for a specific trip during a peak hour, attractiveness of housing unit in a neighbourhood, distance between an origin

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and a destination, closeness of a shopping centre to the route from work back home, etc.

10.2.2. Modelling Multidimensional Search An individual, based on her/his past experience Eit and subjective beliefs Pdit , forms expectations on potential gain (search gain) from behavioural adjustments along each dimension. The decision to search for a new alternative is based on the interplay of subjective search gain and perceived search cost. Let an agent’s generalised cost on the currently used alternative be C. The subjective search gain (gdt) is based on subjective beliefs, Pdit , and defined as the expected improvement in regard to generalised cost savings per trip from an additional search: X gdt = p ⋅ ðC − Cj Þ ð10:4Þ jð∀C < CÞ j j

where C is actually the minimum of all experienced generalised costs because individuals can select from all tried alternatives in dimension d and pick the one with the d lowest costs Cmin . We assume all individuals start with a preferred travel pattern. It can be the stabilized travel pattern with an initial generalised cost C0. Once a policy/ congestion stimulus emerges, travel condition deteriorates. Let us further assume that individuals have the initial beliefs that search and switching to another alternative will lead to a travel condition as good as their original travel condition C0 until they search and experience otherwise. As the search process proceeds, the subjective probability of finding an alternative with C0 after N searches is 1/(N + 1). Therefore, Eq. (10.4) can be further simplified as: gdt =

d Cmin − C0 N þ1

ð10:5Þ

d While C0 remains universal among all dimensions, Cmin , the currently best travel option(s) in dimension d, can differ in each dimension d since the search processes in different dimensions vary and result in diverse outcomes. The subjective search gain gdt evolves and reflects how much value each search can gain based on subjective beliefs. Once gdt is less than or equal to zero, it indicates that search along dimension d is no longer worthwhile and the search process will not initiate. A positive gdt will asymptotically decrease to zero as the number of searches increases and d as a better alternative is found (Cmin getting increasingly closer to C0). Furthermore, the theory formulates satisficing behaviour that even with positive gains, individuals may stop search once the gain is lower than the perceived search cost. The search and information acquisition are no longer free as this theory recognises the inconveniences and risks associated with each behaviour adjustment dimension. This impedance is conceptualised as a search cost for each agent and each dimension. Search cost can be perceived and inferred once individuals’

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searching sequence can be reconstructed using empirical observations collected from survey. The empirical data provides evidence about agents’ search and decision processes. Each individual follows her/his own path along the three dimensions in reaching the final behaviour decisions. When it is observed that an individual ends her/his search in dimension d and has searched N times along that dimension for the time being, it infers that the individual satisfices after N rounds of search in d. The search cost must be lower than gd,t−1 so that the Nth search is meaningful and rewarding. Meanwhile, the search cost must be higher than gdt so that the (N + 1)th search does not occur. Let us denote individual i’s search cost along dimension d as scdi, which is viewed as an innate personal characteristic for individual i. It can be estimated by using the lower and upper bounds: d Cmin;t − 1 − C0 N

ð10:6aÞ

d Cmin;t − C0 N þ1

ð10:6bÞ

1 ðgd;t − 1 þ gdt Þ 2

ð10:6cÞ

scdi ≤ gd;t − 1 = scdi ≥ gdt = sc di =

Note that for each individual, only one of the multidimensional perceived search costs can be perceived from the empirical data. A subsequent regression analysis for all survey subjects and all dimensions thus needs to be estimated in order to empirically model search cost. We specify that the search cost model in dimension d as: scdi = β0 þ β1 C0 þ β2 gender þ β3 fixedsch þ β4 purpose þ β5 income1 þ β6 income2 þ β7 income3 þ β8 distance þ β9 peak þ β10 veh þ ɛi

ð10:7Þ

where C0 is the generalised cost for the originally reported travel experience; distance measures the mileage that the subject travels; Dummy variables include gender (equals to 1 if the subject is female), fixedsch (equals to 1 if the subject has fixed travel schedule), purpose (equals to 1 if the trip purpose is work/school), peak (equals to 1 if the travel is in peak-hour periods) and veh (equals to 1 if household number of vehicles is greater than 2). Different household annual income levels are considered in the model (income1: less than $50,000; income2: $50,000100,000; income3: $100,000150,000; income4: $150,000 and above). In our model, C0 is identified as an instrumental variable (IV) in order to better incorporate the sufficiently high correlation between C0 and other independent variables. We employ the generalised method of moments (GMM) and two-stage least-squares (2SLS) estimator. Denoting the IV as z and the independent variables as x, we can estimate parameters β from the population moment conditions: E½zðscdi − xβÞ = 0

ð10:8Þ

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The estimation result is reported in Table 10.1. The search cost is positively related to the initially experienced generalised cost of the travel. Lower-income agents have higher search costs along the mode dimension. Female agents are more reluctant to search departure times and routes than to search alternative modes. Fixed schedule and travelling during peak-hour increase the search cost for all dimensions. Travel distance has a negative impact on search cost meaning that the longer the travel distance, the more likely she/he will search for alternatives. The coefficients for trip purpose indicate that agents doing commute travels have more incentive to search for alternative modes and departure times. By estimating and applying search cost models, one can make personal/household characteristics endogenous in the search process and model diversified and behaviourally rich multidimensional search. It helps explain why some travellers may adjust routes first while others may adjust departure time first in response to the same stimulus. This feature can potentially provide a rich level of details especially for policy/social equity analysis whence measuring the impacts/benefits by different socio-economic strata of society is of interest. It is hypothesised that agents will search the most rewarding dimension with the highest search gain/cost ratio. Successive unrewarding searches along a particular

Table 10.1: Multidimensional perceived search cost models (generalised method of moments and instrumental variable). Models Variables

Search cost (d: mode)

Search cost (d: departure time)

Coefficients (std. err.) 0.023 (0.010)*** 0.014 (0.088) 0.118 (0.065)*** −0.101 (0.062)*

Coefficients (std. err.) 0.008 (0.001)*** 0.162 (0.071)** 0.194 (0.080)** −0.091 (0.056)*

Generalised Cost C0 Gender (female) Fixed schedule Purpose (work/ school) Annual income 0.188 (0.106)* (2) −0.088 (0.021)*** Constant 1.341 (0.148)***

Search cost (d: route) Coefficients (std. err.) 0.001 (0.000)*** 0.098 (0.046)*** 0.115 (0.045)*** 0.098 (0.048)**

−0.272 (0.201)

−0.299 (0.060)***

−0.285 (0.203)

−0.207 (0.066)***

−0.542 (0.234)**

−0.089 (0.086)

−0.008 (0.001)***

−0.006 (0.000)***

0.112 (0.062*) 0.298 (0.092)*** 0.402 (0.225)*

0.010 (0.041) −0.035 (0.053) 0.384 (0.068)***

*, **, *** significant at 90%, 95%, 99% confidence level.

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behavioural adjustment dimension (e.g. route) will lead to diminishing subjective search gain for that dimension and at a later point cause the search to shift to a different behaviour dimension (e.g. departure time). Once the ratios for all dimensions drop down below one, the multidimensional search process ceases. Since gdt is monotonically decreasing and converges to zero, the search is guaranteed to reach stability. The interplay of these search gains and costs along all feasible behavioural dimensions defines the bounded rationality embedded in the theory. It collectively determines the prospects for profitable searches over finite horizon and guarantees a convergence of behavioural changes. It quantitatively theorizes when individuals start seeking behavioural changes, how they initially change behaviour, how they switch behavioural adjustment dimensions and when they stop the search.

10.2.3. Search Rules and Decision Rules An agent will keep the status quo and repeat her/his habitual behaviour once she/ he decides not to search in any dimension. Once determining a dimension to search, a search process is invoked to find useful alternatives to meet travel demand. Spatial/temporal search is not random and can be biased (Humphreys & Whitelaw, 1979). For instance, if a person currently departs at 8 am and is not satisfied with the resulting travel and/or schedule delay, the person may be more likely to try departing at 7:30 am and 8:30 am than 7 am and 9 am (i.e. an anchor effect). Different knowledge extracting technologies can be applied to mine individuals’ search rules and decision rules. Here we adopt production rules for shorterterm departure time search and route search. For longer-term travel mode search, the process of identifying an alternative mode is theorized as a hidden Markov process. The rule-based method was well documented in Arentze and Timmermans (2004). Here we present the hidden Markov method (the main idea is illustrated in Figure 10.2) which emphasises the dynamic linkages between time periods. As displayed in Figure 10.2, two major components are highlighted in this model: • Hidden states and transitions. The transition is triggered by the evolving travel experience. Starting from an initial state distribution (i.e. at time 1, the probability density function Pr(Hi1) that traveller i is in state Hi1), a sequence of Markov chains is employed to express the likelihood that the level-of-service (LOS) experiences of the habitual mode in the previous periods are strong enough to transition the traveller to another hidden state Pr(Hit|Hit-1). For example, successively experiencing longer waiting time when using transit may cause the traveller to switch to auto-loving state. • State-dependent decision rules. Given the hidden state that a traveller i is in, the probability that she/he will identify mode yit as the alternative in the mode searching stage at time t is determined by Pr(Yit = yit|Hit). Yit is the mode searching decision made by traveller i at time t.

Multidimensional Travel Decision-Making

Traveler i ’s current experienced level-of-service (LOS)

Hidden State: Hit = 1

P(Searcht | Hit = 1)

Hidden State: Hit = 2

P(Searcht | Hit = 2)

223

Yit : Observed mode searching behaviour at time t Hidden State: Hit = h

P(Searcht | Hit = h)

Hidden State: Hit = H

P(Searcht | Hit = H)

Hidden states and transitions

State-dependent searching choice Time t Time t-1...

Figure 10.2: A hidden Markov model of travel mode searching dynamics.

An individual’s decision probabilities are correlated through the underlying path of the hidden states (yi1, yi2, ..., yiN), because of the Markovian properties of the model. Therefore, the joint likelihood function is given as: LðHit Þ = PrðYi1 = Yi1 ; ⋯; YiN = yiN Þ  N N XX X = ⋯ PrðHi1 Þ ∏ PrðHit jHit − 1 Þ ⋅ ∏ PrðYit = yit jHit Þ Hi1

Hi2

HiN

t=2

ð10:9Þ

t=1

More details of this hidden Markov search model regarding model variables and estimation procedure can be found in Xiong and Zhang (2014). After each round of search, a new alternative is identified. Agents either change behaviour to use the new alternative or stay with their habitual behaviour. This is determined by a set of decision rules. Even though during the multidimensional search process many alternatives may be visited, the final decision is assumed to be the outcome of a series of switching decisions. Production rules derived by various machine learning algorithms (Cendrowska, 1987; Cohen, 1995; Quinlan, 1986) are selected here to represent decision rules. Departing from random utility maximisation, this assumption about the search-decision procedure relaxes the unrealistic assumption of human information processing and computational capabilities and incorporates individual-based historical dependencies. It also improves the computational efficiency of agent-based simulation since the execution of production rules only requires minimum computational resources. These search and decision rules are empirically derived for each behavioural dimension and are discussed in greater details elsewhere (Xiong & Zhang, 2013a, 2013c, 2014; Zhang, 2007).

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10.2.4. Empirical Data Collection The development of those quantitative models can be data intensive. This research conducts a stated adaptation experiment administered online to explore possible substitutions to the longitudinal information that is typically missing. This survey method is particularly useful when one seeks answers from respondents on a number of what-if questions such as ‘what would you react if you were faced with specific constraints/conditions’ (Arentze et al., 2004). It helps capture respondents’ multifaceted behavioural responses. Furthermore, it has the capability to infer the procedural decision-making process which embeds the behavioural foundation of the proposed theory and models since respondents will naturally exhibit satisficing behaviour if playing the scenarios repeatedly for a sufficient number of iterations. The survey procedure is reported in Figure 10.3. Reported Most Recent Trip: Eit Policy/Congestion Stimuli Change behaviour?

No

Survey Ends

Yes What is the behaviour dimension you would like to search?

Search Travel Mode

Search Departure Time

Search Route

Identify an Alternative Travel Mode

Identify an Alternative Departure Time

Identify an Alternative Route

No

Switch to this alternative?

Yes Simulated travel experience

Figure 10.3: The stated adaptation experiment flowchart.

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Starting from a self-reported most recent trip, exogenous policy/congestion changes are assumed in each scenario to alter the travel condition for that trip. It is further assumed that each agent will adapt to those changes by searching new modes, departure time and/or routes. The dimensions wherein the behaviour adjustment occurs are asked explicitly in the survey for each subject. The subject then is asked to elaborate the alternative she/he would identify and search along that dimension (this data infers the determination of search rules). Once a search has been recorded by a subject, the programme will feed a corresponding travel condition simulated in the back-end for the subject to consider and make a switching decision between the alternative and the habitual one (this data infers the decision rules). Another round of behaviour adjustment (could be in the same dimension or in another dimension) will occur unless the subject states satisfactory about the travel experience. Iteratively repeating this process, a complete behavioural adjustment sequence of each subject can be observed. Initial samples include 110 University of Maryland staffs and students. They perform adaptations under schemes such as overall congestion increase and road-pricing scenarios.

10.3.

Simulation Results

The proposed multidimensional behavioural theory and models have been estimated and implemented in an agent-based simulation to demonstrate the capability. A toy network with one origindestination pair, three alternative routes and three travel modes (auto, carpool and transit) is employed. The scenario that is analysed in this simulation is an assumed 10 per cent increase in travel demand which creates excessive travel time and cost for the simulated agents and stimulates them to start the multidimensional behaviour adjustments. 90,000 agents are generated in this microsimulation of extended morning peak hours (5:0010:00 am). Agents’ characteristics are synthesised based on Transportation Planning Board (TPB) — Baltimore Metropolitan Council (BMC) Household Travel Survey (2007/2008) data. In the simulation, agents travel from origin to destination, accumulate experience, make behavioural adjustment on one or multiple dimensions, dynamically update beliefs and eventually satisfy on their decisions. The uniqueness of the model brings attention to each agent for whom the interplay of search gain and search cost is dynamically modelled in order to determine the behavioural dimension wherein the search and decision process occurs. Figure 10.4 illustrates the evolving gain/cost ratio for a particular agent. On simulation day 1, the agent initially believes that all dimensions are rewarding (with all gain/cost ratios above one) while the most profitable dimension is the mode dimension. She/he then employs search rules and decision rules to identify and examine one alternative mode. While the subsequent search reveals further information, this agent’s knowledge and subjective beliefs on the mode dimension evolve significantly. And on the second day, the departure time dimension emerges to be the one with the highest gain/cost ratio. A search for alternative departure

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Chenfeng Xiong et al. 2 Gain/cost ratio if searching for mode Gain/cost ratio if searching for departure time Gain/cost ratio if searching for route

1.8

Gain/Cost Ratio

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

5

10 Day (# of iterations)

15

20

Figure 10.4: The evolving gain/cost ratios of multidimensional travel behaviour. time is therefore performed. Iterating this process, the agent forms a time-dependent search path about choosing behavioural adjustment dimensions: mode-departure time-route-mode. On the fifth day, the gain/cost ratios of all dimensions drop down below 1, which indicates that this agent subjectively believes that no more searches are necessary. The agent is thus satisfied and stays dormant afterwards. Once a new turbulence emerges in the transport system, such as new policies and booming travel demand, the agent may be influenced in the way that the gain/cost ratios in certain dimensions grow. And the agent may seek further changes. The convergence of the multidimensional behaviour is illustrated in Figure 10.5a. Overall, the model predicts active and reasonable agent behaviour along the three behavioural dimensions. The convergence processes are smooth. With the innate bounded rationality and satisficing behaviour, agents reach steady state and stop search within 25 search iterations. If each agent travels five days a week and all agents start search at the same time, it would take five weeks for the traffic to stabilise and equilibrate on the network. This is an interesting finding that on the one hand, it allows us to model the gradual behaviour adaptation to exogenous policies (e.g. pricing policy in Stockholm gradually nudge drivers to change behaviour, Bo¨rjesson, Eliasson, Hugosson, & Brundell-Freij, 2012). On the other hand, it suggests potential applicability of the proposed theory in large-scale planning models and simulation since it embeds multidimensional behavioural responses while maintaining a reasonable converging speed. In response to the assumed demand increase, changing route and changing departure time are the most significant ways of behavioural adaptation. The initially high route searching frequency cools down rapidly since agents can hardly identify any better alternative routes under the assumed overall demand increase. Agents quickly learn the fact and update the subjective beliefs, which results in a decreasing search gain in the route dimension. Then agents turn to search alternative modes

Multidimensional Travel Decision-Making × 10

(a)

4

227

(b) 0.8 # of travelers searching for departure time # of travelers searching for mode # of travelers searching for route

Mode Share (Auto D) Mode Share (Auto P) Mode Share (Transit) Mode Share Percentage

Number of Agents

4

3

2

1

0.6

0.4

0.2 0 0

5

10 15 Day (# of iterations)

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25

(c) 3000

0

(d)

5

10 15 Day (# of iterations)

20

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1 0.9

Relative Searching Payoff

Number of Agents

2500 2000 1500 1000 500

Demand before peak spreading Demand after peak spreading

0 4:00

5:00

6:00

7:00 8:00 9:00 Time-of-Day

10:00

11:00

0.8 0.7 0.6 0.5 0.4 0.3 0.2

Payoff/payoffmax if searching for mode

0.1

Payoff/payoffmax if searching for departure time Payoff/payoffmax if searching for route

0 0

5

10 15 Day (# of iterations)

20

25

Figure 10.5: Agent-based experiment of the multidimensional travel behaviour theory. (a) The convergence of the multidimensional behaviour, (b) Agents’ mode search and switching behaviour, (c) Agents’ departure time changes and peak spreading, (d) Agents’ payoff dynamics.

and departure times instead. Thus we can observe in the simulation an increasing number of agents searching for alternative departure times in the second and third simulation days. A few agents search for alternative modes. Agents’ mode searching and switching behaviour is illustrated in Figure 10.5b. Agents’ departure time changes are illustrated in Figure 10.5c. By aggregating the individual behaviour into travel patterns, we can observe that the multidimensional learning and adaptation leads to a slight percentage decrease of auto drivers (Auto D in Figure 10.5b). Those agents switch to auto passengers (Auto P) or transit users. The aggregate mode share of auto drivers drops from 63.4% to 58.3%. After 6 simulation days, the mode share tends to be stabilized even though from the microscopic level, there still exist some 3000 travellers changing their travel modes. The active departure time changes lead to a significant peak spreading effect. The assumed demand increase results in more severe congestion and travel time unreliability especially during peak hours. The excessive travel time, cost and schedule delays make the departure time adjustments necessary in order for the agents to gain an acceptable payoff through search. The model predicts that the dominating behavioural responses to the stimulus are route changes and departure time changes, which

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are in consistency with the existing research (e.g. Arentze et al., 2004). Meanwhile, the model predicts the behavioural dynamics and adaptive processes, which advance our current understanding about multidimensional travel behaviour adjustments. Travellers in the multidimensional agent-based model are not perfectly ‘rational’ in that they do not maximise their utility (or payoff). Instead, they are restrained by information acquisition cost, decision cost, computational limitation, time budget and deadlines. They are not perfectly rational also in the way that they follow different intuitives and heuristic behavioural rules. Figure 10.5d demonstrates that through multidimensional learning and adaptation, agents search and improve their relative searching payoff. This term is defined as the ratio of the cumulative actual search gain and the cumulative subjective search gain (i.e. subjectively believed maximum payoff from the search) for all the searchers. Judging by the curves, the departure time dimension turns out to be the most profitable dimension. Once searching in this dimension, agents are able to retrieve the highest relative searching payoff. However, this learning and adaptation does not ensure them to make decisions that result in maximum payoff. This example demonstrates the bounded rationality of the agents in search and changing their behaviour.

10.4.

Discussion and Conclusion

This chapter introduces a theoretical framework to modelling multidimensional travel behaviour based on artificially intelligent agents, search theory and bounded rationality. For decades, despite the number of heuristic explanations for different results, the fact that ‘almost no mathematical theory exists which explains the results of the simulations’ (Herbert, 1999) remains as one of the largest drawbacks of agentbased computational process approach. This is partly the side effect of its special feature that ‘no analytical functions are required’. Among the rapidly growing literature devoted to the departure from rational behaviour assumptions, this theoretical framework makes an effort to embed a sound theoretical foundation for computational process approach and agent-based microsimulations. The theoretical contributions are three-fold: • A pertinent new theory of choices with experimental observations and estimations to demonstrate agents with systematic deviations from the rationality paradigm. Modelling components including knowledge, limited memory, learning and subjective beliefs are proposed and empirically estimated to construct adaptive agents with limited capabilities to remember, learn, evolve and gain higher payoffs. All agent-based models are based on empirical observations collected via various data collection efforts. • Modelling procedural and multidimensional agent-based decision-making. Individuals choose departure time, mode and/or route for their travel. Individuals also choose how and when to make those choices. A behaviourally sound modelling framework should focus on modelling the procedural decision-making

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processes. This study seeks answers to questions that largely remain unanswered including but not limited to: (1) When do individuals start seeking behaviour changes? (2) How do they initially change behaviour? (3) How do they switch behaviour adjustment dimensions? (4) When do they stop making changes? • The transformation from the static user equilibrium to a dynamic behavioural equilibrium. Traditional solution concepts are based on an implicit assumption that agents have complete information and are aware of the prevailing user equilibrium. However, a more realistic behavioural assumption is that individuals have to make inferences. These inferences can be their subjectively believed search gain (or perceived distributions of travel time and travel cost), the multidimensional alternatives they subjectively identify, and the heuristics they employ to evaluate alternatives. It is the process of making inferences that occupies each individual in making a decision. With search start/stop criteria explicitly specified, this process should eventually lead to a steady state that is structurally different to user equilibrium. The estimation of the proposed agent-based models usually needs additional behaviour process data. Whether or not the increased data needs can be justified by improved model realism and model performance in applications can be a subject for further examination. This chapter empirically estimates the models using data collected from a stated adaptation survey, a similar but different survey structure compared to stated preference experiments. This survey method effectively captures adaptations in response to changing attributes or context and can record behaviour process if implemented in an iterative manner (see e.g. Khademi, Arentze, & Timmermans, 2012). The observed behaviour process actually is a search path possessed by each respondent. This historical information can be applied to further calibrate the knowledge model or the search cost models. Another future research direction may explore how advanced data collection technologies such as GPSsurveys, smartphone applications and social network data can improve the affordability and quality of behaviour process data and further support the proposed modelling framework. The numerical example presented in the paper highlights the capabilities of the proposed theory and models in estimating rich behavioural dynamics, such as multidimensional behavioural responses, day-to-day evolution of travel patterns, and individual-level learning, search and decision-making processes. The computational efficiency of the proposed models needs further exploration through real-world implementations using agent-based simulation techniques. It is believed that the flexible framework, computational efficiency and more realistic assumptions can make the proposed modeling tool extremely suitable for integrated large-scale multimodal planning/operations studies which typically have to cope with millions of agents. This work is primarily exploratory in its conceptualization of a descriptive theory, estimation of quantitative models and demonstration in an agent-based microsimulation. In an era of big-data access, multicore processors and cloud computing, the ambition of transportation demand modellers has never been greater. The hope is that the preliminary findings in this chapter could raise interest in the behavioural foundation

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of multidimensional travel behaviour as well as in microsimulating people’s complex travel patterns in the timespace continuum. Extensive examination of the proposed tool on a larger and more representative survey sample and for real-world studies is necessary before we can conclude that the tool is fully practice-ready.

References Arentze, T. A., Hofman, F., & Timmermans, H. J. P. (2004). Predicting multi-faceted activity-travel adjustment strategies in response to possible congestion pricing using an internet-based stated adaptation experiment. Transport Policy, 11(1), 3141. Arentze, T. A., & Timmermans, H. J. P. (2003). Modeling learning and adaptation processes in activity-travel choice  A framework and numerical experiment. Transportation, 30(1), 3762. Arentze, T. A., & Timmermans, H. J. P. (2004). A learning-based transportation oriented simulation system. Transportation Research Part B, 38(7), 613633. Arentze, T. A., & Timmermans, H. J. P. (2005). Modelling learning and adaptation in transportation contexts. Transportmetrica, 1(1), 1322. Auld, J., & Mohammadian, A. (2012). Activity planning processes in the agent-based dynamic activity planning and travel scheduling (ADAPTS) model. Transportation Research Part A, 46(8), 13861403. Bhat, C. R., & Guo, J. Y. (2007). A comprehensive analysis of built environment characteristics on household residential choice and auto ownership levels. Transportation Research Part B, 41(5), 506526. Bhat, C. R., & Koppelman, F. S. (1999). Activity-based modeling of travel demand. In Handbook of transportation science (pp. 35–61). Boston, MA: Kluwer Academic Publishers. Bo¨rjesson, M., Eliasson, J., Hugosson, M. B., & Brundell-Freij, K. (2012). The Stockholm congestion charges  5 years on. Effects, acceptability and lessons learnt. Transport Policy, 20, 112. Bowman, J. L., & Ben-Akiva, M. E. (2001). Activity-based disaggregate travel demand model system with activity schedules. Transportation Research Part A, 35(1), 128. Cendrowska, J. (1987). PRISM: An algorithm for inducing modular rules. International Journal of Man-Machine Studies, 27, 349370. Cohen, W. W. (1995). Fast effective rule induction. In Proceedings of the twelfth international conference on machine learning, Lake Tahoe, California. Golledge, R. G., Kwan, M.-P., & Garling, T. (1994). Computational-process modelling of household travel decision using a geographical information system. Papers in Regional Science, 73(2), 99117. Herbert, D. (1999). Adaptive learning by genetic algorithms: Analytical results and applications to economic models. New York, NY: Springer-Verlag. Humphreys, J. S., & Whitelaw, J. S. (1979). Immigrants in an unfamiliar environment: Locational decision-making under constrained circumstances. Geografiska Annaler, 61B(1), 818. Khademi, E., Arentze, T. A., & Timmermans, H. J. P. (2012). Designing stated adaptation experiments for changes to activity-travel repertoires: Approach in the context of pricing policies. In Proceedings of European Transport Conference, Glasgow, Scotland.

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Chapter 11

Prospect Theory and its Applications to the Modelling of Travel Choice Erel Avineri and Eran Ben-Elia

Abstract Purpose  This chapter explores Prospect Theory — a descriptive model of modelling individual choice making under risk and uncertainty, and its applications to a range of travel behaviour contexts. Theory  The chapter provides background on Prospect Theory, its basic assumptions and formulations, and summarises some of its theoretical developments, applications and evidence in the field of transport research. Findings  A body of empirical evidence has accumulated showing that the principle of maximisation of expected utility provides limited explanation of travel choices under risk and uncertainty. Prospect Theory can be seen as an alternative and promising framework for travel choice modelling (although not without theoretical and practical controversy). These findings are supported by empirical observations reported in the literature reviewed in this chapter. Originality and value  The chapter provides a detailed account of the design and results of accumulated research in travel behaviour research that is based on Prospect Theory’s observations, insights and formulations. The potential of Prospect Theory for particular decision-making in travel behaviour research is articulated, main findings are presented and discussed, and limitations are identified, leading to further research needs. Keywords: Prospect Theory; risk; uncertainty; expected utility theory; route choice

Bounded Rational Choice Behaviour: Applications in Transport Copyright r 2015 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISBN: 978-1-78441-072-8

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Erel Avineri and Eran Ben-Elia

Introduction

The behavioural assumptions on travellers’ choices, applied in mainstream travel behaviour models, can be traced back to microeconomic theory and the paradigm of rational man. The development of travel behaviour models in what Timms (2008) refers to as the ‘economics era’ (stretching from the 1970s to the present day) has been dominated by neoclassical economic concepts, focussing upon the representation of people as individual rational choice makers, interacting together to form a state of equilibrium. This modelling approach, and the specific application of Random Utility Models (RUM), has been largely inspired by models of consumer choice; the principles of rational behaviour such as individual’s tendency to maximise his/her utilities was applied to choice models to explain travel choices such as destination, mode, route and time. An argument that has emerged in transport studies was that the assumptions of mainstream travel behaviour models are often made without reference to existing theories in behavioural sciences (in particular cognitive psychology) (see, e.g. Ga¨rling, 1998). Moreover, although many theoretical contributions to the field of mainstream choice modelling in transport were developed through empirical studies of observed (or stated) travel behaviours, a clear distinction between normative models and descriptive models has not been always made (Ga¨rling, 1998). Traditional travel choice models have largely assumed a set of logical-rational, and sometimes intuitive, behavioural assumptions — leading to normative models that predict how completely rational travellers should behave, but might fail to capture the behaviour of travellers who do not apply complete rationality in their choice behaviours. In the recent two decades researchers and modellers have become more aware of the limitations and flaws in modelling approaches that are based on the rational man paradigm and become more interested in developing alternative theoretical and empirical frameworks to provide better explanations and descriptions of how travellers make choices, leading to descriptive models of choice behaviour. Although the bounded rationality (a term coined by Simon, 1956) in individual choice making has been observed in early studies of human behaviour, an increased interest in this area (mainly in economic and financial contexts) has emerged from the works of cognitive psychologists, among them Kahneman and Tversky, who indicated that individuals’ choices deviate from the predictions of the rational man paradigm — later on inspiring research on travellers’ bounded rationality. Research in behavioural sciences, especially cognitive psychology, indicates that individuals’ choices in a wide range of contexts deviate from the predictions of economic theory which largely assume a rational behaviour. Some of these deviations are systematic, consistent, robust and largely predictable. One of the most influencing papers in this area was published 35 years ago by Kahneman and Tversky (1979). Their seminal paper on Prospect Theory (PT) appeared in Econometrica and has later become the most cited paper ever to be published in this prestigious journal (with over 30,000 citations of it according to Google Scholar at the time of writing this book chapter). Exploring individual choice behaviour through a series of stated-choice experiments, they found strong evidence

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of systematic deviations from normative models of risky choice making, like the Expected Utility-maximisation (EU) model. This has led them to the development of a descriptive model of choice making, which captured the observed behaviour of individuals in settings that involve risky economic and financial choices. For this work, and its further extension, known as Cumulative Prospect Theory (CPT) (Tversky & Kahneman, 1992), and more generally for ‘having integrated insights from psychological research into economic science, especially concerning human judgment and decision-making under uncertainty’ (The Royal Swedish Academy of Sciences, 2002), Daniel Kahneman was awarded the 2002 Nobel Memorial Prize in Economics for the work he did in collaboration with the late Amos Tversky. Kahneman and Tversky’s critique of expected utility theory as a descriptive model of decision-making under risk, the development of PT as an alternative model of choice making under risk and uncertainty, has influenced the emergence of a new field — behavioural economics. In the last three decades PT and its elements have been applied to explain behaviours in a range of contexts, among them finance, economics, consumer choice and political sciences. As many of the behavioural assumptions and paradigms applied in transport research have emerged from classical microeconomic theory, which largely assumes unbounded rationality, it was only natural for transport researchers to revisit these assumptions and explore the potential of PT in providing alternative explanations and better predictions of travel choices. The result is a growing body of recent studies that apply PT (or related theoretical notions such as loss aversion) to model a variety of travel choicedimensions such as route choices (e.g. Avineri, 2006; Avineri & Prashker, 2003, 2004, 2005), departure time choices (Jou, Kitamura, Weng, & Chen, 2008; Senbil & Kitamura, 2004) and trip generation choices (Schwanen & Ettema, 2009). Dozens of studies on empirical and theoretical applications of PT to route choice, mode choice, departure time choice and other travel choice were published in the transport research literature. A seminar on PT and its applications to transport was held in Delft, the Netherlands, in October 2009 and a follow-up special issue of the European Journal of Transport and Infrastructure Research (EJTIR) (Avineri & Chorus, 2010) which included a collection of developments in this field, particularly some relevant theoretical discussions, potential applications and critical views of PT and its applications in the study of travel behaviour. Several review papers on the theoretical, empirical and applied aspects of PT have been published in the transport literature. These include van de Kaa (2010a, 2010b), Timmermans (2010), Li and Hensher (2011), Ramos, Daamen, and Hoogendoorn (2014), and Rasouli and Timmermans (in press).

11.2.

Making Travel Choices under Risk: The Assumptions of Expected Utility Theory (EUT) and Prospect Theory (PT)

Ben-Akiva and Lerman (1985, p. 32, cited in Avineri, 2012) described the theory of choice as a collection of procedures that define the following elements: (i) decisionmaker, (ii) alternatives, (iii) attributes of alternatives and (iv) a decision rule.

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The attractiveness of an alternative in the mind of the traveller is described as a vector of the attribute values (which is later reduced to a scalar, ‘utility’, as an index of the attractiveness of an alternative). Moreover, travellers are largely expected to act as rational human beings, and specifically to exhibit consistency and transitivity in their choices (Ben-Akiva & Lerman, 1985, p. 38). As realistic travel choices are strongly affected by dynamic and stochastic changes to both demand and supply of transport systems and networks, travel choice decisions usually involve trade-offs between expected attribute values and uncertainty, where travel time and its reliability are rather common attributes of such models. Stochastic route-choice models that account for variable network travel times assuming different risk-taking behaviours were presented by Small (1982), Mirchandani and Soroush (1987), Tatineni, Boyce, and Mirchandani (1997), Chen and Recker (2000) and others. Although a range of measures have been incorporated in models to represent risk and uncertainty of travel attributes, standard deviation of travel times has been commonly used. Where a model allows for risk-taking elements, risk aversion is commonly assumed. This is much in line with normative models of choice making, such as Expected Utility Theory (EUT); when facing choices with comparable returns, a decision-maker is assumed to prefer the less risky alternative (Bernoulli, 1738; Friedman & Savage, 1948), which may be described by a concave utility function. Famous measures of risk aversion were introduced by Pratt (1964) and Arrow (1965). The assumptions of EUT, introduced by Von-Neumann and Morgenstern (1944) and extended by Savage (1954), have largely shaped mainstream modelling of choice modelling under risk and uncertainty. In EUT, it is presumed that for every decision-maker there exists some real-valued function u, defined on the relevant set X of outcomes x1, x2, …, xN, so that if one available action a results in probabilities pi over the outcomes xi (for i = 1, …, N) and another available action b results in probabilities qi over the same outcomes, then the decision-maker (strictly) prefers action a to action b if and only if the statistically expected value of this ‘utility function’ u is greater under a then under b. Formally, the criterion for choosing a is thus N X i=1

pi uðxi Þ >

N X

qi uðxi Þ

ð11:1Þ

i=1

Hence, given travel time distributions, the cognitive process of a traveller would be reduced to a problem of formulating and maximising expectation. The traveller is thus assumed to behave as he/she correctly assigned probabilities to relevant random travel times and choose a route that maximises the expected value of his/her utility (e.g. Minimising the expected travel time). Variations in travel times could result from the differences in the mix of vehicle types on the network for the same flow rates, differences in driver reactions under various weather and driving conditions, differences in delays experienced by different vehicles at intersections, etc. Travellers’ risk-taking behaviour might therefore be described as a function of

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the variance associated with travel time (or other attributes of travel). It is hence quite common to describe and investigate travellers’ risk-taking behaviour in the framework of EUT. It is assumed that decision-maker’s attitude toward risk can be rationalised by an expected utility function. See Bates, Polak, Jones, and Cook (2001), Noland and Polak (2002), Denant-Boe`mont and Petiot (2003), Sun, Arentze, and Timmermans (2005), Ettema, Tamminga, Timmermans, and Arentze (2005), de Palma and Picard (2005), Batley (2007), Asensio and Matas (2008), Chorus, Arentze, and Timmermans (2009), and Batley and Iba´n˜ez (2009) for examples of EUT-maximisation based travel choice studies. Behavioural-science research found that people’s rationality is restricted by their cognitive limitations (Simon, 1957). Experiments in behavioural sciences often find systematic deviations from the predictions of classical theory of utility maximisation and risk (Camerer, 1988; McFadden, 1999). Specifically, human decision-making deviates in many ways from the assumptions of EUT. The development of cognitive psychology provided the first real critique of EUT, as an explanation of decisionmaking behaviour under risk and uncertainty conditions. While retaining the central premise that people actively evaluate risk(s) in terms of costs and benefits, cognitive research during the 1960s and 1970s shifted the emphasis towards a consideration upon ‘errors’ and ‘biases’, in decision-making processes, or lapses from optional rationality (see Coombs, 1975; Fishburn, 1977; Hansson, 1975; MacCrimmon & Larsson, 1979). A principal contribution in this area was the foundational work of Tversky and Kahneman (Kahneman & Tversky, 1979; Tversky, 1969; Tversky & Kahneman, 1974, 1981). They provided extensive empirical, laboratory-based, evidence of instances in which human decision-making deviated from EUT. They concluded that biases, errors and misconceptions typify much of human decisionmaking in the presence of risk. The Allais Paradox (Allais, 1953, 1979) and other deviations from EUT were studied by Kahneman and Tversky (1979). They showed that changing the ways by which alternatives are framed could generate predictable and dramatic shifts in preference. Following an empirical study of decision-making under risk, Kahneman and Tversky (1979) displaced the utility-maximisation model in favour of what they considered to be a more behaviourally realistic formulation: PT. Unlike traditional economic theories, which deduce implications from normative preference axioms, PT takes a descriptive approach. One of the most notable concepts related to PT and other descriptive models of choice is the concept of reference-dependent preferences. How an individual assesses the outcome of a choice is determined in large part by its contrast with a ‘reference point’, as by intrinsic taste for the outcome itself (Kahneman & Tversky, 1979; Ko¨szegi & Rabin, 2006). Reference points have been much discussed in the specific context of loss aversion: it was generally observed that in the evaluation of choices people tend to feel and respond differently to outcomes that are perceived as gains or losses, against a reference point (Kahneman & Tversky, 1979; Thaler, Tversky, Kahneman, & Schwartz, 1997). PT assumes that risky outcomes associated with individual choices (‘lotteries’) are evaluated in a two-step process: an initial phase of editing and a subsequent phase of evaluation. In the editing phase, lottery outcomes

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are mapped as gains or losses relative to some reference point. Such a reference point may be the current asset position, but may be influenced by the presentation of the lottery or expectations of the decision-maker. Under PT, value is assigned to gains and losses rather than to final assets; also probabilities are replaced by decision weights. The evaluation phase utilises a value function vð⋅Þ and a probability weighting function πð⋅Þ Consider a lottery with three outcomes: x with a probability p, y with a probability q, and the status quo with a probability 1 − p − q. The prospect-theoretic value of the lottery is then given by πðpÞvðxÞ þ πðqÞvðyÞ

ð11:2Þ

Note that the argument of the value function is the lottery payoff, which is the change in wealth, and not the level of it. In PT, the carriers of utility are gains and losses measured against some implicit reference point. A strong intuition about preferences in risky environment is that people treat positive outcomes (‘gains’) and negative ones (‘losses’), differently. This observed risk-taking behaviour, called loss-gain asymmetry or loss aversion, refers to the observation that people tend to be more sensitive to decreases in their wealth than to increases (Thaler et al., 1997). The value function (see Figure 11.1), defined on deviations from a reference point, is generally concave for gains (implying risk aversion), convex for losses (implying risk seeking) and is generally steeper for losses than for gains (implying loss aversion) which is consistent with the experimental evidence of Kahneman and Tversky (1979) on domain-sensitive risk preferences. The curvature of the value function is also consistent with the psychometric theory that as the deviations from a reference point increase, they are experienced with diminishing marginal sensitivity. Empirical measurements have generally supported concavity of the value function in gains (whereas less empirical research exists about the convexity in losses).

Value of Outcome

GAINS Lottery Outcome LOSSES

Figure 11.1: A hypothetical value function. Source: Drawing by Shani Avineri (Based on Kahneman & Tversky, 1979).

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This holds both when expected utility is assumed (e.g. Fishburn & Kochenberger, 1979) and when PT is assumed (Abdellaoui, 2000; Fennema & van Assen, 1999). While PT (Kahneman & Tversky, 1979) suggests that losses loom larger than equivalent gains on subjective evaluations of alternative (risky) choices, an alternative model of losses was recently proposed by Yechiam and Hochman (2013) who suggested that losses increase the allocation of attentional resources to the task. In this vein recent studies report evidence of negativity bias in autonomic nervous system arousal (for a review, see Rick, 2010). Another critique of EUT as a descriptive model of decision-making under risk, incorporated in Kahneman and Tversky’s (1979) PT, is related to decision-makers’ responses to the probabilities of outcomes associated with their choices. They observed that people underweight outcomes that are merely probable in comparison with outcomes that are obtained with greater certainty; and that weights associated with probabilities can be captured by a mathematical function of unique characteristics. Kahneman and Tversky’s experimental results imply a reversed-Sshaped probability weighting function. Diminishing marginal sensitivity occurs in this function with respect to the benchmark case of certainty. As probabilities move further away from the 0 and 1 endpoints, the probability weighting function flattens out. Experimental results reveal that this curve tends to lie disproportionately below the 45° line (thus, decision weights are generally lower than the corresponding probabilities, except in the range of low probabilities) as shown in Figure 11.2. Two important implications of the probability weighting function should be noted. First, the overweighting of small probabilities implies that decision-makers will make risk-seeking choices when offered low probability, high-reward lotteries.

π (p) Decision Weight

1 Underweighting of high probabilities

0.5

Overweighting of small probabilities 0 0

0.5

1

Stated Probability: P

Figure 11.2: A hypothetical weighting function. Source: Based on Kahneman and Tversky (1979) and Tversky and Kahneman (1992).

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Second, the extreme underweighting of high probabilities, which fall short of certain outcomes, make positive certain outcomes to be very attractive.

11.3.

Cumulative Prospect Theory (CPT)

Tversky and Kahneman (1992) have later developed a new version of PT that employs cumulative rather than separable decision weights. This version, called CPT, applies to uncertain as well as to risky prospects with any number of outcomes. CPT applies the cumulative functional separately to gains and to losses. An uncertain prospect f is a function from a finite set of states of nature S into a set of outcomes X, that assign each state an outcome. To define the cumulative functional, the outcomes of each prospect are arranged in increasing order of their values. A prospect f is then represented as a sequence of pairs (xi, Ai), which yields xi if Ai occurs, where xi > xj iff i > j and Ai is a partition of S. Positive subscripts are used to denote positive outcomes, negative subscripts to denote negative outcomes, and the zero subscript to index neutral outcome. The positive part of f, denoted f +, is obtained by letting f +(s) = f(s) if f(s) > 0, and f +(s) = 0 if f(s) ≤ 0. The negative part of f, denoted f −, is defined equivalently. CPT asserts that there exist a strictly increasing value function v : X→ℝ, satisfying v(x0) = v(0) = 0, and decision weights functions w+ and w−, such that for f = (xi, Ai), −m ≤ i ≤ n, Vð f Þ = Vð f þ Þ þ Vð f − Þ

Vð f þ Þ =

n X i=0

Vð f − Þ =

ð11:3Þ

π iþ vðxi Þ

ð11:4Þ

π i− vðxi Þ

ð11:5Þ

0 X i= −m

where V( f+) is the prospect gains value, V( f −) is the prospect losses value, π+( f+) = (π+0,…, π+n) are the decision weights of the gains, and π(f −) = (π−m, …, π0) are the decision weights of the losses. If the prospect f = (xi, Ai) is given by a probability distribution p(Ai) = pi, it can be viewed as a probabilistic or risky prospect (xi, pi). In this case, decision weights are defined by: π nþ = w þ ð pn Þ

ð11:6Þ

π −− m = w − ð p − m Þ

ð11:7Þ

Prospect Theory and Modelling of Travel Choice π iþ = w þ ð pi þ ⋯ þ pn Þ − w þ ð pi þ 1 þ ⋯ þ pn Þ

241

0≤i≤n−1

ð11:8Þ

π i− = w − ð p − m þ ⋯ þ pi Þ − w − ð p − m þ ⋯ þ pi − 1 Þ 1 − m ≤ i ≤ 0

ð11:9Þ

where w+ and w− are strictly increasing functions from the unit interval into itself satisfying: π þ ð0Þ = w − ð0Þ = 0

ð11:10Þ

π þ ð1Þ = w − ð1Þ = 1

ð11:11Þ

The following functional form for the value function (Tversky & Kahneman, 1992) fits the CPT assumptions:  α if x ≥ 0 x vðxÞ ð11:12Þ − λð − xÞβ if x < 0 The parameter λ ≥ 1 describes the degree of loss aversion and the parameters α, β ≤ 1 measure the degree of diminishing sensitivity. α = β = 1 represents the case of pure loss aversion. The weighting functions proposed by Tversky and Kahneman (1992) for gains and losses are, respectively: w þ ð pÞ = pγ =½ð pγ þ ð1 − pÞγ 1=γ

ð11:13Þ

w − ð pÞ = pδ =½ð pδ þ ð1 − pÞδ 1=δ

ð11:14Þ

The curvature of the weighting function, as well as the point where it crosses the 45° line, are defined by the parametric values of γ and δ. Decreasing γ and δ causes the weighting function to become more curved and to cross the 45° line farther to the right. Alternative functional forms for the decision weight function, that captures the same behavioural assumptions, were suggested by Goldstein and Einhorn (1987), Prelec (1998) and others (for an overview, see Rasouli & Timmermans, in press). Tversky and Kahneman (1992) estimated the parameters α, β, λ, γ and δ. The values which best fit their experimental results are: α = β = 0.88; λ = 2.25 and γ = δ = 0.61 as median values. Although similar estimations were found in different data sets from different decision-making tasks (Benartzi & Thaler, 1995; Camerer & Ho, 1994; Wu & Gonzalez, 1996), and even in route-choice modelling (Avineri & Prashker, 2004), these values should not be treated as universal and should be

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estimated empirically for specific applied contexts of CPT (for a discussion of this issue see Avineri, 2006; Avineri & Bovy, 2008). In specific, Avineri (2006) argues that it is not trivial to assume that the parameter values estimated by Tversky and Kahneman (1992), or in other studies of economic choices under risk, might be transferable to decision problems in the travel behaviour domain. These estimations have been derived from gambles that incorporated money outcomes, and any attempt to draw conclusions related to travel choice based on these values should be questioned. One may wonder if there is a reason to assume that the violations of EUT will occur also in the context of travel choice. Such an assumption is not trivial due to two main reasons (Avineri & Prashker, 2004). First, there is a significant difference between travel choice decision-making which is commonly associated with expected travel time, and the gambling problems based on monetary values studied by Kahneman and Tversky (1979). One of Kahneman and Tversky’s findings was that people are often much more sensitive to the difference between an outcome and the reference level than about the absolute value of the measured outcome. Unlike monetary outcomes associated with economic and financial decisions, travel time cannot be stored or pooled; therefore, the observation regarding sensitivity to a reference point might be of more relevance and importance in the context of travel choice problems rather than for problems dealing with selection of risky monetary assets. Another characterisation of the route-choice problem, which makes it different from monetary-based problems, is that homogeneity of reference point values in the mind of individual travellers is not very likely. When dealing with monetary gains and losses, $0 may be a common value of the reference point. Dealing with traveller’s experienced travel time, the framing of the resulted travel time as a gain or a loss is based on an individual reference point, which may be based on the traveller’s past experience and expectations. Such a reference point may differ from one traveller to another. To illustrate the relevance of PT to travel choice modelling, a route-choice experiment, inspired by Kahneman and Tversky’s (1979) work, was conducted by Avineri and Prashker (2004). The participants were asked to make stated-choice preferences (SP) in response to route-choice situations. They showed that the preferences revealed in sets of route-choice problems violate EUT assumptions (regardless of the utility function shape) and on the other hand can be explained and predicted by PT. Their findings illustrated the so-called certainty effect, or Allais’ Paradox (Allais, 1953, 1979), captured by the shape of the weighting function: an extreme underweighting of high probabilities, which fall short of certain outcomes, make positive certain outcomes very attractive. Another violation of EUT, the inflation of small probabilities, was also illustrated in Avineri and Prashker (2004).

11.4.

Numeric Example

Applying CPT to a route-choice problem, for each route a cumulative weighted value, associated with a set of probabilistic outcomes, will be calculated according

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to Eqs. (11.3)(11.14). All alternatives will be compared by their cumulative weighted values. Assuming travellers are prospect maximisers, the most attractive route would be the one that carries the highest cumulative weighted value. The travel time distribution of the alternative routes A, B and C are given by probability functions pA(xi) = pAi, pB(xi) = pBi and pC(xi) = pCi, where piA, piB and piC are the probabilities associated with xiA, xiB and xiC travel times in routes A, B and C, respectively. The distributions are presented in Table 11.1. The value of the reference point is assumed to be 31.5 minutes, thus a travel time of less than 31.5 minutes will be perceived as a gain for the traveller, and a travel time of more than 31.5 minutes would be perceived as a loss to him/her. The cumulative weighted value of route A’s prospect will be calculated as following: The expected outcomes, derived from route A’s travel time distribution, are the travel times xi minus the value of the reference point (31.5 minutes). Viewed as a probabilistic prospect, f yields the payoffs (−4.5, −3.5, −2.5, −1.5, −0.5, 0.5, 1.5), with the probabilities (0.05, 0.1, 0.2, 0.3, 0.2, 0.1, 0.05). Thus, f + = (0,0.85; 0.5,0.1; 1.5,0.05), and f − = (−4.5,0.05; −3.5,0.1; −2.5,0.2; −1.5,0.3; −0.5,0.2; 0,0.15). Therefore, VðfA Þ = vðfA þ Þ þ vðfA − Þ = vð0:5Þ½w þ ð0:15Þ  w þ ð0:05Þ þ vð1:5Þ½w þ ð0:05Þ  w þ ð0Þ þ vð − 4:5Þ½w − ð0:05Þ  w − ð0Þ þ vð − 3:5Þ½w − ð0:15Þ  w − ð0:05Þ þ vð − 2:5Þ½w − ð0:35Þ  w − ð0:15Þ þ vð − 1:5Þ½w − ð0:65Þ  w − ð0:35Þ þ vð − 0:5Þ½w − ð0:86Þ  w − ð0:65Þ Using the parameter values estimated in Tversky and Kahneman (1992), the cumulative weighted value of route A prospect is V(fA) = −3.055. The cumulative weighted values that are associated with the prospects of routes B and C were calculated using the same technique: V(fB) = −3.652 and V(fC) = +0.204. Thus, according to the above assumptions and parameter values, a traveller would find route C (which has a positive cumulative weighted value) more attractive than routes A and B. Table 11.1: Travel time distribution of the alternative routes A, B and C. xi pA(xi)

36 0.05

35 0.1

34 0.2

33 0.3

32 0.2

31 0.1

30 0.05

xi pB(xi)

36 0.1

35 0.25

34 0.25

33 0.2

32 0.1

31 0.05

30 0.05

xi pC(xi)

36 0.001

35 0.001

34 0.03

33 0.05

32 0.1

31 0.54

30 0.35

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11.5.

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Incorporating Prospect Theory in Travel Choice Modelling — Application Areas and Evidence

In the last 15 years, empirical and theoretical studies of PT in a range of travel behaviour contexts have appeared in the transport research literature. Perhaps some of the catalysts to the interest in bounded rationality in decision-making outside the traditional interest in behavioural sciences were the relative success of several publications in behavioural economics, or fields related to it, and their applications in explaining human behaviour to the broad public. Many studies provided evidence that generally supported the behavioural assumption of PT. For example, Katsikopoulos, Duse-Anthony, Fisher, and Duffy (2000, 2002); Senbil and Kitamura (2004); Avineri and Prashker (2004); van de Kaa (2010a), Rose and Masiero (2010) and others presented empirical evidence that suggested that travellers exhibit aversion to loss (measured against a reference point) and have a strong tendency to avoid choices associated with losses. Through a systematic review of empirical evidence, including national travel surveys from the Netherlands and the United Kingdom, van de Kaa (2010a) showed that PT can shed more light on some puzzling outcomes from these surveys, such as differences in monetary valuation between travel time gains and losses. Investigating asymmetrical preference formation in willingness to pay estimates in discrete choice models (in the context of choice amongst tolled and non-tolled routes), Hess, Rose, and Hensher (2008) reported asymmetrical response to increases and decreases in attributes describing alternatives in a discrete choice context. Their analysis supports the existence of framing effects in SP data and suggests that preference formation may not relate to the absolute values of the attributes shown in SP experiments, but rather to differences from respondent specific reference points. Avineri and Bovy (2008) and Li and Hensher (2011) (and also see Rasouli & Timmermans, in press) argued that many of the empirical studies in the field of travel behaviour that attempted to apply a PT framework have been limited in scope and methodological approach either because (i) they have not fully integrated the PT features (an element missing from many of these works is the non-linear weighting of probabilities); (ii) the empirical analyses are based on small scale SP surveys, limited to provide validated results; (iii) only little attempt has been done to systematically estimate the parameter values in a consistent manner. On the other hand, other studies provided evidence that the static modelling framework of PT has limitations in explaining and predicting travel choice in a dynamic environment of decision-making (Avineri & Prashker, 2005; Ben-Elia & Shiftan, 2010). The following list (Table 11.2) illustrates the wide range of application areas and methodological aspects associated with PT, and might help the reader in referring to specific applied contexts in the fields of travel behaviour and choice modelling. Although not a comprehensive and systematic review of all the literature in this area, Table 11.1 suggests that much of the research has focused on route-choice modelling more than on the modelling of other types of travel choices.

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Table 11.2: Application areas and methodological aspects associated with Prospect Theory. Application area

Methodological aspects

Route/path choice

Empirical studies; stated (one-shot) preferences (no feedback)

Route/path choice

Empirical studies; dynamic choices (with feedback), an extension of CPT to capture dynamic decision-making Application of PT (and other reference-dependence models) to traffic assignment network equilibrium

Route/path choice

Departure time choice Mode choice; choice between service alternatives Freight transport choice Trip generation Location of small office firms Air travel choices

References Avineri and Prashker (2004); Avineri and Bovy (2008); Hess et al. (2008); Avineri (2009); Gao, Frejinger, and Ben-Akiva (2010); Hensher, Greene, and Li (2011); Xu, Zhou, and Xu (2011); Ramos, Daamen, and Hoogendoorn (2013) Katsikopoulos et al. (2000, 2002); Avineri and Prashker (2003, 2005, 2006); Viti, Bogers, and Hoogendoorn (2005); Ben-Elia and Shiftan (2010) Avineri (2006); Lo, Luo, and Siu (2006); Connors and Sumalee (2009); Sumalee et al. (2009); Tampe`re and Viti (2010); Xu, Lou, Yin, and Zhou (2011); Wang and Xu (2011); Site and Filippi (2011, 2012); Tian, Huang, and Gao (2012); Liu and Lam (in press) Senbil and Kitamura (2004); Jou et al. (2008) Avineri (2004); Michea and Polak (2006); Polak, Hess, and Liu (2008); Stathopoulos and Hess (2012) Masiero and Hensher (2009)

Schwanen and Ettema (2009) Elgar and Miller (2006)

Hess (2008)

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Table 11.2: (Continued ) Application area Design of stated preferences surveys Behavioural change

General

Methodological aspects Differences on the willingness to pay and willingness to accept estimates

References Hess et al. (2008); Rose and Masiero (2010); Masiero and Rose (2013)

Influencing the perceived Han, Dellaert, van Raaij, and value of reference point; Timmermans (2005); Avineri valence framing (loss (2006); Avineri and Waygood aversion); design of tradable (2013); Bao, Gao, Xu, and Yang credit schemes (2014) Review papers van de Kaa (2010a, 2010b); Timmermans (2010); van Wee (2010); Li and Hensher (2011); Ramos et al. (2014); Rasouli and Timmermans (in press)

Taking into account the special characteristics of travel choice, different from economic and financial decision-making, several extensions to the modelling framework of PT have been suggested by transport researchers. In a study of departure time choice, Jou et al. (2008) suggested the use of two reference points rather than a single one (preferred arrival time): one associated with the earliest acceptable arrival time and the other associated with preferred work starting time for a given commuter. The use of multiple reference points (associated with different attributes) was suggested by Schwanen and Ettema (2009) in their study of travel choices associated with parents chauffeuring their children. Stathopoulos and Hess (2012) extended the reference-dependence framework to a multi-attribute framework, allowing contemporarily for gain/loss asymmetry, non-linearity and testing for several possible reference points. Avineri (2009) suggested the extension of the concept of a single reference point to a fuzzy set of reference values, in order to address the vague perception of reference travel times travellers have, and introduced the concept of a fuzzy rather than crisp representation of travel choice outcomes as ‘gains’ and ‘losses’ in the context of route choices. More recently, the application of PT (and other reference-dependence modelling approaches) to traffic assignment and to the specific concept of network equilibrium were investigated in the literature. Avineri (2006) suggested that Wardrop’s (1952) principle of user equilibrium may be extended following the principles of risk-taking behaviour captured by PT: ‘Equilibrium under the condition that no user can increase his/her route prospect value by unilaterally switching routes’. This and other studies of theoretical nature (such as Connors & Sumalee, 2009; Sumalee,

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Connors, & Luathep, 2009) demonstrated the significant effect of the reference point value on the stochastic network equilibrium. Modelling travel behaviour has long been applied in the field of transport to estimate the properties of attributes associated with the travel choices (e.g. travel time, travel cost) and with travellers’ characteristics (e.g. socio-demographics) in order to predict their future behaviour. In addition to their obvious relevance to describe and predict travel choices, some of the empirical findings and theoretical concepts presented in the literature might be relevant to policy context, such the design of stated preferences surveys, pricing of user costs for transport infrastructure (e.g. congestion charges), and the incorporation of PT in the design of behavioural change measures (which is discussed in the next section). Individual’s willingness to pay (WTP) systematically been found greater than his/her willingness to accept (WTA) of a choice, findings that can be explained by loss aversion (for a review of WTA/WTP studies, see Horowitz & McConnell, 2002). In the context of travel choice, results reported by Rose and Masiero (2010) and Masiero and Rose (2013) suggest significant differences on the WTP and WTA estimates obtained from stated-choice experiments. These findings might suggest how to reduce biases in stated preferences surveys through the frame design of how information presented to respondents.

11.6.

Prospect Theory and Behavioural Change

One of the goals of modelling travel behaviour, little addressed in the literature, is to provide theoretic insights on why and how people travel — and applying these insights in the development and implementation of a range of design, planning and policy measures, with an ultimate goal of making effective improvements to social and individual wellbeing (Avineri, 2012). The ‘predicted irrationality’ (a term coined by Ariely, 2008) of individuals could (and some argue that it should) play a role in the design of behavioural change interventions. Thaler and Sunstein (2008) advocate the use of ‘choice architecture’ to influence behavioural change: ‘nudges’, small features designed in the environment of choice making, could help individuals to overcome cognitive biases and to highlight the better choices for them — without restricting their freedom of choice, and without making big changes to the physical environment, the set of choices, or the economic attributes of the choices. Few studies in transport research literature have suggested to apply theoretic insights emerging from PT in the design of measures to change travellers’ behaviour. As travellers exhibit aversion to loss and have a strong tendency to avoid choices associated with losses, Avineri (2006) suggested to ‘nudge’ travellers to a better user equilibrium on a traffic network by changing the perceived value of a reference point associated with travel times on the alternative routes. The reference point may be influenced by the presentation of the outcomes or expectations of the decisionmaker and by social norms. This approach is based on the gain/loss asymmetry, and the shape of PT’s value function.

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Another behavioural change approach associated with gain/loss asymmetry is valence framing. Valence framing is a framing technique that highlights gains or losses associated with the choice outcome. According to PT, loss framing would be more effective than gain framing. Loss framing can be incorporated in the design of a variety of information-based measures to promote travel behaviour change.These could include web-based journey planners, carbon calculators or ‘tailored’ travel information provided to individuals and households who participate in a personal travel plan. For example, Avineri and Waygood (2013), applying an experimental approach, describe how loss framing could increase the differences about the mass of transport-related CO2 emissions, as perceived by survey participants, and improve the effectiveness of such information.

11.7.

Shortcomings and Limitations of Prospect Theory in Modelling Travel Behaviour

As evidence from empirical studies on PT are emerging, in both general and travel behaviour contexts, its shortcoming and limitations are also increasingly being identified. While a robust finding in many SP studies is the gain/loss asymmetry, other features of PT are less explored in a travel behaviour context. Interestingly, findings reported by Hess et al. (2008) do not support the exact supposition put forward by PT that the value function for losses should be convex and relatively steep whilst the value function for gains losses would be expected to be concave and not quite so steep. They found different relationships between gains and losses for different attributes. Avineri and Bovy (2008) described some of the major challenges modellers are faced with when applying PT to travel behaviour context. PT is not a full-fledged theory of decision-making, and there is an on-going debate about its applicability in some economics contexts. In addition to the general complexities in estimating CPT parameter values, transport researchers are faced with additional difficulties and challenges. Due to the different contexts travel journeys are made in, different modes and trip purposes, it is difficult to estimate a set of parameter values that represent a typical decision-maker. Therefore the incorporation of heterogeneity terms associated with perceived values of PT parameters might become a rather complex task. Moreover, due to the lack of definite and in-consensus reference points in travel contexts, modellers may have difficulties in understanding and modelling travel choice outcomes as ‘gains’ and ‘losses’, as observed and framed by the travellers. In addition, some open questions remain about the suitability of PT to describe and model a dynamic environment of travel choice, and the effect of feedback and learning on repeated choice. PT was originally proposed in order to capture description-based decisions in one-shot tasks. Most features of PT, such as reference dependency and gain/loss asymmetry, have been exhibited in a range of behaviours

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tested mainly in static (‘one-shot’) settings, with no (or limited) feedback or incentives. However, some recent studies have questioned this paradigm, mainly in dynamic settings (repeated choice) that provides feedback (and sometimes reward) related to the choice, and features a learning process (see Barron & Erev, 2003; Ert & Erev, 2008; and in a travel behaviour context Avineri & Prashker, 2003; Ben-Elia, Erev, & Shiftan, 2008). Many travel choices may be considered to be a routine (e.g. commuting) and can be therefore described as a repeated choice situation, where feedback (resulted travel time) is involved. There is a relation between the travel utilities experienced by the travellers in past time periods and the current attractiveness of travel choices. The travel time distribution is not likely to be completely known to travellers at the time of choice, and their perception of travel time may be dynamically updated, based on accumulating experience. The effect of this dynamic update of attributes and their distribution, combined with the effect of learning (which by itself is characterised by bounded rationality) might be stronger than loss aversion and other features of PT. Evidence to support this argument are presented in Avineri and Prashker (2003) and Ben-Elia et al. (2008). Avineri and Prashker (2005) showed that CPT fails to predict route-choice feedback-based decisions (although a dynamic generalisation of the CPT static model may provide valid predictions of travel choices). Ben-Elia and Shiftan (2010) also did not find choice behaviour fully consistent with PT (while only the mean travel time was tested as a reference point, absolute travel time differences were found significant). Another open question associated with the dynamic nature of travel behaviour is how the reference point is updated through time as a function of the outcomes in past decisions (Arkes, Hirshleifer, Jiang, & Lim, 2008) — so far this issue has not been addressed in the travel behaviour context.

11.8.

Summary and Conclusions

This chapter provided a review of the theoretical background, applied area and evidence related to the incorporation of PT in travel choice modelling. It also addresses some of the intellectual and practical challenges researchers and modellers might be faced with when encountering the PT framework. While it should not be seen as a comprehensive and systematic review of this emerging field of research, it might provide a useful reference for those who have interests in exploring PT in the applied context of travel behaviour. As mentioned in the chapter, reasonably large amount of emerging evidence of PT in a range of transport-related behaviours has been reported in the literature. More and more academics and professionals are becoming interested in further research and application of PT and its elements in transport contexts. These might be seen as indications that the research community has already reached a certain stage in exploring this field and the need to define further the research agenda. We suggest that as PT is a descriptive model of behaviour, and as the applied areas of

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travel choice have different contexts, there is still a need for further development of both theoretical and empirical studies. In specific while route-choice models have attracted a relatively high level of attention, there is very little research in applying PT to other contexts such as mode choice, trip generation, destination choice and vehicle ownership. It may be concluded from the evidence reviewed in the chapter that travellers are not necessarily utility maximisers; however more research is needed to be done in order to determine whether they are indeed prospect maximisers, or to provide us with a better descriptive model of travel-choice behaviour. Furthermore, as many of our travel choices are dynamic by nature, there is a need to further develop the theoretical and empirical aspects of the dynamic extensions to PT and test its predictive value in dynamic environments that involve risk, uncertainty and ambiguity (Kemel, 2013). As interest in the application of other theoretical frameworks (e.g. regret theory) to the field of travel choice modelling is rapidly growing. There is a need to develop a systematic approach to compare in a consistent way the performance of alternative frameworks in real-life situations (rather than focusing on ‘toy’ problems that might over represent corner situations of behaviours that are not captured by traditional RUM models). Also, as part of a suggested research agenda, a development of a unifying theory of travel choice behaviour, that take into account notions from several theories, is suggested.

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About the Authors

Theo Arentze is associated professor of Urban Planning at the Eindhoven University of Technology. His research interests include activity-based modelling, discrete choice modelling, agent-based modelling, supernetwork modelling, human cognition/learning and traveller information systems for application in transportation and urban planning. He is involved as principle researcher, supervisor or project leader in a constant stream of Ph.D., Postdoc and EU projects on these topics. He is member of the editorial board of several international peer-reviewed journals and acts as an ad hoc reviewer and programme committee member for many journals, conferences and research foundations in transportation, planning, geography and consumer research. Erel Avineri is an Associate Professor at Afeka  Tel-Aviv Academic College of Engineering. He is the Head of Infrastructure Systems Engineering and Management Department and the Head of ACITRAL  Afeka Centre for Infrastructure, Transportation and Logistics. He received his degrees in Industrial Engineering and Management (B.Sc.) and Transportation Sciences (M.Sc., Ph.D) from Technion  Israel Institute of Technology. His main research areas lie in travel behaviour, road safety behaviour, intelligent transport systems, and transport policy. Eran Ben-Elia is a Senior Lecturer at Ben-Gurion University, Israel. He has a Ph.D. in Transportation Science from the Technion  Israel Institute of Technology. His research interests lie in unravelling the interconnections between human space, technologies and mobility. In particular, the elicitation of travel behaviour and the application of information and communication technologies to motivate sustainable travel choices. Xiqun Chen received his B.E. degree and Ph.D. degree in Civil Engineering from Tsinghua University, China, in 2008 and 2012, respectively. In 2011, he visited California PATH Program, University of California at Berkeley as a joint training Ph.D. student. His thesis entitled ‘Stochastic Evolutions of Dynamic Traffic Flow: Modeling and Application’ received the 2013 IEEE Intelligent Transportation Systems Society (ITSS) Best Ph.D. Dissertation Award. He is currently a Faculty Research Assistant in the Department of Civil and Environmental Engineering at the University of Maryland, USA, working on simulation-based optimisation, traffic flow theory and intelligent transportation systems. Dr. Chen serves as the Research Director of National Center for Strategic Transportation Policies,

258

About the Authors

Investments and Decisions. He is a member of IEEE, Chinese Overseas Transportation Association (COTA), and an Associate Member of American Society of Civil Engineers (ASCE). He had published more than 20 peer-reviewed journal papers and over 30 conference papers. Caspar G. Chorus is head of the Transport and Logistics group at TU Delft. His main research aim is to increase the behavioural realism of travel choice models, and by implication to provide a stronger and richer foundation for transport policy development. He is interested in developing and testing conventional (travel) choice models as well as new, more unorthodox model form. On these and related topics, Caspar has published extensively in leading journals in the field of transportation and beyond. Caspar has won several international awards for his work, including the Eric Pas prize. He is editor-in-chief of the European Journal of Transport and Infrastructure Research, which is the only ISI-listed open access e-journal in transportation. He is member of the Traveler Behavior and Values-committee, and of the Travel Survey Methods-committee, of the Transportation Research Board; early 2014, he was elected board member of the International Association for Traveler Behavior Research. Andrew T. Collins is a Lecturer in Transport and Logistics Management at The University of Sydney Business School’s Institute of Transport and Logistics Studies. Andrew’s Ph.D., which examined techniques for handling attribute non-attendance in discrete choice models, was awarded the prestigious 2012 Eric Pas Dissertation Prize by the International Association for Travel Behaviour Research. He has broad research interests, spanning advanced discrete choice modelling methodology and its application across many fields, choice heuristics, stated choice experimental designs, final mile logistics, freight transport and air travel choice. His publication record includes top tier journals such as Transportation and Transportation Research Part B. Andrew is a co-developer of Ngene, a widely used software package which generates stated choice experimental designs. In addition to his academic pursuits, Andrew has consulted for government, industry and banks, in the areas of freight transport, and toll road and public transport evaluation and modelling. Benedict G. C. Dellaert is a professor of marketing at the Erasmus School of Economics. His focus in research and education is on consumer decision-making and (online) consumer-firm interaction. His research has applications in financial services, healthcare management and transportation, among others. Benedict is currently co-director of the Erasmus Centre for Marketing and Innovation (ECMI), a research theme coordinator at the Network for Studies on Pensions, Aging and Retirement (Netspar), a fellow of the Erasmus Research Institute of Management (ERIM), and a research fellow at the Tinbergen Institute. He holds a Ph.D. from Eindhoven University of Technology and his former positions include posts at the University of Sydney in Australia, Tilburg University in the Netherlands and Maastricht University in the Netherlands. His recent research has appeared in journals such as the Journal of Marketing Research, Marketing Science, Information Systems Research, Social Science & Medicine and Transportation Science.

About the Authors

259

Ke Han is a Lecturer in Transport Operations and Logistics at Center for Transport Studies, Imperial College London. He received a Ph.D. degree in Mathematics from the Pennsylvania State University in 2013. Ke Han’s research interests span a wide variety of aspects in transportation science and engineering including traffic flow theory, network modelling, dynamic traffic assignment, traffic operation and control, intelligent transportation systems, network and mechanism design, and sustainable transportation. His research methodologies are transferrable to other problems/disciplines: especially electric power pricing, urban logistics, revenue management, e-commerce and internet-based transactions. Ke Han is a member of Institute for Operations Research and Management Sciences (INFORMS), Chinese Overseas Transportation Association (COTA) and Society for Industrial and Applied Mathematics (SIAM). David A. Hensher is Professor of Management, and Founding Director of the Institute of Transport and Logistics Studies at The University of Sydney. David is a Fellow of the Academy of Social Sciences in Australia (FASSA), Recipient of the 2009 International Association of Travel Behaviour Research Lifetime Achievement Award in recognition for his long-standing and exceptional contribution to IATBR as well as to the wider travel behaviour community, recipient of the 2006 Engineers Australia Transport Medal, recipient of the 2009 Bus NSW (Bus and Coach Association) Outstanding Contribution to Industry Award, and Member of Singapore Land Transport Authority International Advisory Panel (Chaired by Minister of Transport). He has published extensively (over 550 papers) in the leading international transport journals and key journals in economics as well as 14 books. David has advised numerous government industry agencies, with a recent appointment to Infrastructure Australia’s reference panel on public transport, and is called upon regularly by the media for commentary. Oliver Horeni holds a Ph.D. from Eindhoven University of Technology where he did research about mental representations underlying activity-travel choices. As part of his project he developed the computerised Causal Network Elicitation Technique. Oliver is currently employed at the transport association for Dresden and the Upper Elbe Region where he is responsible for the further development of the tariff system. Stephane Hess is Professor of Choice Modelling in the Institute for Transport Studies and Director of the Choice Modelling Centre at the University of Leeds. He is also Honorary Professor in Choice Modelling in the Institute for Transport and Logistics Studies at the University of Sydney, and affiliated Professor in Demand Analysis at KTH Royal Institute of Technology in Stockholm. He also holds a director position at RSG, a leading North American consultancy company. His area of work is the analysis of human decision using advanced discrete choice models, and he is active in the fields of transport, health and environmental economics. He has made contributions to the state of the art in the specification, estimation and interpretation of such models, while also publishing widely on the benefits of advanced structures in actual large-scale transport analyses. His contributions have

260

About the Authors

been recognised by a number of awards, including the 2005 Eric Pas award for the best Ph.D. thesis in the area of travel behaviour modelling. He is also the founding editor-in-chief of the Journal of Choice Modelling, and the founder and steering committee chair of the International Choice Modelling Conference. Davy Janssens is associate professor at Hasselt University, Belgium. He teaches courses in the domain of transportation sciences at the School for Transportation Sciences at Hasselt University. He is also a member of the Transportation Research Institute (IMOB) at Hasselt University, where he is appointed as Program Leader of the Travel Behaviour & Mobility Management research group. His research area of interest is situated within the application domain of advanced quantitative modelling (e.g. data mining) in activity-based transportation modelling and in the analysis of big data. He has published several articles in scientific peer-reviewed journals and conferences such as Accident Analysis and Prevention, Environment and Planning, Transportation, Geographical Analysis, Knowledge Discovery and Data Mining, Transportation Research Record and Information Systems. Within IMOB, he is the coordinator of a variety of national and international research projects, including several EU projects such as FP7 DATA SIM. Ifigenia Psarra is Ph.D. Candidate at the Eindhoven University of Technology. In 2010, she received an engineering degree with honours in Architecture and Urbanism at the Aristotle University of Thessaloniki, Greece. In parallel with her studies, she worked as a trainee in several architectural offices. In January 2011, she joined the Urban Planning Group, in the Department of the Built Environment, at the Eindhoven University of Technology, the Netherlands. She published her research findings in various journals and conference proceedings. Her research interests span from the architectural to the urban and spatial planning scale. Soora Rasouli is Assistant Professor of the Urban Planning Group of the Eindhoven University of Technology, the Netherlands. She has research interests in activity-based models of travel demand, modelling of choice processes under uncertainty and complex systems. She is a member of the editorial board of Journal of Urban Planning and Development, International Journal of Transportation, Modern Traffic and Transportation Engineering Research, International Journal of Urban Science, Journal of Traffic and Transportation Planning and JRCS. She is junior career member of the ISCTSC Board of the International Steering Committee for Travel Survey Conferences, and member of the Transportation Research Board (TRB) Special Committee on Travel Forecasting Resources. Her work has been published in journals such as Environment and Planning A, Environment and Planning B, Networks and Spatial Economics, Transportation Letters, Transportation Research A, Transportation Research Record and the International Journal of Geographic Information Science. She has acted as guest editor for Environment and Planning B, Travel Behavior and Society and the Journal of Choice Modelling. She has received several international awards for her publications. WY Szeto is the Deputy Director of Institute of Transport Studies and Assistant Professor at the Department of Civil Engineering at The University of Hong

About the Authors

261

Kong. He is author of over 70 refereed journal papers related to dynamic traffic or transit assignment, low carbon transport, network design and reliability, and sustainability. He has received many international awards including the World Conference on Transport Research Prize and the Eastern Asia Society for Transportation Studies Outstanding Paper Award. Currently, he is Editor of Central European Journal of Engineering, Editor in Asian Region of International Journal of Transportation, Associate Editor of Journal of Intelligent Transportation Systems, Transportmetrica A and B and Travel Behaviour and Society, and Editorial Board Member of Transportation Research Part B and C, Journal of Advanced Transportation, International Journal of Sustainable Transportation and International Journal of Traffic and Transportation Engineering. He has been Guest Editor of eight journals. Harry Timmermans is Head of the Urban Planning Group of the Eindhoven University of Technology, the Netherlands. He has research interests in modelling decision-making processes and decision support systems in a variety of application domains, including transportation. His main current research project is concerned with the development of a dynamic model of activity-travel behaviour. He is founding editor of the Journal of Retailing and Consumer Services, and serves on the editorial board of several journals in transportation, geography, urban planning, marketing, artificial intelligence and other disciplines. He is Co-chair of the International Association of Travel Behavior Research (IATBR), and member of several committees of the Transportation Research Board. He has also served on many conference committees in transportation, marketing and artificial intelligence. He is (co)-author of more than 500 refereed articles in international journals. Yi Wang has graduated with first-class honours from Hong Kong Polytechnic University. Her graduate design won the Merit Award in the Institution of Civil Engineers Hong Kong Association (ICE HKA) Graduates & Students Division Papers Competition 2011. She is currently a Ph.D. candidate under Dr. W. Y. Szeto’s supervision in the Department of Civil Engineering of the University of Hong Kong. Her research interests are sustainable transportation problems and the applications of operation research in the field of transportation design problem. She is a reviewer of Networks and Spatial Economics. Geert Wets received a degree as commercial engineer in business informatics from the Catholic University of Leuven (Belgium) in 1991 and a Ph.D. from Eindhoven University of Technology (The Netherlands) in 1998. Currently, he is a full professor at the School for Transportation Sciences at Hasselt University (Belgium). He is also Director of the Transportation Research Institute (IMOB). His current research entails transportation behaviour, activity-based transportation modelling, traffic safety and data mining. He has published in several international journals such as Accident Analysis and Prevention, Environment and Planning, Geographical Analysis, Knowledge Discovery and Data Mining, Transportation Research Record and Information Systems.

262

About the Authors

Chenfeng Xiong received his B.S. degree from Tsinghua University, China, in 2009. He received his M.S. in Civil Engineering and M.A. in Economics from the University of Maryland (UMD), USA, in 2011 and 2013. He is now a Ph.D. candidate in the Department of Civil and Environmental Engineering of UMD. Mr. Xiong has co-authored more than 30 peer-reviewed journal and conference papers. His research focuses on travel behaviour and agent-based modelling. He is a member of Transportation Research Board (TRB), American Society of Civil Engineers (ASCE) and Chinese Overseas Transportation Association (COTA). He actively serves as reviewers for refereed journals such as Transportation Research Record and IEEE Transactions on Intelligent Transportation Systems. Mr. Xiong has been a recipient of the UMD A. James Clark Engineering School’s Future Faculty Fellowship, the International Road Federation Best Student Essay Award, Transportation Research Forum Best Student Paper Award and several other fellowship/scholarship honours. Lei Zhang received the B.S. degree in civil engineering from Tsinghua University, China, the M.S. degrees in civil engineering and applied economics from the University of Minnesota, Minneapolis, MN, USA, and the Ph.D. degree in transportation engineering from the University of Minnesota in 2006. He is currently an Associate Professor with the Department of Civil and Environmental Engineering, University of Maryland, College Park, MD, USA, where he also directs the National Center for Strategic Transportation Policies, Investments and Decisions, which employs interdisciplinary approaches to model the interdependency between transportation, land use and economic systems. He has published more than 90 peer-reviewed journal and conference papers on topics including transportation planning, transportation economics and policy, travel behaviour, advanced travel demand modelling, transportation data and survey methods and traffic operations. His research interests are in the areas of transportation systems analysis, transportation and land use planning, transportation economics and policy and mathematical, statistical, and agent-based modelling. Junyi Zhang is a Professor at Graduate School for International Development and Cooperation, Hiroshima University, Japan. Focusing on the various issues related to city, transportation, environment and tourism, as of October 2014, he published more than 300 refereed academic papers, about the development of methodologies (e.g., surveys, modeling, policy evaluation, and planning) and applications of the methodologies from the interdisciplinary perspective in journals such as Transportation Research Part B, Journal of Transport Geography, Tourism Management, Annals of Tourism Research, Energy Policy, and Environment and Planning B. One of his core research interests related to this book is the modeling of individual and group decision-making using advanced econometric models. He’s currently serving as the editor-in-chief of Asian Transport Studies and on the board of several other international journals. He’s also on the board of several academic committees of organizations such as IATBR, TRB, and EASTS. His work was awarded more than 10 times by international academic organizations.

About the Authors

263

Wei Zhu is currently associate professor of Department of Urban Planning, Tongji University, China. For over 10 years, he has been studying pedestrian behaviour in urban environments using field surveys, stated preference experiments, choice modelling, multi-agent simulation and other quantitative methods. He has special interest in modelling bounded rationality. His recent research focuses on the planning and promotion of green transportation, particularly public bicycle systems, cycling and walking. He teaches courses in Methods for Urban System Analyses, Urban Geography, Urban Comprehensive Planning and Urban Detailed Planning, and tries to incorporate quantitative methods into pedagogical development as well as planning practices.

Index

Activation level, 196197, 199200, 202205, 210 Activity-based model, 126, 139141, 190, 214 Activity-based travel demand models, 2 Activity-travel behaviour, 117, 158, 189210 Activity-travel choice task, 122 Advantage maximization, 10 Agent-based models, 213, 228229 Albatross, 138, 140, 151154, 157159 Aspiration levels, 7 Aspirations, 189191, 195, 198201, 204206, 209210 Association pattern technique, 120 Attribute, 1, 312, 1420, 22, 2425, 3237, 46, 5556, 63, 66, 68, 7376, 7985, 8789, 9193, 96105, 111, 119121, 123, 125126, 129130, 147, 157, 193194, 196, 198, 236, 246 Attribute non-attendance (ANA), 1, 1718, 7381, 8392 Attributes, 110, 1321, 2325, 3133, 3539, 41, 46, 51, 53, 59, 62, 66, 6869, 7384, 8689, 92, 96, 98102, 105, 115, 117121, 123, 125131, 157, 192194, 197198, 215218, 229, 235237, 244, 246249 Awareness, 195, 199200 Awareness reinforcement parameter, 195 Awareness retention rate, 195, 200

Bayesian Decision Network, 118 Bayesian learning, 218 Bayesian networks, 137138, 141142, 144146, 149, 152156, 158159 Bayesian updating, xviii Behavioural change, 222, 246248 Behavioural economics, 56, 235, 244 Behavioural equilibrium, 190, 229, 247 Beliefs, 102103, 105107, 116, 191, 193, 199200, 215219, 225226, 228 Benefits, 115119, 121, 123, 125126, 128131, 221, 237 Bias, 59, 76, 81, 92, 122, 239 BNT classifier, 137, 143, 149150, 152159 Bounded rationality, 1, 3, 57, 911, 13, 15, 17, 19, 21, 23, 2526, 51, 6869, 74, 9596, 98, 102, 111, 137141, 143, 145, 147, 149, 151, 153, 155, 157, 163165, 167, 169171, 173, 175, 177, 179, 181, 189, 191, 209, 213, 215, 222, 226, 228, 234235, 244, 249 Causal knowledge, 115, 118 Causal network, 115, 117118, 120, 129 Causal Network Elicitation Technique (CNET), 115, 117, 121122, 132 CB-CNET, 122 Censored normal distribution, 75 Centrality of variables, 129 Choice complexity, 7475

266

Index

Choice set, 1, 38, 10, 1214, 1921, 2325, 38, 5055, 5758, 6264, 66, 6869, 97, 111, 119, 123, 126, 131, 190198, 200, 202204, 206, 208, 210 Choice set formation, 19, 23, 25, 191192, 194 Choice set generation, 2324, 5455, 6263 CNET card game, 122 Cognitive mapping, 119 Cognitive maps, 116 Cognitive responses, 193 Cognitive space, 4, 68, 20, 25 Cognitive subsets, 123, 125126, 128130 Compensatory choice, 25, 75 Competing destination model, 11 Compromise alternatives, 10 Conjunctive decision rule, 68, 20, 23 Consideration set, 4, 6, 15, 1921, 2324 Constrained latent class model, 76 Construal Level Theory, 119 Context-dependent choice models, 1011 Cumulative Prospect Theory (CPT), 142143, 235, 240242, 245, 248249 Decision process, 14, 19, 38, 77, 9598, 100101, 111112, 117118, 140, 215, 220, 225 Decision rule, 1, 68, 10, 20, 23, 25, 3133, 38, 41, 46, 97, 137138, 141, 146147, 149, 154, 156, 158159, 213217, 222223, 225, 235 Decision rules, 1, 67, 10, 23, 25, 3133, 38, 41, 46, 97, 137138, 141, 146147, 149, 154, 156, 158159, 213217, 222223, 225 Decision strategies, 74, 107, 111 Decision trees, 137138, 146, 149, 152154, 157159 Descriptive theory, 163, 213, 215, 229

Dirichlet distribution, 218 Disjunctive rule, 78, 14, 96 Dynamics, 2, 163, 168, 189193, 195, 197199, 201, 203, 205, 207, 209210, 213, 223, 227229 Dynamic traffic assignment, 163165, 167, 169, 171, 173, 175, 177, 179, 181 Dynamic user optimal, 163, 169, 172 E-Commerce, 123128, 130131 Elimination by aspects, 38 Emotional responses, 191, 193194, 199, 206, 210 Entropy measure, 74, 149 Error variance, 7475 Expected utility, 23, 137, 193194, 196197, 200201, 203205, 207208, 210, 233, 235237, 239 Expected Utility-maximization (EUT), 137138, 193194, 235237, 239, 242 Exploitation, 197198, 200, 202205, 207209 Exploration, 132, 191, 197198, 200, 202210, 229 Feasibility, 2, 177, 179 Forgetting, 189, 191, 199200, 202, 209, 216, 218 Four-step model, 2, 214 Gap function, 175, 177 G-RRM, 3339, 41, 46 Habit formation, 189, 191, 196, 199 Hard laddering, 120 Heterogeneity, 4, 18, 25, 3133, 41, 46, 53, 5860, 69, 7677, 8081, 92, 9596, 111, 116117, 131132, 248 Heuristic, 15, 20, 2225, 74, 9597, 99107, 109, 111112, 140141, 171, 177, 214, 228 Hidden Markov chain, 213, 222223

Index Hierarchical value maps, 120 Hyperbolic response curve, 9 Inertia, 190, 196 Inferred attribute non-attendance, 73 Influences on attribute non-attendance, 7393 Information load, 7377, 7981, 8385, 8789, 9192 Instrumental Variable, 220221 Interview protocol, 121 Laddering, 116, 120, 132 Latent Class, 5, 1718, 3133, 3739, 46, 54, 74, 7678, 96, 116 Latent class model, 17, 54, 74, 76, 96 Learning, 132, 137138, 140143, 145146, 152, 163, 189, 191, 199, 209, 213218, 223, 227229, 248249 Lexicographic rule, 1, 1417, 23, 75, 9597, 100, 111 Lexicographic choice, 14, 23, 75 Lexicographic semi-order, 15 δ-logit, 19 Lognormal distribution, 81, 84, 91 Long-term change, 190191, 198199, 207 Maximax decision rule, 9 Maximin decision rule, 9 Means-end-chain, 120121 Means-end-chain theory, 120 Memory decay, 189, 195 Memory retrieval ability, 195 Mental effort, 25, 95, 97, 102103, 105107, 111112, 119, 217 Mental maps, 116 Mental representations, 115117, 119123, 125127, 129132 Mind reading, 119 Minimax decision rule, 9 Minimum awareness level, 195 Minimum difference lexicographic rule, 15 Mixed Logit, 116

267

Mixture model, 3133, 37 Multidimensional decisions, 214 Multinomial logit, 23, 5, 10, 2224, 50, 52, 77, 95, 98 Multiple context dependency, 49 Needs, 33, 91, 103105, 115119, 125126, 131132, 151, 159, 192, 198, 200, 210, 214, 217, 220, 229, 233 Network equilibrium, 245247 Non-compensatory model, 23, 96 Nonlinear complementarity problem, 163, 171, 174 Nudge, 226, 247 Number of alternatives, 20, 25, 54, 7375, 80, 8687, 89, 92, 206 number of attribute levels, 15, 73, 75, 80, 87, 92 number of attributes, 18, 32, 7375, 8082, 8687, 89, 92, 96, 123, 125126 number of choice tasks, 80, 8687, 89 range of attribute levels, 81, 89 Online shopping, 122, 123, 126, 130, 131, 132 Ordered heterogeneous logit model, 75, 87 Passive bounded rationality model, 74 Perceived search cost, 217, 219221 Predicted irrationality, 247 Preference heterogeneity, 7677, 8081, 92, 116 Priming, 132 Prospect, 49, 51, 56, 64, 6668, 233235, 237241, 243250 Prospect Theory (PT), 51, 56, 64, 233235, 237250 Random parameters attribute nonattendance (RPANA) model, 7374, 7678, 8081, 8388, 9092

268

Index

Random Utility Models (RUM), 25, 31, 3339, 41, 4546, 49, 51, 132, 234, 250 Rationally adaptive model, 7475, 92 Recall techniques, 119, 120, 132 Recognition techniques, 119 Reference point, 5, 9, 13, 4952, 6869, 237238, 242244, 246249 Regret, 1, 3, 913, 25, 3137, 39, 41, 43, 45, 49, 56, 6468, 103, 250 Regret-based choice models, 3 Regret minimization, 13 Regret-weight, 3237 Reinforcement, 195 Relative advantage model, 13 Relative utility, 10, 1314, 25, 4969 Reluctance to change, 196 Reporting error, 73, 75, 84 Risk aversion, 236, 238 Risk/loss asymmetry, 238 Route choice principle, 169, 172, 174 Route-swapping algorithm, 177 RRM, 3239, 41, 4546, 6469 Rule complexity reduction, 137 RUM, 31, 3339, 41, 4546, 132, 234, 250 SAM, 156157 Satisficing states, 7 Satisficing theory, 195 Shopping behavior, 95, 97, 112 Short-term change, 196, 198 Similarity, 1012, 58, 6061, 68, 74, 119, 157 Situational dependence, 117, 131 Stated adaptation, 210, 224, 229 Stated adaptation experiment, 210, 224 Stated attribute non-attendance, 17, 75, 83, 84 Stated choice, 17, 31, 46, 66, 75, 8081, 92 Stress, 189191, 195201, 210

String recognition algorithm, 122 Subjective search gain, 217, 219, 222, 228 Supervised learning, 137, 146 Taste heterogeneity, 18, 46, 116 Threshold utility value, 7 Tolerance, 163, 171174, 176177, 179181, 191, 196197, 199, 210 Tolerance based principle, 181 Traffic flow component, 166167, 174 Transformation function, 173, 176177 Travel behavior, 13, 31, 138, 139, 141, 190, 234, 235, 244, 248249 Travel behavior forecasting, 2, 214 Type I strategy, 74 Type II strategy, 74 Unsupervised learning, 146 User equilibrium, 164, 168172, 179, 181, 229, 246247 Utility, 13, 511, 1320, 2225, 3133, 35, 37, 39, 41, 43, 45, 4970, 7879, 9597, 104, 111112, 116, 137, 140, 158, 169, 193198, 200201, 203205, 207210, 214215, 223, 228, 233239, 242, 250 Utility maximization theory, 13, 158 Utility-maximizing behavior, 111 Utility space, 68, 13, 15, 20, 25 Valence framing, 246, 248 Value function, 109, 238, 240241, 247248 Value judgments, 56 Weight function, 241 Willingness to accept (WTA), 246247 Willingness to pay (WTP), 7374, 83, 88, 9091, 244, 246247 Working memory, 117118