Planning Behavior: Theories and Experiments [1 ed.] 1527537838, 9781527537835

Table of contents :
Dedication
Table of Contents
List of Figures
List of Tables
Foreword • Lewis D. Hopkins
Preface
1. Differential Effects of Outcome and Probability on Risky Decisions in Gams and Losses
2. Frame and Contingent Utility for Risky Choices
3. Land Development Decisions and Lottery Dependent Utility
4. Isolation Effect as a Source of Myopic Planning for Urban Development
5. Comparison of Regimes of Policies for Urban Development: A Social Welfare Approach
6. Effects of Urban Containment Policy on Land Development Decisions
7. Toward a Solution to the Voting Dilemma
8. Information Structure Exploration as Planning for a Unitary Organization
9. An Anatomy of Time Explicit Planning Behavior for Urban Complexity
10. Toward a Behavioral Planning Theory
11. Why Plans Matter for Cities
12. Managing Urban Complexity through Decision Coordination
Index

Citation preview

Planning Behavior

Planning Behavior: Theories and Experiments By

Shih-Kung Lai

Cambridge Scholars Publishing

Planning Behavior: Theories and Experiments ByShih-KungLai This book first published 2019 Cambridge Scholars Publishing Lady StephensonLibrary, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright© 2019 by Shih-KungLai All rights for this book reserved. No part ofthis book may be reproduced, stored in a retrieval system, or transmitted, in any fonn or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior pennission ofthe copyright owner. ISBN (10): 1-5275-3783-8 ISBN (13): 978-1-5275-3783-5

To My Mentor Lewis D. Hopkins

TABLE OF CONTENTS

List of Figures ............................................................................................. ix List of Tables ............................................................................................... x Foreword .................................................................................................... xi

by Lewis D. Hopkins Preface ...................................................................................................... xiv

1. Differential Effects of Outcome and Probability on Risky Decisions in Gams and Losses ..................................................................................... 1

2. Frame and Contingent Utility for Risky Choices .................................. 35 3. Land Development Decisions and Lottery Dependent Utility ............... 53 4. Isolation Effect as a Source of Myopic Planning for Urban Development ............................................................................................. 69 5. Comparison of Regimes of Policies for Urban Development: A Social Welfare Approach ....................................................................... 83

6. Effects of Urban Containment Policy on Land Development Decisions ................................................................................................. 105 7. Toward a Solution to the Voting Dilemma .......................................... 139 8. Information Structure Exploration as Planning for a Unitary Organization ............................................................................................ 150 9. An Anatomy of Time Explicit Planning Behavior for Urban Complexity .............................................................................................. 178 10. Toward aBehavioral Planning Theory .............................................. 203

viii

Table of Contents

11. Why Plans Matter for Cities .............................................................. 215 12. Managing Urban Complexity through Decision Coordination .......... 234 Index ........................................................................................................ 249

LIST OF FIGURES

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 2.1 3.1 4.1 5.1 5.2 5.3 5.4 5.5 5.6 6.1 6.2 6.3 6.4 9.1

The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 1 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 2 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 3 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 4 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 5 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 6 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 7 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 8 Two utility functions using the PE and CE methods Effects ofUGBs as developable land control An example of the value function in prospect theory (Kahneman and Tversky, 1979) The game tree for player 1 adopting Tit for Tat for iterations k ~ 1 and 2 The space-time plot of the rule of 10000000 The space-time plot of the rule of 10100000 The space-time plot of the rule of 10100100 The space-time plot of the rule of 11101100 The space-time plot of the rule of 11111110 Conceptual diagram illustrating the role ofUCBs Land demand and supply under UCBs Value functions within and outside UCBs Value functions within and outside UCBs with different reference points Conceptual framework of the planning procedure in each time interval

LIST OF TABLES

1.1

Comparison of effects of payoff and probability on subjects'

2.1

A summary of the elicitation questions originally designed in prospect theory The experimental design A summary of the percentages of the subjects choosing prospect A orB Comparison of two expected utility models in explaining choice

choices through gray relation analysis

2.2 2.3 2.4 3.1

3.2 3.3

4.1 4.2 4.3 5.1

behavior The t test of the comparison of utilities of monetary values across frames for the NTPU experiment (N ~ 38)

The ttest of the comparison of utilities of monetary values across frames for the pretest (N ~ 25) The ttest of the comparison of utilities of monetary values across frames for the Zhejiang experiment (N ~ 26) The distribution of independent decisions for questions 1 and 3 The distribution of independent decisions for questions 2 and 4 The distribution of sequential decisions The standard payoff table for the two-person, iterated prisoner's dilemma game

5.2

The simplified payoff table for the two-person, iterated prisoner's dilemma game

5.3 6.1

6.2 6.3 6.4 6.5 7.1. 7.2.

Transition rules corresponding to the values of b Questionnaire statistics Respondents summary for question 1 Reference points inside and outside UCBs Risk tolerance within UCBs before and after delineation Risk tolerance outside UCBs before and after delineation A hypothetical voting matrix The hypothetical voting matrix ifA3 withdrew

FOREWORD LEWIS D. HOPKINS DEPARTMENT OF URBAN AND REGIONAL PLANNING UNNERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

Shih-Killlg Lai is building a body of work that combines deduction from

axiomatic concepts with empirical investigations and experiments. This approach is unusual in the plarming literature, which increases the challenge of pursinng it. Building on decision theory, prospect theory, and behavioral economics, Lai brings these ideas and the research approach to questions that arise in the planning of cities. This work includes aspects of what we have begilll to call "The Illinois School of Thinking about Plans" (Hopkins and Knaap, 2019). In particular,

the Illinois School considers that concepts from economics, operations research, and regional science are useful in thinking about how plans work,

including the dynamics of adjustment, uncertainty, and sequential decisions. Lai uses abstract framing

ill

mathematical telTIls, logical

deduction, and numerical examples to illustrate his ideas in relation to plarming and urban phenomena. Lai developed the idea of "framed rationality-how the problem is framed

and represented in the decision maker's head" as way to reconcile normative, descriptive, and prescriptive claims about decision making. He uses framed rationality in an attempt to understand the contradictions within subjective expected utility, bounded rationality, and prospect theory.

xii

Foreword

Initial empirical work seeks to untangle utility as separate from probability rather than the usual lumping the two together in lottery questions for experimental subjects. If we think of the decision maker as making rational decisions given a particular frame, we may be able to devise new techniques for aiding prediction and prescription about decisions. To make decision theory pertinent to plarming, we need to consider linked decisions. Most decision theory focuses on isolated decisions, but plans are inherently about more than one decision. Decisions can be framed within varying degrees of isolation in time, space, and decision maker. Planners are also concerned with individual payoffs to predict payoff and aggregate payoffs to evaluate outcomes and game theory provides one opportunity for investigation, both theoretical and empirical. Lai finds concepts from decision theory, often framed as well-defmed puzzles, and builds on them to cope with more complex plarming situations. This approach has been used successfully in related fields, such as the tit-for-tat strategy in the prisoner's dilemma game as an analogue for how players who have already interacted many times will behave differently from players in first time interactions. The empirical and experimental results presented here remain tentative, in part because few other researchers in planning are pursuing a similar approach so there is little

replication

or

complementary work

to

enable

cumulative

interpretations. The chapters in this book provide a launching point for such further research.

Planning Behavior: Theories and Experiments

xiii

Reference Hopkins, L. D., and Knaap, G.-I. 2019. "The 'Illinois School' of Thinking about Plans," Journal of Urban Management 8(1): 5-11. doi:DOI: 10.1 016/j .jum.2019.02.001

PREFACE

There are many social and natural phenomena now recognized as complex systems, such as cities, economies, ecologies, political entities, and societies. The crux across many disciplines in social sciences is to ask the question of how to make rational decisions and plans in such systems. Decision making as manifested in the perfectly rational choice theory of economics is insufficient in dealing with such complex systems, especially when decisions are interdependent, indivisible, irreversible, and with imperfect foresight (Hopkins, 2001). The most accepted paradigm of rationality, the subjective expected utility theory or the SED model, has been challenged by psychologists (Hogarth and Reder, 1987) and experimental economists (Ariely, 2008) as being unable to describe how people actually make choices, at least in experimental settings. In addition, the traditional distinction among descriptive, normative, and prescriptive theories of choice provides more confusion than explanations in dispelling how people make decisions. This seeming misconception of rational choice is mainly caused by a simplified, mechanical view of the world where the relationship between causes and effects is clear cut and systems tend to equilibrium. In particular, a general law of the SED approach to choice theory renders behavioral deviations as anomalies that might actually be rational in particular frames, resulting in the redundancy of the descriptive, nOlmative, and prescriptive distinction. In this book, we provide a fresh look at rationality, namely, framed rationality. Much has been said about decision making under uncertainty. Among other paradigms, subjective expected utility theory and prospect theory are

Planning Behavior: Theories and Experiments

xv

both derived from decision analysis. Subjective expected utility theory considers that if the outcome of a choice is uncertain, then the traditional method of calculating expected monetary value fails to measure the decision maker's preferences among alternatives. The concept of utility is needed, and expected utility is calculated instead to measure the decision maker's preferences. Based on subjective expected utility theory, the rational decision maker will choose the alternative that has the highest expected utility when faced with uncertain alternatives. In 1979, Kahneman and Tversky designed a set of decision questions and used them to conduct psychological experiments. They found that different ways of framing questions caused preference reversals. The results violate the postulation of subjective expected utility theory that the decision maker should make consistent choices. This phenomenon is called the framing effect. Frames are defined as the choice conditions under which the decision maker behaves. Frames of questioning may influence the choice conditions conceived by the decision maker, who cannot penetrate the underlying logic of these decision questions if asked in different ways. Kahneman and Tversky proposed prospect theory to effectively explain this phenomenon. However, there is no explanation in prospect theory as to whether the choice made by the decision maker corresponds to the principle of utility maximization. The question remains of whether prospect theory can replace subjective expected utility theory in explaining real choice behavior. In this book, we argue that the decision maker is rational in the same sense as defined by subjective expected utility theory, regardless of how frames are defined, and we call this explanation of choice behavior framed rationality. An experiment has been conducted that replicates the experiments that Kahneman and Tversky designed and conducted in 1979, in that subjects revealed preference reversal when

xvi

Preface

questions were framed differently (see Chapters 1, 2, 3, and 4). However, our experiment takes this one step further by measuring the subjects' utilities in making these choices, and confitms the hypothesis of framed rationality in that when using the same questions as in Kahneman and Tversky's experiments, we find that a statistically significant number of subjects maximize subjective expected utilities in making choices, regardless of how the elicitation questions are framed. In other words, preference reversal does not violate the SED model; rather, it validates the model subject to particular frames. This finding might provide a starting point to reconsider or redefine rationality by reconciling currently conflicting views of decision theories. For example, observed preference reversal phenomena might be caused by the framing effects, but tbey do not necessarily violate the SED model in the framed rationality sense, as refuted by its variants, including bounded rationality (Simon, 1955) and prospect theory (Kahneman and Tversky, 1979). The traditional distinction among descriptive, nonnative, and prescriptive views seems redundant from the framed rationality point of view in that these conflicting views are reconciled if we look at these theories from the corresponding frames. That is, the nonnative view argues for the SED model as tbe standard of rationality, and it purports to describe how people should make choices. Any behavioral violation of that model is considered as an anomaly and thus falls into the descriptive view of how people do make choices. As sho\Vll in our experiment, this distinction is unsound if we also consider the anomaly as rational but in a particular frame, or framed rationality. If this logic holds, the prescriptive view that helps the decision maker to choose to confonn to the standard of rationality is unnecessary because there is really no such distinction

Planning Behavior: Theories and Experiments

between the normative and descriptive views.

xvii

Finally, unlike bounded

rationality and prospect theory, framed rationality as proposed here rejects the idea of comprehensive, perfect rationality that is usually assumed by neo-classic economic theory and derived from the positivist philosophy of science, and thus enhances the universality of the SED model (or

something similar) within particular frames.

We argue that the

increasingly recognized conception that our world is complex and far from equilibrium might prompt a paradigm shift in explaining rational choice,

and that framed rationality might be a good start. This book provides a set of collected essays on behavior in planning and decision making. It is compiled on the belief that plans and decisions are intertwined in our acting in the face of urban complexity. Together, these essays aim toward a behavioral plarming theory that is focused on a decision-centered view of plarming behavior. We hope the reader can gain

insights from these essays as to how to plan and act in a complex world. I am grateful to Ching-Pin Chiu, Thong-You Huang, Shu Hung, Shiu-Chien Kuo, Li-Hung Tsai, Hsi-Peng Tseng, and Li-Guo Wang for their help in conducting the research presented in this book. I would also like to tharik Cambridge Scholars' Adam Rummens for his assistance in making this

book possible.

References Ariely, D. 2008.Predictably Irrational: The Hidden Forces That Shape Our Decisions. New York: HarperCollins. Hogarth, R. M. and M. W. Reder. 1987. Rational Choice: The Contrast between Economics and Psychology. Chicago: The University of Chicago Press. Hopkins, L. D. 2001. Urban Development: The Logic of Making Plans. Washington, D.C.: Island Press.

xviii

Preface

Kahneman, D., and A. Tversky. 1979. "Prospect Theory: An Analysis of Decision under Risk," Econometrica XLVII: 263-291. Simon, H. A. 1955. "A Behavioral Model of Rational Choice," The Quarterly Journal ofEconomics 69(1): 99-118.

1. DIFFERENTIAL EFFECTS OF OUTCOME AND PROBABILITY ON RISKY DECISIONS IN GAINS AND LOSSES!

Introduction Probability and payoff (outcome) relevant preference are tbe two main factors that affect how choices are made by the decision maker under uncertainty. Mainstream theories of decision making under uncertainty, most significantly subjective expected utility theory (Savage, 1972; von Neumann and Morgenstem, 1953) and prospect theory (Kahneman and Tversky, 1979), require the decision maker to express probabilistic and preferential judgments separately, based on which probabilities and utilities (values) are derived accordingly and then combined into a composite, weighted measure for comparison to make a choice. In the practice of utility elicitation, subjects are asked to express probabilistic or preferential judgments so tbat they are indifferent to a standard lottery, without knowing the relative contributions of probability and payoff to these judgments. In particular, these mainstream tbeories imply tbat utility (value) is invariant across lotteries or frames. However, recent experiments show that utility is lottery dependent in tbat the utility for the same monetary value is different if elicited in different lotteries (Lai et aI., 2017). The natural question to ask is: How do probability or payoff each contribute to 1

This chapter has been published in Applied Economics Letter 2019,

https:lldoi.orgll 0.1080113504851.2019.1602699

2

1. Differential Effects ofOutcorne and Probability on Risky Decisions in Gains and Losses

the choice of lotteries by the decision maker? Some studies have partially addressed this question (cf., Slovic and Lichtenstein, 1968; Cohen et aI., 1987; Nygren et aI., 1996; Kuhberger et aI., 1999; Rao et aI., 2012). The answer to this specific question may systematically improve our understanding of the underlying cognitive mechanism through which choices under uncertainty are made. We report here an experiment to answer the question. In Section 2, we introduce the experimental design. Section 3 reports our experimental results. We conclude in Section 4.

Experimental Design In their classic paper, Kahneman and Tversky (1979) reported a series of 14 lotteries and came up with the well-known prospect theory. For comparison purposes, we use the first eight pairs of elementary lotteries to explore the relative contributions of probability and payoff to decision making under uncertainty because these eight pairs of elementary lotteries are more transparent and constitute the main body of the lotteries based on which prospect theory is constructed. The remaining six pairs of lotteries are designed to test specific psychological traits, such as the isolation effect, and thus are not quite relevant to our purposes here. Note that we are interested in the measurement of utility in subjective expected utility theory rather than value in prospect theory. The eight pairs of lotteries are given below. Note that the lotteries were presented in their elementary forms. For example, ($4,000, 0.80) stands for a lottery in which there is an 80% of probability that the player would gain $4,000 and a 20% that he or she would gain nothing. The subjects were asked to select the lottery they preferred in a lottery pair. There were 50 subjects participating in our experiment with 40 effective questionnaires. The questionnaires of the remaining 10 subjects were incompletely filled out and were thus excluded.

Planning Behavior: Theories and Experiments

3

Each subject was asked to make a series of choices from lottery pairs. A sample size of 40 is greater than the minimum sample size of 30 needed to

fulfill the requirements of the central limit theorem to allow us to conduct meanuigful statistical tests (Chang et aI., 2008). All the subjects were undergraduate students from the Department of Real Estate and Built Environment at National Taipei University, Taiwan, and each subject was

rewarded NTD 200 as remuneration. One US dollar is roughly equivalent to 30 NT dollar. Lottery Pair 1:

Lottery Pair 2:

A: ($4,000, 0.80)

A: ($4,000,0.20)

B: ($3,000, 1.00)

B: ($3,000,0.25)

Lottery Pair 3:

Lottery Pair 4:

A: ($6,000, 0.45)

A: ($6,000,0.001)

B: ($3,000,0.90)

B: ($3,000,0.002)

Lottery Pair 5:

Lottery Pair 6:

A: (-$4,000, 0.80)

A: (-$4,000, 0.20)

B: (-$3,000, 1.00)

B: (-$3,000,0.25)

Lottery Pair 7:

Lottery Pair 8:

A: (-$6,000, 0.45)

A: (-$6,000, 0.001)

B: (-$3,000,0.90)

B: (-$3,000,0.002)

4

1. Differential Effects ofOutcorne and Probability on Risky Decisions in Gains and Losses

For each pair of lotteries, a series of 20 pairs of lotteries were designed by systematically varying probability and payoff respectively, while keeping the difference in expected values of each pair of lotteries constant. Through such a design, we could record and observe how probability and payoff affected the 40 subjects in making choices in the 20 pairs of lotteries by minimizing noise from unknown factors. A list of all such 20 pairs of lotteries for Lottery Pairs 1 through 8 is provided in Appendix 1.

Results For each of the original lottery pairs, we plotted the percentages of the 40subjects who chose A or B in the 20 systematic pairs of lotteries. As shown in Figure 1.1, one can observe that the percentages of the 40 subjects who chose A or B for the first 10 pairs of lotteries remain stable where the probabilities are held constant (probability-constant lottery pairs). In contrast, the percentages of the 40 subjects who chose A or B for the last 10 pairs of lotteries fluctuate significantly where the payoffs are held constant (payoff-constant lottery pairs). This observation shows that the subjects are more sensitive to the variation of probability than payoff. Appendix 2 shows all the distribution diagrams of the percentages of the 40 subjects choosing A or B for the remaining sets of systematic pairs of lotteries. One can immediately observe that except for Lottery Pairs 6 and 7, all other original lottery pairs show significant differences in distribution of percentages between the probability-constant lottery pairs and the payoffconstant lottery pairs. Note that the payoffs in Lottery Pairs 6 and 7 are all negative, or losses, and that though the payoffs in Lottery Pair 5 and Lottery Pair 8 are also losses, their associated probabilities are to some

5

Planning Behavior: Theories and Experiments

extent extremes of one or negligibility. On the face of it, we could conclude that when faced with gains, the decision maker is more sensitive to probability tban payoff in making choices under uncertainty and that when faced with losses, he or she is equally sensitive to probability and payoff in making such choices. Probability-Constant 1-10

Payoff-Constant 11-20

100%

100%

80%

80%

60%

~

40% 20%

60%

~

40% 20% 0%

0% 12345678910

12345678910

Figme 1.1 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 1

In a closer look at how probability or payoff would affect tbe subjects' choices, we adopted Grey Relation Analysis (Liu et aI., 2017) to compare the arrays of choices made by each subject in tbe probability-constant pairs of lotteries and the payoff-constant pairs of lotteries derived from each original lottery pair. Appendix 3 depicts how Grey Relation Analysis is conducted, and Table 1.1 shows the results of the Grey Relation Analysis. Based on the Grey Relation Analysis, we can conclude that when faced with gains, the decision maker pays more attention to probability in making choices under uncertainty (Lottery Pairs 1 to 4), whereas when faced with losses, the decision maker seems equivalently sensitive to probability and payoff in making choices under uncertainty (Lottery Pairs

6

1. Differential Effects ofOutcorne and Probability on Risky Decisions in Gains and Losses

5 to 7). However, when the probability is negligible and faced with losses, the decision maker focuses more on probability than payoff in making choices under uncertainty (Lottery Pair 8). Table 1.1 Comparison of effects of payoff and probability on subjects' choices through Grey Relation Analysis Lottery Pair

Affected by Payoff

Affected by Probability

Affected Equally by Payoff and Probabilitv

Total

1 2 3 4

5

31 29 24 38 21 16 19 36

4 3 9 1 9 14 8 1

40 40 40 40 40 40 40 40

5

6

8 7

1 10 10

7

13

8

3

Conclusions Gains and losses are two distinct behavioral regimes in which decisions are made dramatically differently. In the gains regime, the decision maker is risk averse, less sensitive to values, and focuses more on probability in making choices under uncertainty; whereas in the losses regime, the decision maker becomes risk seeking, more sensitive to values, and is equivalently sensitive to probability and payoff in making these choices. Previous attempts at answering the question of how probability and payoff contribute to decision making under uncertainty have been inconclusive. We report here an experiment to address this fundamental question. According to the experimental results, we argue that when faced with gains, the decision maker pays more attention to probability than payoff in making choices under uncertainty and that when faced with losses, he or

Planning Behavior: Theories and Experiments

7

she seems equivalently sensitive to probability and payoff in making these choices. In addition, when tbe probability is negligible and faced with losses, the decision maker focuses more on probability than payoff.

References Chang, H-J, C-H Wu, J-F Ho, and P-Y Chen. 2008. "On Sample Size in Using Central Limit Theorem for Gamma Distribution." International Journal of Information and Management Sciences 19(1): 153-174. Cohen, M., J-V. Jaffray, and T. Said. 1987. "Experimental Comparison of Individual Behavior under Risk and under Uncertainty for Gains and for Losses." Organizational Behavior and Human Decision Processes 39(1): 1-22. Huang, J-Y. and S-K. Lai. 2017. "Exploring Cognitive Biases in Making Plans for Urban Development through tbe Isolation Effect of Prospect Theory." Research Report of Ministry of Science and Technology, Taiwan (Grant Number:l03WFA1300061). Kahneman, D. and A. Tversky. 1979. "Prospect Theory: An Analysis of Decision under Risk." Econometrica 47(2): 263-291. Kuhberger, A., M. Schulte-Mecklenbeck, and J. Pemer. 1999. "The Effects of Framing, Reflection, Probability, and Payoff on Risk Preference in Choice Tasks." Organizational Behavior and Human Decision Process 78(3): 204-231. Lai, S-K., J-Y. Huang, and H. Haoying. 2017. "Land Development Decisions and Lottery Dependent Utility." Real Estate Finance 34(2): 39-45. Liu, S., Y. Yang, and J. Forres!. 2017. Grey Data Analysis: Methods, Models and Applications. Singapore: Springer Science+Business Media. Nygren, T. E., A. M. Isen, P. J. Taylor, and J. Dulin. 1996. "The Influence of Positive Affect on the Decision Rule in Risk Situations: Focus on Outcome (and Especially Avoidance of Loss) Ratber Than Probability." Organizational Behavior and Human Decision Processes 66(1): 59-72. Rao, L., and S. Li, T. Jiang, and Y. Zhou. 2012."ls Payoff Necessarily Weighted by Probability When Making a Risky Choice? Evidence from Functional Connectivity Analysis." Plos One 7(7): e41048. Slovic, P. and S. Lichtenstein. 1968. "Relative Importance of Probabilities and Outcomes in Risk Taking." Journal of Experimental Psychology 78(3, P!.2): 1-18.

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1. Differential Effects ofOutcorne and Probability on Risky Decisions in Gains and Losses

Savage, L. J. 1972. The Foundations of Statistics. New Yark: Dover Books. van Neumann, J. and O. Margenstern. 1953. Theory of Games and Economic Behavior. Princeton, NI.: Princeton University Press.

Planning Behavior: Theories and Experiments

9

Appendix 1: Systematic Lottery Pairs Derived from the Original Lottery Pairs The original Lottery Pair 1 is as follows: Lottery Pair 1: A: ($4,000, 0.80) B: ($3,000, 1.00) Based on this original lottery, we designed 20 subsequent lotteries. The

°

first 1 pairs of lotteries were designed by fixing the probabilities and varying the payoffs incrementally so that the difference of the expected values between any pair of lotteries remained the same. The last 10 pairs

of lotteries were designed by fixing the payoffs and varying the probabilities incrementally so that the difference of the expected values between any pair of lotteries remained the same. By eliminating possible unknO\vn noise, this design, and the ensuing analyses, allow us to observe

how probability and payoff each affect the subjects' choices in the lotteries. The systematic pairs of lotteries for the remaining seven original pairs of lotteries were designed in a similar way as sho\Vll below.

Lottery 1-1:

Lottery 1-2:

A: ($1,500, 0.80)

A: ($2,750,0.80)

B: ($1,000, 1.00)

B: ($2,000, 1.00)

Lottery 1-3:

Lottery 1-4:

A: ($4,000, 0.80)

A: ($5,250,0.80)

B: ($3,000, 1.00)

B: ($4,000, 1.00)

10

1. Differential Effects ofOutcorne and Probability on Risky Decisions in Gains and Losses

Lottery 1-5:

Lottery 1-6:

A: ($6,500, 0.80)

A: ($7,750,0.80)

B: ($5,000, 1.00)

B: ($6,000, 1.00)

Lottery 1-7:

Lottery 1-8:

A: ($9,000, 0.80)

A: ($10,250,0.80)

B: ($7,000, 1.00)

B: ($8,000, 1.00)

Lottery 1-9:

Lottery 1-10:

A: ($11,500,0.80)

A: ($12,750, 0.80)

B: ($9,000, 1.00)

B: ($10,000, 1.00)

Lottery 1-11:

Lottery 1-12:

A: ($4,000,0.125)

A: ($4,000, 0.20)

B: ($3,000,0.10)

B: ($3,000,0.20)

Lottery 1-13:

Lottery 1-14:

A: ($4,000,0.275)

A: ($4,000, 0.35)

B: ($3,000,0.30)

B: ($3,000, 0.40)

Lottery 1-15:

Lottery 1-16:

A: ($4,000,0.425)

A: ($4,000, 0.50)

B: ($3,000,0.50)

B: ($3,000,0.60)

Planning Behavior: Theories and Experiments

Lottery 1-17:

Lottery 1-18:

A: ($4,000,0.575)

A: ($4,000, 0.65)

B: ($3,000,0.70)

B: ($3,000,0.80)

Lottery 1-19:

Lottery 1-20:

A: ($4,000,0.725)

A: ($4,000, 0.80)

B: ($3,000,0.90)

B: ($3,000, 1.00)

Lottery Pair 2: A: ($4,000,0.20) B: ($3,000,0.25) Lottery 2-1:

Lottery 2-2:

A: ($1,500, 0.20)

A: ($2,750,0.20)

B: ($1,000,0.25)

B: ($2,000,0.25)

Lottery 2-3:

Lottery 2-4:

A: ($4,000, 0.20)

A: ($5,250,0.20)

B: ($3,000,0.25)

B: ($4,000, 0.25)

Lottery 2-5:

Lottery 2-6:

A: ($6,500, 0.20)

A: ($7,750,0.20)

B: ($5,000,0.25)

B: ($6,000,0.25)

11

12

1. Differential Effects ofOutcorne and Probability on Risky Decisions in Gains and Losses

Lottery 2-7:

Lottery 2- 8:

A: ($9,000, 0.20)

A: ($10,250,0.20)

B: ($7,000,0.25)

B: ($8,000,0.25)

Lottery 2-9:

Lottery 2-10:

A: ($11,500,0.20)

A: ($12,750, 0.20)

B: ($9,000,0.25)

B: ($10,000,0.25)

Lottery 2-11:

Lottery 2-12:

A: ($4,000,0.0875)

A: ($4,000, 0.20)

B: ($3,000,0.10)

B: ($3,000,0.25)

Lottery 2-13:

Lottery 2-14:

A: ($4,000,0.2375)

A: ($4,000, 0.3125)

B: ($3,000,0.30)

B: ($3,000,0.40)

Lottery 2-15:

Lottery 2-16:

A: ($4,000,0.3875)

A: ($4,000, 0.4625)

B: ($3,000,0.50)

B: ($3,000,0.60)

Lottery 2-17:

Lottery 2-18:

A: ($4,000,0.5375)

A: ($4,000, 0.6125)

B: ($3,000,0.70)

B: ($3,000,0.80)

Planning Behavior: Theories and Experiments

Lottery 2-19:

Lottery 2-20:

A: ($4,000,0.6875)

A: ($4,000, 0.7620)

B: ($3,000,0.90)

B: ($3,000, 1.00)

Lottery Pair 3: A: ($6,000,0.45) B: ($3,000,0.90)

Lottery 3-1:

Lottery 3-2:

A: ($2,000, 0.45)

A: ($4,000,0.45)

B: ($1,000,0.90)

B: ($2,000,0.90)

Lottery 3-3:

Lottery 3-4:

A: ($6,000, 0.45)

A: ($8,000,0.45)

B: ($3,000,0.90)

B: ($4,000,0.90)

Lottery 3-5:

Lottery 3-6:

A: ($10,000,0.45)

A: ($12,000, 0.45)

B: ($5,000,0.90)

B: ($6,000, 0.90)

Lottery 3-7:

Lottery 3- 8:

A: ($14,000,0.45)

A: ($16,000, 0.45)

B: ($7,000,0.90)

B: ($8,000,0.90)

13

14

1. Differential Effects ofOutcorne and Probability on Risky Decisions in Gains and Losses

Lottery 3-9:

Lottery 3-10:

A: ($18,000, 0.45)

A: ($20,000, 0.80)

B: ($9,000,0.90)

B: ($10,000, 1.00)

Lottery 3-11:

Lottery 3-12:

A: ($6,000, 0.05)

A: ($6,000,0.10)

B: ($3,000,0.10)

B: ($3,000, 0.20)

Lottery 3-13:

Lottery 3-14:

A: ($6,000, 0.15)

A: ($6,000,0.20)

B: ($3,000,0.30)

B: ($3,000,0.40)

Lottery 3-15:

Lottery 3-16:

A: ($6,000, 0.25)

A: ($6,000,0.30)

B: ($3,000,0.50)

B: ($3,000,0.60)

Lottery 3-17:

Lottery 3-18:

A: ($6,000, 0.35)

A: ($6,000,0.40)

B: ($3,000,0.70)

B: ($3,000, 0.80)

Lottery 3-19:

Lottery 3-20:

A: ($6,000, 0.45)

A: ($6,000,0.50)

B: ($3,000,0.90)

B: ($3,000, 1.00)

Planning Behavior: Theories and Experiments

Lottery Pair 4: A: ($6,000,0.001) B: ($3,000,0.002) Lottery 4-1:

Lottery 4-2:

A: ($2,000,0.001)

A: ($4,00, 0.001)

B: ($1,000,0.002)

B: ($2,000, 0.002)

Lottery 4-3:

Lottery 4-4:

A: ($6,000,0.001)

A: ($8,000, 0.001)

B: ($3,000,0.002)

B: ($4,000,0.002)

Lottery 4-5:

Lottery 4-6:

A: ($10,000, 0.001)

A: ($12,000, 0.001)

B: ($5,000,0.002)

B: ($6,000,0.002)

Lottery 4-7:

Lottery 4-8:

A: ($14,000,0.001)

A: ($16,000, 0.001)

B: ($7,000,0.002)

B: ($8,000, 0.002)

Lottery 4-9:

Lottery 4-10:

A: ($18,000,0.001)

A: ($20,000, 0.001)

B: ($9,000,0.002)

B: ($10,000,0.002)

15

16

1. Differential Effects ofOutcorne and Probability on Risky Decisions in Gains and Losses

Lottery 4-11:

Lottery 4-12:

A: ($6,000,0.001)

A: ($6,000, 0.10)

B: ($3,000,0.002)

B: ($3,000,0.20)

Lottery 4-13:

Lottery 4-14:

A: ($6,000, 0.15)

A: ($6,000,0.20)

B: ($3,000,0.30)

B: ($3,000, 0.40)

Lottery 4-15:

Lottery 4-16:

A: ($6,000, 0.25)

A: ($6,000,0.30)

B: ($3,000,0.50)

B: ($3,000,0.60)

Lottery 4-17:

Lottery 4-18:

A: ($6,000, 0.35)

A: ($6,000,0.40)

B: ($3,000,0.70)

B: ($3,000,0.80)

Lottery 4-19:

Lottery 4-20:

A: ($6,000, 0.45)

A: ($6,000,0.50)

B: ($3,000,0.90)

B: ($3,000, 1.00)

Planning Behavior: Theories and Experiments

Lottery Pair 5: A: (-$4,000, 0.80) B: (-$3,000, 1.00)

Lottery 5-1:

Lottery 5-2:

A: (-$1,500, 0.80)

A: (-$2,750, 0.80)

B: (-$1,000, 1.00)

B: (-$2,000, 1.00)

Lottery 5-3:

Lottery 5-4:

A: (-$4,000, 0.80)

A: (-$5,250, 0.80)

B: (-$3,000, 1.00)

B: (-$4,000, 1.00)

Lottery 5-5:

Lottery 5-6:

A: (-$6,500, 0.80)

A: (-$7,750, 0.80)

B: (-$5,000, 1.00)

B: (-$6,000, 1.00)

Lottery 5-7:

Lottery 5- 8:

A: (-$9,000, 0.80)

A: (-$10,250, 0.80)

B: (-$7,000, 1.00)

B: (-$8,000, 1.00)

Lottery 5-9:

Lottery 5-10:

A: (-$11,500, 0.80)

A: (-$12,750,0.80)

B: (-$9,000, 1.00)

B: (-$10,000, 1.00)

17

18

1. Differential Effects ofOutcorne and Probability on Risky Decisions in Gains and Losses

Lottery 5-11:

Lottery 5-12:

A: (-$4,000, 0.125)

A: (-$4,000, 0.20)

B: (-$3,000,0.10)

B: (-$3,000,0.20)

Lottery 5-13:

Lottery 5-14:

A: -($4,000, 0.275)

A: (-$4,000, 0.35)

B: (-$3,000, 0.30)

B: (-$3,000,0.40)

Lottery 5-15:

Lottery 5-16:

A: (-$4,000, 0.425)

A: (-$4,000, 0.50)

B: (-$3,000,0.50)

B: (-$3,000,0.60)

Lottery 5-17:

Lottery 5-18:

A: (-$4,000, 0.575)

A: (-$4,000, 0.65)

B: (-$3,000,0.70)

B: (-$3,000,0.80)

Lottery 5-19:

Lottery 5-20:

A: (-$4,000, 0.725)

A: (-$4,000, 0.80)

B: (-$3,000, 0.90)

B: (-$3,000, 1.00)

Planning Behavior: Theories and Experiments

Lottery Pair 6: A: (-$4,000, 0.20) B: (-$3,000,0.25)

Lottery 6-1:

Lottery 6-2:

A: (-$1,500, 0.20)

A: (-$2,750, 0.20)

B: (-$1,000, 0.25)

B: (-$2,000,0.25)

Lottery 6-3:

Lottery 6-4:

A: (-$4,000, 0.20)

A: (-$5,250, 0.20)

B: (-$3,000,0.25)

B: (-$4,000,0.25)

Lottery 6-5:

Lottery 6-6:

A: (-$6,500, 0.20)

A: (-$7,750, 0.20)

B: (-$5,000,0.25)

B: (-$6,000,0.25)

Lottery 6-7:

Lottery 6-8:

A: (-$9,000, 0.20)

A: (-$10,250, 0.20)

B: (-$7,000, 0.25)

B: (-$8,000,0.25)

Lottery 6-9:

Lottery 6-10:

A: (-$11,500, 0.20)

A: (-$12,750,0.20)

B: (-$9,000,0.25)

B: (-$10,000,0.25)

19

20

1. Differential Effects ofOutcorne and Probability on Risky Decisions in Gains and Losses

Lottery 6-11:

Lottery 6-12:

A: (-$4,000, 0.0875)

A: (-$4,000, 0.20)

B: (-$3,000,0.10)

B: (-$3,000,0.25)

Lottery 6-13:

Lottery 6-14:

A: (-$4,000, 0.2375)

A: (-$4,000, 0.3125)

B: (-$3,000, 0.30)

B: (-$3,000,0.40)

Lottery 6-15:

Lottery 6-16:

A: (-$4,000, 0.3875)

A: (-$4,000, 0.4625)

B: (-$3,000,0.50)

B: (-$3,000,0.60)

Lottery 6-17:

Lottery 6-18:

A: (-$4,000, 0.5375)

A: (-$4,000, 0.6125)

B: (-$3,000,0.70)

B: (-$3,000,0.80)

Lottery 6-19:

Lottery 6-20:

A: (-$4,000, 0.6875)

A: (-$4,000, 0.7625)

B: (-$3,000, 0.90)

B: (-$3,000, 1.00)

Planning Behavior: Theories and Experiments

Lottery Pair 7: A: (-$6,000, 0.45) B: (-$3,000,0.90)

Lottery 7-1:

Lottery 7-2:

A: (-$2,000, 0.45)

A: (-$4,000, 0.45)

B: (-$1,000, 0.90)

B: (-$2,000,0.90)

Lottery 7-3:

Lottery 7-4:

A: (-$6,000, 0.45)

A: (-$8,000, 0.45)

B: (-$3,000,0.90)

B: (-$4,000,0.90)

Lottery 7-5:

Lottery 7-6:

A: (-$10,000, 0.45)

A: (-$12,000,0.45)

B: (-$5,000,0.90)

B: (-$6,000,0.90)

Lottery 7-7:

Lottery 7-8:

A: (-$14,000, 0.45)

A: (-$16,000,0.45)

B: (-$7,000, 0.90)

B: (-$8,000,0.90)

Lottery 7-9:

Lottery 7-10:

A: (-$18,000, 0.45)

A: (-$20,000,0.45)

B: (-$9,000,0.90)

B: (-$10,000,0.90)

21

22

1. Differential Effects ofOutcorne and Probability on Risky Decisions in Gains and Losses

Lottery 7-11:

Lottery 7-12:

A: (-$6,000, 0.05)

A: (-$6,000, 0.10)

B: (-$3,000,0.10)

B: (-$3,000,0.20)

Lottery 7-13:

Lottery 7-14:

A: (-$6,000, 0.15)

A: (-$6,000, 0.20)

B: (-$3,000, 0.30)

B: (-$3,000,0.40)

Lottery 7-15:

Lottery 7-16:

A: (-$6,000, 0.25)

A: (-$6,000, 0.30)

B: (-$3,000,0.50)

B: (-$3,000,0.60)

Lottery 7-17:

Lottery 7-18:

A: (-$6,000, 0.35)

A: (-$6,000, 0.40)

B: (-$3,000,0.70)

B: (-$3,000,0.80)

Lottery 7-19:

Lottery 7-20:

A: (-$6,000, 0.45)

A: (-$6,000, 0.50)

B: (-$3,000, 0.90)

B: (-$3,000, 1.00)

Planning Behavior: Theories and Experiments

Lottery Pair 8: A: (-$6,000, 0.001) B: (-$3,000,0.002)

Lottery 8-1:

Lottery 8-2:

A: (-$2,000, 0.001)

A: (-$4,000, 0.001)

B: (-$1,000, 0.002)

B: (-$2,000,0.002)

Lottery 8-3:

Lottery 8-4:

A: (-$6,000, 0.001)

A: (-$8,000, 0.001)

B: (-$3,000,0.002)

B: (-$4,000,0.002)

Lottery 8-5:

Lottery 8-6:

A: (-$10,000, 0.001)

A: (-$12,000, 0.001)

B: (-$5,000,0.002)

B: (-$6,000,0.002)

Lottery 8-7:

Lottery 8- 8:

A: (-$14,000, 0.001)

A: (-$16,000, 0.001)

B: (-$7,000, 0.002)

B: (-$8,000,0.002)

Lottery 8-9:

Lottery 8-10:

A: (-$18,000, 0.001)

A: (-$20,000, 0.001)

B: (-$9,000,0.002)

B: (-$10,000,0.002)

23

24

1. Differential Effects ofOutcorne and Probability on Risky Decisions in Gains and Losses

Lottery 8-11:

Lottery 8-12:

A: (-$6,000, 0.001)

A: (-$6,000, 0.10)

B: (-$3,000,0.002)

B: (-$3,000,0.20)

Lottery 8-13:

Lottery 8-14:

A: (-$6,000, 0.15)

A: (-$6,000, 0.20)

B: (-$3,000, 0.30)

B: (-$3,000,0.40)

Lottery 8-15:

Lottery 8-16:

A: (-$6,000, 0.25)

A: (-$6,000, 0.30)

B: (-$3,000,0.50)

B: (-$3,000,0.60)

Lottery 8-17:

Lottery 8-18:

A: (-$6,000, 0.35)

A: (-$6,000, 0.40)

B: (-$3,000,0.70)

B: (-$3,000,0.80)

Lottery 8-19:

Lottery 8-20:

A: (-$6,000, 0.45)

A: (-$6,000, 0.50)

B: (-$3,000, 0.90)

B: (-$3,000, 1.00)

25

Planning Behavior: Theories and Experiments

Appendix 2: Distribution Diagrams of Percentages of Subjects' Choices for All Lottery Pairs Probability-Constant 1-10 100%

90% 80% 70%

60%

I-Bl

50%

~

40%

30% 20% 10%

0% 23

4

5678

9

10

Payoff-Constant 11-20 100%

90% 80% 70%

60%

I-Bl

50%

~

40% 30% 20%

10% 0%

2345678910 Figme 1.2 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 2

Figme 1.3 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 3

Figme 1.4 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 4

Figme 1.5 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 5

Figme 1.6 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 6

Figme 1.7 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 7

Figme 1.8 The percentages of subjects choosing A or B in the 20 systematic pairs of lotteries for lottery pair 8

32

1. Differential Effects ofOutcorne and Probability on Risky Decisions in Gains and Losses

Appendix 3: Example of Grey Relation Analysis A Grey Relation Analysis (Liu et aI., 2017) is conducted to demonstrate how the relationship between two arrays of data is computed by Equations 1 and 2. This appendix shows how the Grey Relation Analysis is carried out. The subjects' responses to each systematic lottery pair were re-coded

and calculated according to Equations 1 and 2 to derive the Grey Relation Coefficients to detelTIline whether his or her response to the original

lottery pair was mainly affected by probability or payoff. The greater Grey Relation Coefficient implies tbat tbe associated array (by eitber fixing probability or payoff) has a greater impact on the subjects' responses. For details of the recoding and calculation, the reader is encouraged to refer to

Huang and Lai (2017).

(1)

(2) 1. Let tbe target array Xo and two internal relation arrays Xl andX2 be as follows:

Xo

= (1.0,3.0,6.0,10.0,15.0,21.0, 2B.O, 36.0)

X,

= (1.0,2.5,4.5, 7.0, 10.0, 13.5, 17.5, 22.0)

X2

= (1.0, 1.0, 4.0, 5.0, 10.0, 12.0, 19.0, 22.0)

Planning Behavior: Theories and Experiments

33

2. TransfOlTIl the three arrays to unit-free arrays as follows:

X'o

= (1.0,3.0,6.0,10.0,15.0,21.0,28.0,36.0)

X',

= (1.0, 2.5, 4.5, 7.0, 10.0, 13.5, 17.5, 22.0)

X'z

= (1.0, 1.0,4.0,5.0,10.0,12.0,19.0,22.0)

3. Compute differences between the arrays.

L11

= (0.0, 0.5,1.5, 3.0, 5.0, 7.5, 10.5, 14.0)

L12

= (0.0, 2.0, 2.0, 5.0, 5.0, 9.0, 9.0, 14.0)

4. Select the maximum and minimum values in the difference arrays.

M

= Max~,Max~kL1,(k) = 14; m = Min~,Min~kL1,(k) = 0

5. Compute Grey Relation Coefficients for X'o andX' l by setting

0.5. f(x o(l),x, (l))

= 1.00;

f(xo(3),X,(3))

= 0.82;r(xo(4),x, (4)) = 0.70;

f(xo(5),x1(5))

= 0.58;

f(xo(7),X,(7))

= 0.40; r(xO(8),X1(8)) = 0.33;

f(x o(2),x, (2))

f(xo(6),X1(6))

= 0.93;

= 0.48;

S to

be

34

1. Differential Effects ofOutcorne and Probability on Risky Decisions in Gains and Losses

6. Compute Grey Relation Coefficients for X'o and X2 by setting 0.5.

rex o(1),x,(l))

= 1.00;

T(xo(3),x,(3))

= 0.78; r(xo(4),x,(4)) = 0.58;

rexo(5),x,(5))

= 0.58;

T(xo(7),x,(7))

= 0.44; r(xo(8),x,(8)) = 0.33;

T(x o(2),x,(2))

T(xo(6),x,(6))

= 0.78;

= 0.44;

7. Compute Grey Relation Degree for Xo andXL

T(xo, x,)

= (1.00+0.93+0.82+0.70+0.58+0.48+0.40+0.33)/8 '" 0.66 8. Compute Grey Relation Degree for Xo and X2.

T(xo, x,)

= (1.00+0.78+0.78+0.58+0.58+0.44+0.44+0.33)/8 '" 0.62 9. Rank the Grey Relation Degrees.

S to be

2. FRAME AND CONTINGENT UTILITY FOR RISKY CHOICES

Introduction The principle of utility maximization has dominated economic theory in general and decision science in particular for more than seven decades. Since the 1950s when Herbert Simon first proposed bounded rationality in an attempt to rectify behavioral anomalies distinct from the utility maximization principle, alternative theories have been proposed that are claimed to be more effective in explaining choice behavior than standard utility theory, including, among others, locally global planning (Pollock, 2006) and goal-based models of choice (Krantz and Kunreuther, 2007). Regardless of these recent attempts, standard utility theory and its variants will continue to play a dominant role in the development of economic theory and decision science for the foreseeable future. Standard utility theory is a purely mathematical construct with a convincing logic, rather than being derived from empirical observation or verification (e.g., Kirman, 2011). It mimics "the best examples of theoretical physics" (von Neumann and Morgenstem, 1944, ix). It is therefore internally valid, but subject to external verification. Standard utility theory and its variants, such as prospect theory, assume that the decision maker's utility functions are invariant across different decision contexts or frames. We argue that these utility functions change with these contexts or frames, rather than being invariant. If this argument is accepted, the explanation and formulation by prospect theory begs a close

36

2. Frame and Contingent Utility for Risky Choices

reexamination as to why preference reversals occur for the decision maker to make choices among questions that are logically equivalent but framed

differently. More specifically, Kahneman and Tversky (1979) were able to show that the decision maker was prone to selecting certain outcomes rather than

making risky choices - the certainty effect -which violated expected utility theory. Consider the following problems. Problem 1: Choose between A:

2,500 with probability

.33,

2,400 with probability

.66,

owith probability

.01.

B:

2,400 with certainty.

Problem 2: Choose between C:

2,500 with probability

.33, D: 2,400 with probability

owith probability

.67. 0 with probability

.34,

.66.

Most subjects in the experiment preferred B to A and C to D. The first selection pattern between A and B implies that u(2,400) > .33u(2,500) + .66u(2,400) or .34u(2,400) > .33u(2,500) while the second selection pattern implies the reversed inequality.

Note that the utility functions are invariant between the two choice problems, that is, u. Given the observation of preference reversal,

Kahneman and Tversky (1979) and Tversky and Kahneman (1981) proposed an alternative model to account for this discrepancy from

Planning Behavior: Theories and Experiments

37

expected utility theory by replacing probabilities with a weighting function and the invariant utility function with a value flUlction, that is, if(x,p;y, q) is a regular prospect, where x and y are the outcomes and p and q are the associated probabilities, then vex, P; y, q)

~

rr(p)v(x) + rr(q)v(y),

where rr and v are defined on probabilities and outcomes and V is defmed on prospects. We argue, however, that the decision maker's utility functions in Problem 1 and Problem 2 might not be invariant; rather, they might be contingent on the contexts or frames of the decision situations. More specifically, let u rand u rr be the two contingent utility functions for Problem 1 and

Problem 2. The first selection pattern between A and B implies that 0.34u'(2,400) > .33u'(2,500) or u '(2,500)/u '(2,400) < 1.03,

whereas the second selection pattern between C and D implies that .33u"(2,500) > .34u"(2,400) or u "(2,500)/u "(2,400) > 1.03.

Both inequalities satisfy the monotonically increasing pattern of utility flUlctions and thus do not violate the utility maximization principle. The present chapter is to report experiments that attempt to replicate the preference reversal phenomenon as observed in prospect theory. However, distinct from the experiment conducted by Kahenman and Tversky (1979), in each pair of the problems we explicitly elicited the contingent utility functions for each problem in order to verify that the principle of utility maximization might still hold. In other words, we hypothesize that the preference reversal in each pair of problems is caused not by violation of

38

2. Frame and Contingent Utility for Risky Choices

the utility maximization principle but rather by using an invariant utility function across the problems as preferential constructs, whereas the utility

functions associated with the two problems might be contingent and subject to framing effects.

The Experiment The hypothesis of the experiments is that the decision maker maximizes expected utility subject to particular frames. That is, in contrast to prospect theory that attempts to rectify subject expected utility theory in describing

how the decision maker makes risky choices, we hypothesize that the utility maximization principle is valid empirically and universal subject to particular frames. In order to test this hypothesis, two experiments were conducted with the same procedure and subjects, but with different experimental instruments. In the first experiment, the subjects used paper and pencil to answer the elicitation questions, whereas in the second experiment, a computer interface was designed so that the subjects answered these questions on the computer. Both experiments were divided into two parts: replication of the findings in prospect theory and the elicitation of utilities. The elicited utilities were used to calculate the expected utilities in order to test our hypothesis that the subjects would maximize their expected utilities in making risky choices regardless of how the elicitation questions originally designed in prospect theory were framed.

Replication ofprospect theory This part of the elicitation questions followed those originally designed by Kahneman and Tversky (1979), but translated into Chinese. These questions are summarized in Table 2.1, and the reader is encouraged to refer to Kahneman and Tversky's work for the complete description of

Planning Behavior: Theories and Experiments

39

these questions. Note that all the numerical prospects are transfOlmed in the standard format in that if (x, P; y, q) is a regular prospect, then x and y are the outcomes and p and q are the associated probabilities, respectively. Table 2.1 A summary of the elicitation questions originally designed in prospect theory Question munber 1 2 3 3' 4 4' 5 6 7 7' 8 8' 9 9' 10

11

12

13 13' 14 14'

Prospect A

Prospect B

(2,500, .33; 2,400, .66) (2,400) (2,500,33) (2400,34) (3 000) (4000.80) (-4000 .80) (-3,000) (4000.20) (3,000 .25) (-4000 .20) (-3 000 .25) A one-week tom of 50% chance to win a three-week tom of England, France, and Italy England, with certainty 5% chance to win a three-week tour of 10% chance to win a oneEngland, France, and Italy week tour of England (6,000 .45) (3 000 .90) (-6000 .45) (-3,000 .90) (6000.001) (3 000 .002) (-3,000, .002) (-6000, .001) Insme Not insure (-7000.25'-174 .D25· -87 .95) (-700005) Consider the following two-stage game. In the first stage, there is a probability of .75 to end the game without winning anything and a probability of .25 to move into the second stage. If you reach the second stage, you have a choice between (4,000, .80) and (3,000) Your choice must be made before the game starts, i.e., before the outcome of the first stage is known. In addition to whatever you mVli, you have been given 1,000. You are now asked to choose between (1,000, .50) and (500) In addition to whatever you mVli, you have been given 2,000. You are now asked to choose between (-1000, .50) and (-500) (600025) (4000 .25' 2 000 .25) (-6,000 .25) (-4,000, .25; -2 000 .25) (5000 .001) (5) (-5) (-5,000, .001)

40

2. Frame and Contingent Utility for Risky Choices

Note that Kahneman and Tversky (1979) designed these elicitation questions to explain the preference reversal phenomenon and derive an alternative theory to account for various anomalies violating standard

utility tlieory, such as the reflection effect and the isolation effect. Our purposes here are to replicate the findings of prospect tlieory and to test our hypothesis that the decision maker maximizes expected utility regardless of how the questions are framed. Note that Question 9' is designed specifically in order to elicit the utilities for Question 9 and to test the framing effects.

Elicitation of utilities Two elicitation modes for utility were designed in order to measure the

subjects' utility functions: the probability equivalent metliod (PE), and tlie certainty eqinvalent metliod (CE) (Wakker and Deneffe, 1996). The PE and CE methods are similar in that both repetitively ask the subjects to respond with probabilities and outcomes so that the decision maker is indifferent to a pair of prospects and the utility of a particular outcome can

be calculated. More specifically, we can distinginsh between the PE and CE metliods by tlie following formula: PE: (w)'" (X,l!.; y) and CE:

60'" (x,a; y),

(1)

where w, x, and y are Qutcomes,a is the probability associated with x, and ~

symbolizes that the decision maker is preferentially indifferent between

the prospects on the left and right sides. The PE method asks the decision maker what the probability a would be, given w, x, and y, so that he or she is indifferent between a certain w and a

prospect (x,a; y). On tlie other hand, the CE method asks the decision

Pbming B ehavior: Theorie, ,.,d Experiment'

maker what the outcome w would be, given x, y, and a, so that he or she is indifferent between a certain w and a prospect (x,aiY). If X andy are the best and wocst outcomes whose utilities are set to one and zero respectively, then a is equivalent to the utility ofw. Based OIl the PE and CE methods in combination with the mid-value splitting technique (Keeney and Raiffa, 1993), it is possible to elicit two utility fimctiOlls foc each O1lbject as shown in Figure 2.1.

1.0

.75

.50

.25

.00

o

2,500 5,000 7,500 10,000

FiglIe 2.1 Two utility functicns using the PE and CE methods

Based on the PE method, we can COllstruct the utility functiOll with white dots. Foc example, we can ask the decision maker to estimate each a when w is set to 2,500, 5,000, and 7,500 respectively, given that x andy are set to zero and 10,000 respectively. )n a similar way, for the C:E method, we can COIlstruct the utility function with dark dots by asking the decisiOll maker to estimate each w, when a is set to .25, .50, and .75 respectively,

42

2. Frame and Contingent Utility for Risky Choices

given that x and y are set to zero and 10,000 respectively. Note that all outcomes are measured in dollars. Based on the two utility functions derived from the PE and CE methods as shown in Figure 2.1, we can construct a hybrid, averaged utility function by calculating the averaged utilities of each outcome and use the averaged utility function as the final benchmark to compute the utilities given a certain outcome. The averaged

utility function would avoid judgmental biases derived from the PE and CE methods because it takes into account both judgments on probabilities and certain outcomes.

Estimation of decision weights The original standard utility theory postulates that the decision maker should make risky choices in order to maximize expected utility BD as follows: EU

~

L, u(x,)p"

(2)

where u is a utility function, associated with

Xl.

Xi

is the outcome i, and P I is the probability

In the light of the preference reversal phenomenon

found in their experiments, Kahneman and Tversky (1979) argued for a replacement of the probabilities with decision weights and of the value function with the utility function to account for these anomalies that violate standard utility theory. Because our hypothesis is that the decision maker maximizes expected utility regardless of how the elicitation questions are framed, we maintain that utilities should be the carriers of risky preferences as shown in (2). However, it would be interesting to examine the effectiveness of the weighting function in combination with the utility function; therefore, we designed a weighted utility function as shown below:

Planning Behavior: Theories and Experiments

43

(3) where rr is the weighting flUlction transfonning probabilities into decision

weights, which are not probabilities and do not obey the probability axioms. They should not be interpreted as measures of degree of belief as

exposited in standard utility theory (Savage, 1954). Following Tversky and Kahneman (1992), the estimation of decision weights is based on the differences between cumulative probabilities for outcomes in relation to the one lUlder consideration. More specifically, consider a risky prospect (Xl, PI). Let all outcomes in the prospect be ranked in ascending order so that -m ::; i ::;n and if i . ~ k " . . ""'" t..,..,. ;• • " • .." " "'Y (bboo" .... ""~kj,

".,,)

Th=.,." fr£« theoc (T

(O+b )12. Tbe two requirements being satisfied, the logic of the

two-person, iterated prisoner's dilemma game is thus retained. The beauty of this simplified version of the two-person, iterated prisoner's dilemma game lies in its simplicity of reducing the analytic structure to a single parameter b, making the exact comparison of different interactive strategies possible.

Planning Behavior: Theories and Experiments

87

The Exact Comparison Four commonly adopted interactive strategies are compared deductively

based on the payoff table depicted in Table 5.2, namely, always defect (AD), always cooperate (AC), Tit For Tat (TFT), and random action (RA). As the label expresses, AD (AC) indicates that the player adopting this strategy will take the action of defection (cooperation) no matter what action the other player takes. In particular, for AD, player I cooperates initially with a probability of one, and will keep defecting once player 2 defects. If player two defects in iteration I, then player I will defect in return in iteration 2 with a probability of one. Starting from iteration 2,

player 2, noticing that player I defected in iteration I and to avoid being exploited, will also defect in the ensuing iterations, and the process enters

into a punishment phase where both players defect (Dixit and Skeath, 2002). The player adopting the TFT strategy will cooperate in the first iteration, and then depending on the other player's action, the TFT player

will copy that action in the ensuing iterations. The RA player simply cooperates or defects in a random fashion, regardless of what action the

other player takes. Elsewhere (Chiu and Lai, 2008), we have shown that TFT is the best strategy yielding the highest expected payoff for the

individual player who adopts that strategy among all the four strategies, for limited and unlimited numbers of iterations. However, we show here

that, under some conditions, TFT is also the best strategy yielding the highest overall expected payoff across the two players, or the social welfare, for limited and unlimited numbers of iterations. The four strategies can be represented as game trees. Consider the game

tree for player 1 adopting Tit For Tat as shown in Figure 5.1. The thick branch indicates the evolution of possible paths throughout the game tree.

88

5. Comparison of Regimes of Policies for Urban Development

At the beginning of iteration 1, player 1 cooperates first and player 2, whose strategies are unknO\vn, can either cooperate or defect. In iteration 2, if player 2 cooperated in the previous iteration, then player 1 will cooperate and player 2 can either cooperate or defect. On the other hand, if player 2 defected in iteration 1, then player 1 will defect and player 2 can either cooperate or defect in iteration 2. This logic applies to the ensuing iterations. All the other three strategies can be represented as game trees. Note that this game tree can also be used to represent player 2's payoffs if we replace the payoffs at the leaves with those from player 2's perspective. What is remarkable is that all four strategies are recursive in the game trees, which makes the calculation of the overall expected payoffs tractable. More specifically, let Te denote a sub-tree in the TFT game tree for iteration k. Referring to Figure 5.1, a closer examination shows that the following recursive expected payoff fimctions for both players 1 and 2 obtain.

T3

~

qT2 + (1 - q)T2

~

T2,

Therefore, for TFT, Tn = Tn_1 = Tn_2 = ... = T3 = T2 • Since the payoffs for players 1 and 2 inT1 are q and q + b - bq respectively and for those in T2 are both q2+ qb - q2 b, it can easily be shown that the sum of the overall expected payoffs across the two players up to iteration k = n is Stt ~ (2q + b - bq) + 2(n _1)(q2+ bq - bq2).

(1)

Planning Behavior: Theories and Experiments

89

o

Figure 5.1 The game tree for player 1 adopting Tit For Tat for iterations k = 1 and 2. C = cooperates; D = defects; p is the probability player one will cooperate and q the probability that player two will cooperate; b is a payoff value where 1 < b < 2.

5. Comparison of Regimes of Policies for Urban Development

90

Let Pk denote a sub-tree in the game tree of AD for iteration k. The recursive expected payoff functions for both players 1 and 2 are:

P2

~

qP 1 + (1 - q)O

~

qP 1,

Pj

~

qP2 + (1 - q)O

~

qP2 ,

p" ~ qP"-l + (1 -

q)O~

qP"-l.

Since the payoffs for players 1 and 2 in? 1 are q and q + b-qb respectively, we have the sum of the overall expected payoffs across the two players up to iteration k = n as

Sp, ~ (2q+ b-- bq) (,,-_q;)

(2)

Let Fk denote a sub-tree in the game tree of AC for iteration k. The recursive expected payoff functions for both players 1 and 2 are:

Therefore, for AC, Fn = Fn_1 = Fn_2 = ... = F3 = F2 = F 1• Since the payoffs

for players 1 and 2 inFl are q and q + b-- O.

Assume that (Sff - Sp,) > 0 for k ~ n, that is, [(2q + b - bq) + 2(n-l)(q2+ bq-hq2)] - (2q+ b -bq) (1_ qn» 0 1-q

When k= n + 1, we have

(5)

5. Comparison of Regimes of Policies for Urban Development

92

2

2

(l_qn+1)

Sff - Sp, ~ [(2q + b - bq) + 2n (q + bq-bq )]- (2q+ b -bq)l"'-"-----'1-q ~

[(2q + b - bq) + 2(n - 1) (q2+ bq-b q 2) + 2(q2+ bq-b q 2)]-[(2q+ b -

qnl + (2qn+ b q n-1_b q n)] bq)(11-q qnl

~ [(2q + b - bq) + 2(n - 1) (q2+ bq-b q 2) - (2q+ b _bq)(1]+ [2(q2+ 1-q

qnl

~ [(2q + b - bq) + 2(n - 1) (q2+ bq-b q 2) - (2q+ b _bq)(1]+[2q 2 (1 1-q

Based on the logic of induction, we can conclude that TFT is better than

AD for k? 2, in terms of the overall expected payoffs across the two players. We now turn to the comparison between TFT and AC. Referring to Equations (1) and (3), when k~ 2 we have Sff - Sft ~ [(2q + b - bq) + 2(q2+ bq-b q 2)] - 2(2q + b - qb) ~ 2(q2+ bq-

bq2) - (2q + b - bq) ~ (1 - q)(2bq - b - 2q)

Whether (2bq - b - 2q) > 0 depends on the probability q, and a closer examination shows that for this inequality to hold, 2q ~> 1, where 1 < b-1

b < 2.

Assume that this inequality holds and that when k ~ n, Sff - Sft > 0; that is, [(2q + b - bq) + 2(n-1)(q2+ bq-b q 2)]- n (2q + b - qb) ~ 2(n-1)(q2+ bqbq2) - (n - 1)(2q + b - qb» 0

(6)

93

Planning Behavior: Theories and Experiments

Let k

~

b

n + 1 and assume that 2q >-> 1, and we have b-1

Sff - Sfi ~ 2n (q2+ bq--bq2) - n (2q + b - qb) ~ [2(n-l)(q2+ bq-b q 2) - (n -

1)(2q + b - qb)] + [2(q2+ bq-b q 2) -(2q + b - qb)]

~

[2(n- 1)(q2+ bq-b q 2)

- (n - 1)(2q + b - qb)] + (1 - q)(2bq - b - 2q) > 0

Based on the logic of induction, we can conclude that TFT is better than AC for k::: 2, in terms of the overall expected payoffs across the two . .

b

players, but under the condItIOn that 2q >-> 1. b-1

Finally, we compare TFT with RA. Referring to Equations (1) and (4), when k = 2, we have Sff - Sm' ~ [(2q + b - bq) + 2(q2+ bq-bq 2)] - 4(Pq + pb - pbq) ~ 2q + b +

2q2+ bp - 2bq2- 4pq - 4pb + 4pbq

Thus, if(2q-4p + 1)

~

(2q - 4p + 1)[b(1 - q) + q] + q.

-q ,or(q-2p»-{ q b(l-q)+q 2[b(1-q)+q]

+I} , thenTFT

is better than RA. Now, assume when k ~ n and the following inequality holds: Sff - Sfi ~ [(2q + b - bq) + 2(n- 1)(q2+ bq-bq 2)]- 2n(pq + pb - pbq) > O.

(7) Letk=n + 1, we have Sff - Sfi ~ [(2q + b - bq) + 2(n- 1)(q2+ bq-bq 2)] - 2n(pq + pb - pbq) +

2(q2+ bq-b q 2) - 2(Pq + pb - pbq) > 0, because 2(q2+ bq-b q 2) - 2(Pq + pb -pbq)

~2(q-p)[b(l-q)

+q] > 0 if(q-p) > O.

94

5. Comparison of Regimes of Policies for Urban Development

Based on the logic of induction, we can conclude that TFT is better than RA for k? 2, in terms of the overall expected payoffs across the two players, but under the condition that (q - p) >

o.

The case of an unlimited number of iterations For the comparison between TFT and AD, when k = n we know that the difference of the overall expected payoffs under the two strategies is given in Inequality (5). \¥hen n~

00,

implying an unlimited number of iterations,

the overall expected payoff difference between TFT and AD is expressed as

lim {[(2q + b - bq) + 2(n- 1)(q2+ bq-b q 2)]- (2q+ b _bq)(1-

qnl

l-q

n-->oo

~lim {[(2q + b - bq) + 2(n - l)q(q+ b -bq)]- (2q+ b _bq)(1-

qnl

l-q

n-->oo

~oo - (2q+ b -bq)...!L> 1-q

j

j

o.

We can conclude that TFT yields the higher overall expected payoff across the two players than AD in the case of an unlimited number of iterations. For the comparison between TFT and AC, the difference of the overall expected payoffs across the two players is given in Inequality (6). When

n -->

00,

the overall expected payoff difference between TFT and AC

becomes

lim {[(2q + b - bq) + 2(n - 1)(q2+ bq _b q 2)]_ n (2q + b - qb)} n~oo

~ lim n~oo

{(n - 1)[2(q2+ bq _b q 2) -(2q + b - qb)]}

95

Planning Behavior: Theories and Experiments ~lim

[en - 1)(1 - q)(2bq - b - 2q)]

n~oo

For (2bq - b - 2q) > 0, we have 2q>-'l-> 1. Therefore, TFT is better than b-1

AC if this inequality holds. Finally, we compare TFT with RA. The difference of the overall expected payoffs across the two players is shown in Inequality (7). When n-'>

00,

the

overall expected payoff difference between TFT and RA is shown as lim {[(2q + b - bq) + 2(n - 1)(q2+ bq _b q 2)]_ 2n(pq + pb - pbq)) n~oo

~

lim{(2n -1) (q - p)[b(1 - q) + q] + q} n~oo

Thus, if (q - p) > 0, then TFT is better than RA. It is clear that the answer really depends to the question of whether TFT outperfOlTIlS the other tree strategies, namely, AD, AC, and RA, in telTIlS of the overall expected payoffs across the two players in the two-person, iterated prisoner's dilemma game. More specifically, TFT unconditionally

outperforms AD. But for AC and RA, if 2q>_b_> 1 and (q -2 p) > b-1

{

q 2[b(1-q)+q]

+ I} respectively for the case of a limited number of iterations,

and if 2q>-'l-> 1 and (q -p) > b-1

°

respectively for the case of an unlimited

number of iterations, TFT still perfolTIls the best. In other words, in order

for TFT to outperform AC, q>~, meaning that player 2 is inclined to 2

cooperate. For RA, the implication for the probability distribution of p and

q for the two players is difficult to derive in the case of a limited number of iterations. However, in the case of an unlimited number of iterations,

96

5. Comparison of Regimes of Policies for Urban Development

the condition that TFT outperforms RA implies that player 2 is more inclined to cooperate than player 1.

Some Implications for Urban Development The game-theoretic analysis exposited in the present chapter can be used to draw implications for urban development. For example, mixed use may occur regardless of whether a zoning system for single use is imposed on a city. In particular, we argue that as long as the zoning system allows mixed use to a limited extent, as in the cases of Taiwan and Japan, the system would result in fractal rallier than Euclidean structural morphology. Firstly, according to Schelling (2006), segregation of neighborhoods emerges when there is mild prejudice against different etlmic groups. On the other hand, we argue that if there is no discrimination against mixed use, which would be the case in most Asian cities, segregation in land use would not occur. Secondly, mixed use is a more natural development pattern that might yield higher target profits than single use because of increasing returns to complementarity of uses. Compare mixed use and single use patterns of commercial and residential developments. In the mixed use pattern, the adjacency between commercial and residential uses would result in higher property right capture lliat is left in llie public domain as manifested by potential gains due to better accessibility between the two uses than when they are segregated, thus resulting in the total profits of the mutual development, which could be formulated as a prisoner's dilemma game as sho\Vll below. Consider only two land uses, commercial and residential, which could be adopted by two developers on two neighboring parcels in a community. The development decisions adopted by the two developers can be

Planning Behavior: Theories and Experiments

97

represented as a two-person, iterated prisoner's dilemma game in that each developer can either cooperate to develop the land for residential use or

defect for commercial use. Nowak and May (1993) design a simplified version of the two-person, iterated prisoner's dilemma game that serves as a basis for our deductive comparison. In their formulation, Nowak and

May (1993) reduce the payoff table of the two-person, iterated prisoner's dilemma game into one that contains only one parameter b as follows.

C

D

C

1

D

b

o o

In the above payoff table, the values represent the payoffs received by each player when that player (as shown in the rows) takes a certain action, while the other player (as shown in the columns) takes another action. For example, if player one cooperates and player two also cooperates, then

player one will receive a payoff of unity. Ifplayer one defects while player two cooperates, then player one will receive a payoff of b. For the simplified version of the two-person, iterated prisoner's dilemma game to be equivalent to the original one, b must be greater than unity and less than two, so that the Nash equilibrium settles on the combination of actions

where both players would defect. Chiu and Lai (2008) compare four strategies in the simplified version of the two-person, iterated prisoner's dilemma game, namely, always

defect/cooperate (AD or AC), Tit for Tat (TFT), and random actions (RA). Intuitively, in the land development context, AD or AC would result in

single use and TFT and RA would result in mixed use of spatial pattern. Chiu and Lai (2008) are able to show that in the case of an either limited

98

5. Comparison of Regimes of Policies for Urban Development

or unlimited number of iterations, the ranking of strategies in terms of the overall expected payoff obtained by a single player is TFT, RA, AC, and AD, implying that the strategies resulting in the outcome of mixed use are dominant. In addition, in the present chapter we show that TFT conditionally outperfOlTIlS other strategies even when the overall expected payoff is calculated across the two players. Note, however, that the TFT strategy might be difficult to adopt in practice because of the significant cost of revising a development decision, or irreversibility (Hopkins, 2001). Indeed, Hopkins (1979) argues that quadratic models reminiscent of mixed use are more effective in plan making than linear-programming models reminiscent of single-used zoning. Both empirical and simulated data demonstrate that the spatial pattern of land uses in Taipei is fractal and mixed use, rather than Euclidean and single use (Lin and Lai, 1998; Lai and Chen, 2006). Whether a single or mixed use pattern would prevail could be detelTIlined, on the other hand, by the parameter b. This question can be investigated through one-dimensional cellular automata (Wolfram, 2002). Consider an array of cells. In the land development context, suppose that each cell represents a block and there are only two pelTIlissible types of land use for each block, residential or commercial. The payoff matrix for the land development is written as follows.

R

E

R

1

E

b

o o

where R stands for residential and E symbolizes connnercial with b as the parameter in the original payoff table of the two-person, iterated prisoner's

Planning Behavior: Theories and Experiments

99

dilemma game. The above payoff matrix assumes that the payoff of a block is 1 when both blocks are in residential use and b when a block in commercial use interacts with another in residential use. In other situations,

the payoff is O. In the elementary cellular automaton with two neighbors, there are a total of eight arrays for the values of a cell and its two neighbors: 111, 110, 101, 100, 011,010,001 and 000. When each of the two neighbors interacts with its outward neighbor, there are four possibilities for each array, as shown below. The numbers in the brackets

denote the payoffs of three cells in the simplified prisoner's dilemma game, as shown in the above matrix. The last row shows the sum of the payoffs of the four possibilities of each of the three cells arrays. OIDO

OllQO

0.lQl0

01000

(b 0 b)

(b b2)

(2b l2b)(2b 2 3) (2b b)

(2 2b 2) (3 2 2b) (3 3 3)

OIDl

OllQ1

0.lQl1

00101

(b 00)

(b b1)

lIDO

01001

OQUO

00100

00010

00000

00011

00001

(2b 1 b) (2b 22) (2 b 0)

(22b 1) (32b)

(332)

lllQO

1.lQl0

11000

1QUO

10100

10000

(00 b)

(0 b 2)

(b 1 2b) (b 2 3)

(1 b b)

(1 2b 2) (22 2b) (23 3)

1ID1

lllQ1

1.lQl1

11001

1QU1

10101

10011

10001

(000)

(0 b 1)

(b 1 b)

(b 2 2)

(1 b 0)

(1 2b 1) (2 2 b)

(232)

OQU1

10010

(2b 0 2b)(2b 4b 6)(6b 4 6b)(6b 8 10)(6 4b 2b)(6 8b 6)(10 8 6b)(10 12 10)

100

5. Comparison of Regimes of Policies for Urban Development

It can easily be shown that when the value of b falls in different intervals

as sho\Vll in Table 5.3, different transition rules reign for the elementary cellular automaton, and the space-time plots corresponding to these rules show different patterns. According to Wolfram's classification, a Class 1 rule results in a homogeneous pattern with all white or black cells (see Figures 5.2, 5.3, and 5.6), reminiscent of a single use neighborhood, while a Class 2 rule comes up with a pattern of fixed structures (see Figures 5.4

and 5.5, reminiscent of a mixed use neighborhood). Note that in this particular example, a white cell symbolizes "1" as commercial use and a black cell represents "0" as residential use and that which transition rule reigns depends in turn on the value of b, which could be controlled through policy making, such as imposing a price system on land (Hopkins, 1974). Table 5.3 Transition rules corresponding to the values of b Binary Code

Wolfram's Classes

b 0; 2/3 2/3 0)

Assume that you are developing land outside the UCBs. Which of the following standards of success is true?

c.

Whether the project achieves the expected profits

d.

As long as there are profits (i.e., profit> 0)

Questions 2 through 5 use the actual profits as the reference point for making choices. Questions 6 and 7 were intended to test whether the reference point for land development inside the UCBs is the same as that

for land development outside the UCBs. According to the subjects' responses as sho\Vll in Table 6.3, for land development inside the UCBs,

most of the subjects (89%) used the expected profits as the reference point, while for land development outside the UCBs, most of the subjects (67%) used the status quo of zero as the reference point (see Table 6.3). This result was consistent with that obtained from the previous pair of questions in that the developers were mostly concerned with gains outside the UCBs

and losses inside the UCBs. Combined with the effect of differences in value judgments, the resulting value functions derived from prospect

theory for land development inside and outside the UCBs are shown in Figure 6.4.

Planning Behavior: Theories and Experiments

127

Table 6.3 Reference points inside and outside DeBs

Whether Expected Profits Are Achieved As Long as There Are Profits

Inside 102 (89%)

Outside 12 (11% )

38 03%)

76 (67%)

Value

Outside UCBs

-----Within UCBs Losses

Gains

Status Quo

Expected Value

Figme 6.4 Value fimctions within and outside UCBs with different reference points

Questions 8 and 9: Differences in risk tolerance among developers regarding land development on both sides of the DeBs If there were no delineation of the UCBs such that there were no

distinction between land development "inside" and "outside" the UCBs,

128

6. Effects of Urban Containment Policy on Land Development Decisions

and assuming uncertainties and risks in land acquisition exist, how much risk would you tolerate in order to acquire the land?

a.

Only when the risk is below _____ % would I consider acquiring the land.

Once the UCBs are delineated by the local government such that there is a distinction between land development "inside" and "outside" the UCBs, and assuming uncertainties and risks in land acquisition exist, how much risk would you tolerate in order to acquire the land?

b.

Only when the risk is below _____ % would I consider acquiring land inside the UCBs.

c.

Only when the risk is below _____ % would I consider acquiring land outside the UCBs.

The purpose of Questions 8 and 9 was to test whether there are differences in risk tolerance levels between land development with the delineation of

UCBs and land development without the delineation of UCBs. The results showed that before the delineation of the UCBs, the mean of the tolerated risk was 37.70%, with a standard deviation of 16.13; after the delineation of the UCBs, the mean of tolerated risk for land acquisition inside the

UCBs was 37.39%, with a standard deviation ofl7.91, while the mean of tolerated risk for land acquisition outside the UCBs was 41.706%, with a standard deviation of 21.26. A t test for the risk tolerance before and after the delineation of the UCBs showed that there was no significant difference (see Table 6.4, a

=

0.05). However, a closer examination

between land acquisition inside and outside the UCBs in tenns of risk tolerance showed that there was a significant difference between the two locations, with the risk tolerance for acquisition outside the UCBs higher

Planning Behavior: Theories and Experiments

than that for acquisition inside the DeBs (see Table 6.5, a

129 ~

0.05). This

implies that after the delineation of UCBs, the average developer was more tolerant toward the risks of land development outside the DeBs, a risk seeking tendency toward this type of land development. Table 6.4 Risk tolerance within DCBs before and after delineation Paired Variance Mean

Standard Deviation

Standard Deviation of Mean

Confidence Level (95%) Lower Upper Bmmd Bound

.316

14.621

1.369

-2.397

Degree t

of

Significance

Freedom

3.029

.231

113

.818

Table 6.5 Risk tolerance outside DCBs before and after delineation Paired Variance Mean

3.360

1.

Standard Deviation

Standard Deviation of Mean

Confidence Level (95%) Lower Upper BOlmd BOlmd

13.621

1.276

-5.887

-.832

Degree t

of

Significance

Freedom

2.633

113

.010

Discussion

The main purpose of the delineation of DeBs is to contain urban growth so that developers are confined to conducting land development inside the DeBs. The results of the questionnaire survey showed tliat after tlie delineation of DeBs, about half of the developers would seek land outside the UCBs for development. The main reason for this was that developers view land development inside and outside tlie DeBs differently. In other words, they "frame" tlie land development problem differently witli regard to the land inside versus outside the UCBs, causing some of them to

6. Effects of Urban Containment Policy on Land Development Decisions

130

pursue land development outside the UCBs for at least the following three reasons: 1)

Property rights capturing: To prevent the landowners from receiving the windfalls due to the delineation of the UCBs, some of the developers indicated that they would capture the property rights left in the public domain by turning to land development outside the UCBs to avoid property rights losses.

2)

Loss aversion in land development behavior inside the UCBs: After the delineation of the UCBs, the developers indicated that they would mainly be concerned with whether they could achieve the expected level of profits, leading to loss aversion for land development inside the UCBs and "pushing" them outward to look for land outside the UCBs for development.

3)

Risk seeking for land development behavior outside the UCBs: After the delineation of the UCBs, the developers indicated that they would tolerate more risk for land development outside the UCBs than for development inside the UCBs, leading to risk seeking for land development outside the UCBs and "pulling" them outward to look for land outside the UCBs for development.

Given these reasons for seeking land development outside of UCBs, in order to prevent urban development from sprawling, if the local government could prevent or reduce the level of loss aversion for land development inside of UCBs in addition to risk seeking for land development outside the UCBs, this should enhance the effectiveness of the delineation of UCBs. To do this, a local government would need to eliminate the endowment effect on the developers who conduct land

Planning Behavior: Theories and Experiments

131

development inside UCBs. For example, in granting construction pelTIlits to developers, the local government should be very clear in specifying the allowed floor area ratio (FAR) so that the developers would not expect that they could acquire more FAR than allowed as their endowment. On the other hand, in order to prevent or reduce risk seeking for land development outside of UCBs, a local government should control the information on land development outside the UCBs and effectively implement legal systems to do so. In addition, if the local government is not clear about land development outside the UCBs, developers would not stop taking the related risks through rent seeking by asking the local government to relax the control of land development outside the UCBs through rezoning. The usual viewpoint is that the implementation of plans requires legislation: for example, to control land development and implement plans to ensure such control, a local government needs to establish a zoning system to fulfill the goals set up in these plans. If plan makers do not understand the effects of these plans, the ensuing law making process would ignore the plans, resulting in a discrepancy between the plans and the associated regulations. This may have been manifested in the unexpected effects of the delineation of UGBs on land development behavior (Richardson and Gordon, 2001; Cox, 2001; Jun, 2004) because these policies did not anticipate developers' behavioral responses to the delineation of the UGBs. The effects of plans are an important, but difficult, topic that is wortliy of investigation (Hopkins, 2001). Plans and regulations are two distinct notions in that plans are intentional actions while regulations delineate rights. Plans affect individual agents'

132

6. Effects of Urban Containment Policy on Land Development Decisions

behaviors through infOlmation, while regulations enforce such restricted behaviors through the delineation of rights (Hopkins, 2001). The delineation of UCBs can be considered a plan in the present chapter, but one usually associated with regulations and control. Therefore, it is very difficult in reality to distinguish, via assessment, the effects of plans from those of regulations. Therefore, in the design of the questionnaire, we attempted to make clear the description of the scenario. However, due to subjective interpretations, the subjects would still confuse plans with regulations. The empirical results could not eliminate the effects of control as well. Ideally, the subjects would have consisted entirely of individuals who practice land development in the real world, though our questionnaire was simple enough to eliminate this shortcoming. More realistic questionnaires for more rigorously designed experiments on practicing developers call for future work.

Conclusions Based on the perspective of property rights, the present chapter attempts to explain both theoretically and empirically the effects of UCBs on developers' behaviors. Through a theoretical exposition, we argued that the delineation of UCBs would result in land development outside the UCBs, rather than containing it inside the UCBs. The main reasons for this would be that the developers inside and outside the UCBs frame the development problems differently, as a result of property rights capturing, loss aversion for land development inside the UCBs, and risk seeking for land development outside the UCBs. The questionnaire survey results confirmed these explanations of why the delineation of UCBs would exacerbate contradictory land development behaviors, causing urban sprawl rather than stopping it. Our findings provide strong evidence as to

Planning Behavior: Theories and Experiments

133

why many cities expand regardless of urban containment policies, in particular cities in China, such as Beijing (Wang et aI., 2014). These findings also constitute new knowledge as to how UGBs in general, and UCBs in particular, would affect developers' behaviors, namely their inclination to seek land outside the zones for development. To prevent such phenomena from happening, the policy implications are twofold: 1) elimination of the endowment effect by specifying clearly allowed building bulks inside tbe UCBs and 2) reduction of risk seeking by controlling development information and effectively implanting legal systems outside of the UCBs. These recommendations can be applied to countries where UCBs are practiced, such as China, or where UGBs are implanted, but not effectively enforced, such as some cities in the U.S.A.

134

6. Effects of Urban Containment Policy on Land Development Decisions

Appendix: Design ofthe Questionnaire In order to ensure that the subjects understood the telTIlS used in the questionnaires, the following definitions of telTIlS were fully explained

before the subjects filled out the questionnaires: 1)

Urban Growth Boundaries (UGBs): A particular form of urban containment policy that designates delineated boundaries for managing urban growth in order to confme urban growth in compact fOlTIls. In Taiwan, the delineated boundaries between urban development and non-urban development land uses, urban construction boundaries

(UCBs), are similar to the notion ofUGBs. 2)

Inside UCBs: Urban development land use, including residential, commercial, and industrial zones for construction purposes.

3)

Outside UCBs:

Non-urban

development

land

use,

including

agricultural and protected zones inside plarmed areas and non-urban

land in regional plans. 4)

Uncertainties and risks: There exist uncertainties in land acquisitions

which can be represented by probabilities as risks. 5)

Expected profits: The profits expected by the developer in advance of acquiring land, less possible transaction costs.

6)

Realized profits: The actual profits resulting from land development afterwards, less realized transaction costs.

7)

Land price: Because of the delineation of UCBs, land supply decreases, causing the price of land inside the UCBs to be higher than that of land outside them, as predicted in the literature (Phillips and Goodstein, 2000; Cho, Wu, and Boggess, 2003; Cho, Chen, and Yen,

2008). 8)

Governmental attitude: Local governments prefer to lead developers

Planning Behavior: Theories and Experiments

135

to develop land inside the UCBs, which is the main purpose for delineating the UCBs. 9)

Public facilities: These facilities are better provided inside the UCBs than outside. Local governments tend to invest more in constructing public facilities to make efficient use of resources in order to encourage developers to develop land inside tbe UCBs.

10) Information: There is more information available inside the UCBs than outside. Developers can acquire more infOlmation inside the UCBs than outside, including the intents of local governments and other developers, implying tbat developers know that otber developers or competitors possess more infOlmation inside the UCBs than outside. 11) Development intensity: Assume tbat the development intensity is tbe same inside and outside the UCBs. This assumption is made to prevent the subjects from discriminating development preferences between botb sides of the UCBs due to legal considerations.

There were five sections in the questionnaire, with a total of nine questions. The first section (Question 1) was mainly intended to test the validity of the theoretical explanation as depicted in this chapter. The second section (Questions 2 and 3) was intended to test the subjects' preferences in terms of gains and losses between areas inside and outside the UCBs. The third section (Questions 4 and 5) was intended to test whetber the subjects valued land acquisition differently between areas inside and outside the UCBs. The fourth section (Questions 6 and 7) was intended to test whether the subjects selected reference points differently between areas inside and outside the UCBs. The fifth section (Questions 8 and 9) was intended to test the subjects' risk tolerance capability between the areas inside and outside the UCBs once the UCBs were delineated.

136

6. Effects of Urban Containment Policy on Land Development Decisions

References Baerwald, T. 1981. "The Site Selection Process of Suburban Residential Builders." Urban Geography 2(4): 339-57. Barnett, J. 2007. Smart Growth in a Changing World. Chicago: American Planning Association Planners Press. Barzel, Y. 1991. Economic Analysis of Property Rights. New York: Cambridge University Press. Bengston, D. N., J. O. Fletcher, and K. C. Nelson. 2004. "Public Policies for Managing Urban Growth and Protecting Open Space: Policy Instruments and Lessons Learned in the Uinted States." Landscape and Urban Planning 69(2-3): 271-286. Bengston, D. N. and Y. C. Youn. 2006. "Urban Containment Policies and the Protection of Natural Areas: The Case of Seoul's Greenbell." Ecology and Society 11(1): Article No. 3. Berke, P. R., D. R. Godschalk, and E. J. Kaiser. 2006. Urban Land Use Planning, 5th ed. Urbana, IL.: University of Illinois Press. Cho, S. H., Z. Chen, and S. T. Yen. 2008. "Urban Growtli Boundary and Housing Prices: The Case ofKnox County, Tennessee." The Review of Regional Studies 38(1): 29-44. Cho, S. H., J. J. Wu, and W. G. Boggess. 2003. "Measuring Interactions among Urban Development, Land Use Regulations, and Public Finance." American Journal ofAgricultural Economics 85: 988-999. Couch, C. and J. Karecha. 2006. "Controlling Urban Sprawl: Some experiences from Liverpool." Cities 23(5): 353-363. Cox, W. 2001. American Dream Boundaries: Urban Containment and Its Consequences. Atlanta, GA.: Georgia Public Policy Foundation. DeGrove, J. M. and D. A. Miness. 1992. The New Frontier for Land Policy: Planning and Growth Management in the States. Cambridge, MA.: Lincoln Institute of Land Policy. Gennaio, M. P., A. M. Hersperger, and M. Burgi. 2009. "Containing Urban Sprawl-Evaluating Effectiveness of Urban Growtli Boundaries Set by the Swiss Land Use Plan." Land Use Policy 26(2): 224-232. Gore, T. and D. Nicholson. 1991. "Models of the Land-Development Process: A Critical Review." Environment and Planning A 23: 705-730. Hopkins, L. D. 2001. Urban Development: The Logic of Making Plans. Washington D.C.: Island Press. Jun, M. J. 2004. "The Effects of Portland's Urban Growtli Boundary on Urban Development Patterns and Commuting." Urban Studies 41(7): 1333-1348.

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Kahneman, D., J. Knetsch, and R. Thaler. 1991. "Anomalies: The Endowment Effect, Loss Aversion, and Status-Quo Bias." Journal of Economic Perspectives 5(1): 193-206. Kahneman, D. and A. Tversky. 1979. "Prospect Theory: An Analysis of Decision under Risk." Econometrica 47: 263-291. Kline, J. and R. Alig. 1999. "Does Land Use Planning Slow the Conversion of Forest and Farm Lands?" Growth and Change 30: 3-22. Knaap, G. J. and L. D. Hopkins. 2001. "The Inventory Approach to Urban Growth Boundaries." Journal ofAmerican Planning Association 67(3): 314-326. Lai, S-K. 2001. "An Empirical Study of Equivalence Judgments versus Ratio Judgments in Decision Analysis." Decision Sciences 32(2): 277302. Leung, L. 1987. "Developer Behavior and Development Control." Land Development Studies 4: 17-34. Lucy, W. H. and D. L. Phillips. 2000. Confronting Suburban Decline: Strategic Planning for Metropolitan Renewal. Washington, D.C.: Island Press. Millward, H. 2006. "Urban Containment Strategies: A Case-Study Appraisal of Plans and Policies in Japanese, British, and Canadian Cities." Land Use Policy 23(4): 473-485. Moharned, R. 2006. "The Psychology of Residential Developers: Lessons from Behavioral Economics and Additional Explanations for Satisficing." Journal of Planning Education and Research 26: 28-37. Nelson, A. C. and J. B. Duncan. 1995. Growth Management Principles and Practices. Chicago: American Plarming Association Planners Press. Nelson, A. C. and T. Moore. 1993. "Assessing Urban Growth Management: The Case of Portland, Oregon, the USA's Largest Urban Growth Boundary." Land Use Policy 10: 293-302. O'Sullivan, A. 2007. Urban Economics, 6th ed. New York: McGraw-Hill Press. Patterson, J. 1999. "Urban Growth Boundary Impacts on Sprawl and Redevelopment in Portland, Oregon." Working Paper, University of Wisconsin-\Vhitewater. Pendall, R., J. Martin, and W. Fulton. 2002. Holding the Line: Urban Containment in the United States. Washington, D.C.: The Brookings Institution Center on Urban and Metropolitan Policy. Phillips, J. and E. Goodstein. 2000. "Growth Management and Housing Prices: The Case of Portland, Oregon." Contemporary Economic Policy 18: 334-344.

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Porter, D. R. 1986. Growth Management Keeping on Target? Washington, D.C.: Urban Land Institute and Lincoln Institute of Land Policy. Porter, D. R., R. T. Dunphy, and D. Salvesen. 2002. Making Smart Growth Work. Washington, D. C.: Urban Land Institute. Richardson, H. W. and P. Gordon. 2001. "Portland and Los Angeles: Beauty and the Beas!." Paper presented at the 17'" Pacific Regional Science Conference, Portland, OR. Samuelson, W. and R. Zeckhauser. 1988. "Status Quo Bias in Decision Making." Journal ofRisk and Uncertainty 1: 7-59. Schaeffer, P. V. and L. D. Hopkins. 1987. "Behavior of Land Developers: Planning and the Economics of Infmmation." Environment and Planning A 19: 1221-1232. Simon, H. A. 1955. "On A Class of Skew Distribution Functions." Biometrika 52: 425-440. Stein, J. M. 1993. Growth Management: The Planning Challenge of the 1990s. N ewbury Park, CA.: Sage Publications. Szold, T. S. and A. Carbonell. 2002. Smart Growth: Form and Consequences. Cambridge, MA.: Lincoln Institute of Land Policy. Thaler, R. 1980. "Toward A Positive Theory of Consumer Choice." Journal ofEconomic Behavior and Organization 1: 39-60. Urban Land Institute. 1998. Smart Growth Economy, Community, Environment. Washington, D.e.: Urban Land Institute, von Neumann, J. and O. Morgenstern. 1944. Theory of Games and Behavioral Research. Cambridge: Cambridge University Press. Wang, L-G., H. Han, and S-K. Lai. 2014. "Do Plans Contain Urban Sprawl? A Comparison of Beijing and Taipei." Habitat International 42: 121-130.

7. TOWARD A SOLUTION TO THE VOTING DILEMMA4

Introduction Policies or plans for urban development are usually made collectively in a

democratic society (Hopkins, 2001).

As a result, environmental

management can only be carried out effectively through a reasonable institutional

design

that

aggregates

individual

preferences

ill

a

representative government (Haefele, 1973). For the theoretical part, multiattribute decision making techniques are widely applied in many fields as a way of aggregating preferences (cf., Yoon and Hwang (1995) for an

introduction and overview), partly because the decision maker needs to make tradeoff judgments frequently among conflicting attributes or objectives (Keeney and Raiffa, 1993). Most of the applications of such

techniques have been introduced to help individual decision makers to make choices among given alternatives (e.g., Hammond et aI., 2002). It is relatively less known that multi-attribute decision making techniques have profound implications for group decision making in general (Sarin and Dyer, 1979) and social choice in particular (Arrow and Raynaud, 1986). In addition, Laukkanen et al. (2002) apply voting theory in natural resource management in telTIlS of multi-criteria group decision making. We focus

here on specific preference aggregation methods that are commonly applied, namely the analytic hierarchy process (ARP) and multi-attribute utility theory (MAUT). Multi-attribute decision making and collective 4

This chapter has been published in Review a/Social Sciences 2012, Vol. 6, No. 1,

pages 59 72.

140

7. Toward a Solution to the Voting Dilemma

choice share a common logic of preference aggregation, except that the fonner deals with individual decision making in relation to attributes while

the latter focuses on group decision making in relation to individuals. The present chapter addresses the issues of collective choice through multiattribute decision making methods.

Two commonly applied multi-attribute decision making techniques are multi-attribute utility tbeory (MAUT) (Keeney and Raiffa, 1993) and the analytic hierarchy process (AHP) (Saaty, 1986). Though developed independently, the two techniques mean implicitly the same tbing. Lai (1995) has shown that, if applied in an appropriate way, the decision rules in the two techniques are mutually pennissibly transfOlmable. That is, the weights of attributes and value functions of these attribute levels are mutually transformable from one technique to the otber so tbat the relative wortbs of alternatives are retained. In particular, Lai and Hopkins (1995) designed a variant scaling procedure of AHP, AHP', that combines the merits of MAUT and AHP and simplifies further the elicitation questions for weights and value functions by making them more concrete and meaningful. The formal proof for tbe validity of AHP' is given by Lai (1995). There is a large literature on the relationship between MAUT and AHP that we do not intend to delve into in the present chapter. However, one particular piece of work we want to single out for our purposes here is Perez's (1995) demonstration on how multi-district proportional elections can be interpreted in terms of MAUT and AHP. The voting dilemma was well formulated by Perez (1995) and will be introduced in detail in Section 2. In essence, based on a consistent multi district proportional election mechanism, the dilemma implies that the election outcome would be different if a candidate decided to abstain

Planning Behavior: Theories and Experiments

141

before the voting took place, and we will show that this inconsistency can be resolved by the proposed AHP' preference aggregation method. Multidistrict proportional elections are widely applied in many countries, in particular parliamentary elections. For example, in Taiwan the election of legislators in the Legislative Yuan adopts a multidistrict proportional election in that each district shares a fixed number of the total seats competed for by more than one candidate from different political parties. We shall first review Perez's voting model in the context of multidistrict proportional elections, pinpointing how the rank reversal phenomenon in AHP renders any universal election procedure impossible. We shall then demonstrate how AHP' developed earlier can help resolve this dilemma and [mally discuss its implications.

The Dilemma According to Perez (1995), consider n divisions (attributes) (DJ, D 2 ,

... ,

D") wanting to set up a procedure for the election of an assembly of 200

representatives from m parties (alternatives) (Ai, A 2 ,

...

,Am). In addition,

let Vji

= the number of votes obtained by candidature Aj in division DJ,

Vi ~ the total number of votes obtained by candidatureAJ , W l = the total number of votes cast in division DJ, El

= the electorate (potential votes) of division D l ,

E, V = the total electorate and the total number of votes cast respectively, sji

= the number of seats obtained by candidature Aj in division D l , and

SJ

~

the total number of seats obtained by candidature AJ.

142

7. Toward a Solution to the Voting Dilemma

There are two extreme solutions for this voting procedure problem: proportional election and multi-district proportional election. In the proportional election, each candidature Aj obtains a number of seats If; proportional to the total number V; of votes obtained, that is, v· 20o-L.

(1)

v

In the multi-district proportional election, each division D J is assigned a fixed number of seats, r i, proportional to its electorate El. Thus, r ) = 200~. E

For each division D J, each candidature Aj obtains a number of seats, sji, proportional to the number v ji of his or her votes in D J, that is, Sj

"n Sji = L..i=l "n ri V]i "n 200 V]i El = L..i=l Wi = L..i=l Wi E

(2)

Perez (1995) proposed a more general solution to this voting problem: Let f3 be a parameter with values in the interval [0, 1]. Each division D, is assigned a variable number of seats,

r/ , inside the

interval [0, 200],

allocating the 200 seats among the divisions in proportion to the coefficients c, ~ f3W, + (1 -f3)E,. Thus,

~200PW,+(1-PlE'. PV+(l-PlE

(3)

In addition, for each division DJ, each candidature Aj obtains a number of seats proportional to the number of his or her votes in DJ as in Equation (2), and we have (4)

Planning Behavior: Theories and Experiments

It can easily be shown that when p ~

143

°

or!, the general solution is reduced

to the multi-district proportional election as in Equation (2) and tbe proportional election as in Equation (1), respectively. What is remarkable is tbat the voting problem of collective choice corresponds to MAUT and AHP of multi-attribute decision making when

p

is equal to 1 and 0, respectively. If we interpret candidatures as

alternatives and divisions as attributes, "the evaluation of global election results is a simple but proper multicriteria decision problem." (perez, 1995, p. 1093). Consider MAUT first in tbe context of the voting problem. Let

v;

be the best level of attribute I

MaXj=1,2 ....,m{vjd , and

V"i

across all alternatives, or

the worst level of that attribute across all

alternatives, or Minj=1,2, ...,m{vjd. Since each vote is treated as equally important, ajl = Vi*i-V*ievaluates the value obtained by A j in DJ in an interval Vi -v*i

scale and

Wl =

Vi~*V*i is

the weight of Dl, where ["= l:Y=l(V;

-

v"a. Thus,

the aggregate evaluation of Aj is

(5) Equation (5) is the same as Equation (l)up to an admissible transformation, meaning tbat tbe extreme solution in (1) or P ~ 1 in (4) is notbing but tbe application of MAUT to tbe voting problem. Put differently, Equation (5) transforms the vote counts into an MAUT scale of multi-attribute utility. Now, consider AHP in the same voting problem of collective choice. Let aji= vii ,

w,

which evaluates the values of Aj for D l , but in a ratio scale. The

144

7. Toward a Solution to the Voting Dilemma

fixed number r ) = ~ is the weight of DJ. Thus, the aggregate evaluation of E

A] becomes Sit-HP ]

= l:1!1=1 aJ1.. ~E = l:1!1=1 E117]1 Wt "

(6)

E

Equation (6) is the same as Equation (2), up to an admissible transformation, meaning that the extreme solution in (2) or f3

~O

in (4) is

nothing but the application of AHP to the voting problem. Put differently, Equation (6) transforms the vote counts into an AHP scale of multiattribute score. Ideally, MAUT and AHP would reach the same election outcome when no candidate withdraws. However, when a candidate abandons the election and all the followers of that candidate abstain, MAUT and AHP come up with different election outcomes. This inconsistency between MAUT and AHP is equivalent to the rank reversal

debate that occurred in the 1990s due to the deletion or addition of alternatives (e.g., Dyer, 1990a; 1990b; Harker and Vargas, 1990), but Perez (1995) was able to present it in the context of a voting problem. For concreteness and following Perez (1995), consider the following voting matrix with m = 3 candidates, n = 2 divisions, with the votes cast and potential electors given as shown in Table 7.1.

Table 7.1. A hypothetical voting matrix

Aj A2 A3 Total votes cast Potential votes

Dj

D2

500 260 240 1,000 (W,) 1000 (E ,)

520 745 735 2,000 (W,) 2000 (E,)

Total 1,020 (V,) 1 005 (V,) 975 (V,) 3,000 (V) 3000 (E)

Planning Behavior: Theories and Experiments

Applying Equation (4) and since W, ~ E, for i will show that S,

~

68, S,

~

67, and S3

~

~

145

1, 2, a closer examination

65 and A , wins. MAUT and AHP

agree. However, if A3 withdrew before the election took place and all his

or her followers abstained, then when

f3

~l

(i.e., the adoption of the

MAUT procedure), A , would obtain approximately 101 seats and win, but when f3

~

0 (i.e., the adoption of the AHP procedure), A, would obtain

approximately 101 seats and win, and, as a result, the voting dilemma of rank reversal occurs. The voting dilemma is a general phenomenon caused

by the different aggregation procedures as manifested by MAUT and AHP as noted in the literature on rank reversal (e.g., Dyer, 1990a; 1990b; Harker and Vargas, 1990).

A Solution As argued by Lai (1995), the rank reversal phenomenon of AHP is caused by the decision maker applying the wrong weights to attribute levels or values, both being measured in different scales. One way to resolve this phenomenon is to rescale the weights or attribute levels so that the two values are measured on a consistent scale. In particular, Lai and Hopkins

(1995) proposed a variant scaling procedure of AHP, AHP', later proved formally as valid by Lai (1995), that requires the decision maker to make interval judgments between attributes to derive attribute weights in MAUT, make ratio judgments within attributes to derive attribute values in AHP,

and then rescale the attribute values in AHP proportionally so that the best attribute value within an attribute across all alternatives is equal to one. The resulting evaluation outcome should be consistent with either MAUT or AHP, if applied in an appropriate way.

146

7. Toward a Solution to the Voting Dilemma

More formally, using Perez 's (1995) language and referring to Equation (5), we have the attribute weights

v ~ -v·

Wi

for MAUT as.:.L....:...!.! l*

v?-v· = 1:n=1(17 * -v*r)

referring to Equation (6), we have the attribute values

Rescaling

ap

so that

ap

r

H

1

1

ajiaji

=

;

v-,

~. .

,

=

attribute values with the associated MAUT weights, we have (7) v-

Note that in the voting dilemma case, the rescaling factor~ would not V i -17* 1

make the best attribute value within an attribute across all alternatives equal to one; it simply restores the MAUT interval scale from the AHP ratio scale so that both the weights and attribute values are expressed in the same scale. Apparently, S/HP' is the same, up to an admissible transfOlmation, as the scale If; of Equation (1). For concreteness, returning to our voting matrix, i£4 3 withdraw before the election took place and all his or her followers abstained, we have the following revised voting matrix as shown in Table 7.2: Table 7.2. The hypothetical voting matrix if A, withdrew

Aj A2 Total votes cast Potential votes

Dj 500 260 760 (W,) 1,000 (E ,)

D2 520 745 1265 (W,) 2,000 (E,)

Total 1,020 (V,) 1,005 (V,) 2025 (V) 3,000 (E)

A , obtains the proportionality of c,; applying the AHP' procedure, and

since l'~ 240 + 225

~

465, we have

Planning Behavior: Theories and Experiments

~x 500-260 500-260

+~ X

465

745-520

147

745-250 ~2 193 465 ..

A, obtains tbe proportionality of st-

HP

'

= ~~x 500-260 + ~ X

S;:HP'+St-HP'

500-260500-260

Thus, Al obtains 200 x

2.193 2.193+2.161

465

745-520

745-250

~2.161

465

'" 101 seats and A, obtains 200 x

2.161

:-:-:""'':''-:-:- '" 99 seats. Note tbat, compared to the situation when A3 2.19 3 +2.161

participates, not only the rank but also the proportionality among the seats obtained from the candidatures are preserved in the application of the AHP' election procedure.

Conclusions Derived from tbe detailed exposition, Perez (1995) was only partially correct by arguing that no general preference "aggregation method is expected to be suited for every situation." (p. 1095) In our view, tbis claim is a manifestation of Arrow's (1963) linpossibility Theorem that no preference aggregation method exists for at least three alternatives (or candidatures in the voting context), that simultaneously satisfies four conditions: non-dictatorship, Pareto principle, unrestricted scope, and independence of irrelevant alternatives (MacKay, 1980). However, we have sho\Vll that a variant scaling procedure of AHP, AHP', can partially resolve this voting dilemma or rank reversal in the context of AHP and MAUT as framed by Perez (1995). It does not require a priori an institutional design that might impose additional administrative costs as usually perceived by selecting the parameterI'. All AHP' requires is to count the votes and apply the aggregation procedure as shown in Equation

148

7. Toward a Solution to the Voting Dilemma

(7) and in the numerical example, regardless of the withdrawal or addition of candidatures. To simplify, we assume strictly in our analysis that when a candidature withdraws, all hisJher followers abstain. In order to retain some realism, it is possible to extend the current fOlTIlUlation to allow for shifts in voting when this situation occurs. Our focus here is, however, to

demonstrate the logic of the voting dilemma and propose a possible solution. Environmental management in a democratic society in general,

and a representative government in particular, would be made more effective by adopting reasonable preference aggregation procedures as presented here.

References Arrow, K. J. 1963. Social Choice and Individual Values. New Haven: Yale University Press.

Arrow, K. J. and H. Raynaud.1986. Social Choice and Multicriterion Decision-Making. Cambridge, MA.: The MIT Press. Dyer, J. S. 1990a. "Remarks on the Analytic Hierarchy Process." Management Science36: 249-258. Dyer). S. 1990b."A Clarification of 'Remarks on the Analytic Hierarchy Process'."Management Science36: 274-275. Hammond,J. S., R. L. Keeney, and H. Raiffa. 2002. Smart Choices: A Practical Guide to Making Better Decisions. New York: Broadway. Harker, P. T. and L. G. Vargas. 1990. "Reply to 'Remarks on the Analytic Hierarchy Process'."Management Science36: 269-273. Haefele, E. T. 1973. Representative Government and Environmental Management. Baltimore, Maryland: The Johns Hopkins University Press.

Hopkins, L. D. 2001. Urban Development: The Logic of Making Plans. Washington, D.C.: Island Press. Keeney, R. L. and H. Raiffa. 1993.Decisions with Multiple Objectives: Preferences and Value Tradeoffs. New York: Cambridge University Press.

Lai, S-K. 1995. "A Preference-Based Interpretation of AHP," Omega: Internationa/Journal ofManagement Science23(4): 453-462.

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Lai, S-K. and L. D. Hopkins.1995. "Can Decisionmakers Express Multiattribute Preferences Using AHP and MUT? An Experiment." Environment and Planning B: Planning and Design. 22: 21-34. Laukkanen, S., A. Kangas, and J. Kangas. 2002. "Applying Voting Theory in Natural Resource Management: ACase of Multiple-Criteria Group Decision Support."Journal of Environmental Management 64(2): 127137. MacKay, A. F. 1980. Arrow's Theorem: The Paradox of Social Choice: A Case Study in the Philosophy of Economics. New Haven: Yale University Press.

Perez, J. 1995. "Some Comments on Saaty's AHP."Management Science41(6): 1091-1095. Saaty, T. L. 1986. "Axiomatic Foundation of the Analytic Hierarchy Process."Management Science32: 841-855. Sarin, R. K. and J. S. Dyer. 1979. "Group Preference Aggregation Rules Based on Strength ofPreference."Management Science25(9): 822-832. Yoon, K. and C-L. Hwang. 1995Multiple Attribute Decision Making: An Introduction. Thousand Oaks, CA: Sage.

8. INFORMATION STRUCTURE EXPLORATION AS PLANNING FORA UNITARY ORGANIZATIOW

Introduction The chapter provides a theoretical basis for an improved understanding of the logic of planning, taking into account infonnation processing. Much has been said about planning methods, i.e., how planning problems should be approached. The behavior of plarmers and decision makers in making and carrying out plans has seldom been addressed (exceptions include Schaeffer and Hopkins, 1987; Sheshinsky and Intriligator, 1989; Knaap et aI., 1998). In previous attempts to explore planning behavior, tbe proposed plarming procedure has not been described explicitly in terms of how information is processed.

The central theme of the chapter is that in

general planning, like decision making, can be seen as the practice of gathering

information from which judgments on preference and

probability are made. Planning behavior, as in choice behavior, has thus to do witb probabilistic and value judgments (March 1978). The crucial distinction between plarming and decision making is tbat plarming considers a set of related decisions at the same time while decision making chooses from a set of alternatives based on some criteria. Decision making is an economic activity in that it searches for the best or optimal use of limited resources to attain objectives (Marschak and Radner, 1978, p.3). 5 This chapter has been published in Planning and Markets 2002, Vo!. 5, pages 32 41.

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Since planning is concerned with gathering infonnation and organizing decisions accordingly, plarming behavior is also an economic activity. Planning and organizational design are partially substitutable. Both are ways to coordinate actions to achieve desired outcomes. The ways in which actions are related in the two fonns of activity are different. In an organization, actions are coordinated through explicit structures of decision making and problem solving (Cohen et aI., 1972). For example, different decision makers have access to different decision making occasions, which tend to vary in tenns of capabilities of resolving different problems. In plarming, there are no such explicit structures confining which decision makers can attend to which decision making occasions and which problems are resolved by which decision making occasions. Plarming tends to focus on relatedness of actions in time and space, i.e., temporal and spatial decision making. There is no sharp conceptual distinction between planning and organizational design in that most planning takes place in organizational contexts, such as corporations, finns, and governments. It is difficult to define planning in any exclusive way, but at least we can identify that plarming enhances decision making by reducing uncertainty (Hopkins, 1981; Friend and Hickling, 1987). Orgainzations are artifacts designed for coordinating actions, i.e., arranging sequences of actions to achieve certain goals. A plan is a set of related, contingent decisions; therefore, plarming and organization are in part interchangeable (Lai and Hopkins, 1995). It is crucial, however, to distinguish between plarming with respect to substantive decisions and plarming with respect to planning activities (Hopkins, 1981). Lai provided a typology of planning behavior taking into account this distinction (Lai, 1994). In that typology, eight types of

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8. Information Structure Exploration as Planning for a Unitary Organization

plarming behavior were distinguished in terms of whether problems are internal or external to the organization; whether the decision making entity is single or multiple persons; and whether decisions are about substantive problems or about plan making. Lai then ran a set of simulations focusing on a particular type of plarming for an organization resolving external problems through making substantive decisions, to explore the effects of plarming on organizational choice behavior. Interesting findings were reported by Lai in that plarming might increase decision making efficiency but not necessarily resolve more problems. In this chapter, based on a different approach, I explore planning behavior of a similar type. In particular, I focus on plarming in the simplest fmm of organization, a unitary, coherent organization of individuals with consistent preferences, as a starting point to explore how planning should take place in organizational contexts. Incremental improvements on the present model can be made to match more realistic situations and tested empirically in real or laboratory settings. The approach I take in the chapter is information economics. Viewing plarming as information gathering and organizations as settings of information exchange among their members for making decisions, I investigate how such activities should be carried out optimally to yield desired outcomes. Using infmmation economics to describe plarming behavior for land development has been studied (Schaeffer and Hopkins, 1987). Instead of leaving detailed transformations of information structures unspecified as fmmulated earlier, I take one step further here to describe how infmmation should be processed and plans made in plarming for a unitary organization.

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Background The theoretical basis for interpreting planning behavior is utility theory. Plarming is defined here as a sequence of infOlmation gathering activities to reduce different kinds of uncertainties (Hopkins, 1981) and to solve problems encountered by plarmers to improve decision making by coordinating related decisions in time and space based on such information. The results from these activities could be either plans (or sets of decisions) or no action, depending on whether tbe planning yields benefit. A plan is a set of related, contingent decisions taking into account these uncertainties (e.g., Hopkins, 1981; Schaeffer and Hopkins, 1987). Though planning is not equivalent to decision making as argued earlier, from a nOlmative point of view, both are problem solving activities, and therefore their theoretical underpinnings must share some commonality. A more comprehensive understanding of plarming begs a long research agenda. Without pretending to provide such an agenda, the research is aimed at addressing plarming in a narrower context. The interpretation presented here is thus based on nOlmative decision theories because these theories prescribe how people make decisions and are deductively justified (e.g., von Neumann and Morgenstem, 1947; Savage, 1972, utility theory). I am, therefore, interested in how planning should take place, rather than how plarming does take place, for the decision maker. In the simplest telTIls, decision making selects from a set of options the best alternative that maximizes expected utility, while plan making coordinates a set of contingent decisions so that making plans yields benefits. A planner is a person who makes plans and acts accordingly. The descriptive aspects of plarming are, of course, relevant and need future work. Similar attempts have been made to describe plarming behavior for

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8. Information Structure Exploration as Planning for a Unitary Organization

planners (e.g., Schaeffer and Hopkins, 1987; Knaap et aI., 1998). The approach taken here considers how infOlmation structures should be searched yielding the maximum expected utility or payoff. The interpretation will be based mainly on Savage's choice theory under uncertainty (Savage, 1972). I focus here on a simplified planning situation consisting of a unitary group of plarmers and the environment, or the grand world and the corresponding small worlds (in Savage's language) in which events occur. The unitary group also makes collective decisions as will be introduced shortly. To simplify, I call such a group a planner. In Savage's choice theory, a sequence of assumptions and postulates are made to derive, for an idealized person or for that matter a coherent group, individual probabilities and utilities in making choices among acts under uncertainty. These assumptions and postulates are the substance of rationality. Savage introduces the notions of 'small worlds' and the 'grand world,' and the decision maker is to choose the best act in a small world. The concept of small worlds provides a useful way of describing the logic of planning. For example, a small world consists of a set of states or descriptions about the world and the actions available to the decision maker with the associated consequences given each state. A small world is, in a sense, the cognitive representation of the problem that the decision maker is to solve. Events are subsets of the set of states. Individual probabilities are assigned to these events. In the planning context, small worlds are the plarmer's perspectives about planning problems. The plarmer makes decisions in his or her

0\Vll

small worlds, which are defined in the grand world. The

plarmer makes plans and then acts accordingly, reminiscent of orders sent from the observer to the actor in a two-person tearn (Marschak, 1974),

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except that the observer and the actor are the same person in the unitary organization case. In following these plans, the plarmer chooses among the alternative actions suggested by the plans to solve problems. The planner could revise these plans if unexpected events occur. All such plarming activities can be described in terms of primitives as will be discussed in more detail. Most of these primitives are defmed rigorously by Savage (1972); I introduce them here in the context of planning.

Basic Concepts The planner as a decision maker is rational in Savage's sense. That is, he or she behaves according to Savage's assumptions and postulates of rationality and is a utility maximizer. In the simplified plarming situation of the unitary organization, two worlds are considered: the grand world and the plarmer's world. The plarmer's world is, in Savage's language, a microcosm so that the criterion of maximizing expected utility can be applied to making choices among acts. Each of the two worlds is described by a collection of states, actions available to the decision maker, and the consequences resulting from these actions. The states in small worlds are subsets of the states in the grand world, or the states in the grand world are tlie elementary states for small worlds. Let the grand world be, for example, the outcome of tossing two coins simultaneously. A small world would be the outcome of one of the two coins, regardless of the outcome of the other. The state of the small world, a head for example, is a subset of two states in the grand world (i.e., the outcome of the other coin can be either a head or a tail). Events are subsets of the states in a small world. In the example of tossing a coin, the outcome of a head is an event. The outcome of either a head or

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8. Information Structure Exploration as Planning for a Unitary Organization

a tail is also an event. Subjective probabilities are assigned to these events to express the plarmer's degrees of belief in the occurrence of such events. For example, the probability that a head occurs is usually 112. The probability that either a head or a tail occurs is 1. A decision is the plarmer's choice of acts (or actions) from a given set in a small world to solve problems and to anticipate desired consequences. An act will result in consequences that in turn are expressed as a function of the act. The consequences resulting from the act chosen in the small world can be realized in the grand world as acts that in turn result in consequences in the grand world. A small world could evolve resulting in transfOlmations from one world to another. All these concepts can be expressed in functional telTIlS as will be sho\Vll in the following sections. A planning problem is thus a problem of modifying the current states in the plarmer's world to achieve the desired states, taking into account future contingencies. This modification would require a set of acts, or decisions or a plan, which would in turn result in unexpected consequences. Strictly speaking, a plan is a set of related decisions or choice alternatives conditional on events with various probability distributions. A planning activity is a set of actions taken by the planner in the planning process. I consider the case in which plarming occurs in a discrete time frame in that a plan is made in each time period for solving the planning problems anticipated during that period. A new plan mayor may not be formed for the next time period while the old one is discarded. The process continues until the planner is satisfied with the current states or the plarmer's resources are depleted. A discussion of the dynamic plarming problem is beyond the scope of the chapter. Because of limited space I focus here instead on infolTIlation processing for a single time period; that is, I study

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how information should be gathered in making plans for the unitary organization.

Small Worlds and the Grand World Before constructing my model, some basic ideas on small worlds and the grand world are introduced. A small world is a confmed decision situation derived from the grand world. In the small world there exists a mapping from a set of states into consequences through actions. Savage did not defme explicitly and mathematically what states are, except for a verbal explanation that a state of the world is a description of the world leaving no relevant aspect undescribed (Savage, 1972, p. 9). I provide a broader definition of states here as follows:

Definition 1: States States are the realized outcomes of a set of independent random variables. For example, let X be a set of m independent random variables, X 1,X2,X3,

•••

,Xm , of which the values are real numbers, representing the

grand world. The vector

(Xl,X2,X3 , . . . , Xm)

is a state of the grand world.

Assume the number of possible values for each random variable is finite. That is, each random variable can be defined on a particular finite sample space. It follows that the number of the total states is also finite. A state of a small world is a subset of the elementary states in the grand world. For example, the outcomes resulting from the vector of independent random variables

(*

world where

,XJjX!+l, ... ,X!+h

* )are the states

of a particular small

* denotes the random variables that are irrelevant to the

current decision situation and can be ignored, although they do exist. The grand world can thus be represented by a Cartesian product across the set

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8. Information Structure Exploration as Planning for a Unitary Organization

of independent random variables under consideration, i.e.,Xjx X 2 x X3X. ..

Alternatively, the state in a small world can be any subset of the states in the grand world. Consider throwing a pair of dice. If the 36 outcomes represent all the states in the grand world, the events that the two dice show up even points simultaneously are the states in a particular small world, and are subsets of the states in the grand world. States in a small world are in some sense events in the grand world. The decision maker's degree of belief that a state is realized depends on tbe joint probability distribution across the independent random variables under consideration. Therefore, for each world, whether a small world or the grand world, there is a probability distribution characteristic of the states in the world. The derivation of such a joint probability distribution is beyond the present scope. An act is action committed to a decision and taken by the decision maker capable of implementing the action in order to yield expected consequences. Consequences are anything that could happen to the decision maker and that he or she is concerned with. More fOlmally,

Definition 2: Acts Acts are functions that map states into consequences. Strictly speaking, there is uncertainty in the mapping between acts and consequences because that mapping is contingent on situations imposed by nature. For simplicity, I assume that the functions are detelTIlinistic, eliminating stochastic factors of such mappings.

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According to Savage, a small world is an isolated decision situation in that only subsets of the elementary states in the grand world concern the decision maker, and that only a set of admissible acts are available for the decision maker resulting in various consequences for given states (Savage, 1972, p. 14). Uncertainty plays an important role in deciding on an act because no prior knowledge exists about which small world state would obtain. The best the decision maker can do is to select the act yielding the maximum expected utility among those in the available set. Measurements of probability distributions of all possible states and utilities of consequences are central tomaking such a decision. Savage provides a general axiomatic system proving that under strict conditions, such measurements are theoretically attainable and that the decision maker should act accordingly (Savage, 1972). He coined the term 'microcosm' to represent a small world in which such probability and utility measurements derived from the axiomatic system exist. In my exposition of pi arming behavior, I treat small worlds as microcosms so that subjective expected utility theory can be applied in my theorems. There is a close interplay between a small world and the grand world. More specifically, a small world consequence is an act in the grand world that triggers further consequences in the grand world. Therefore, an act in the small world will lead indirectly to the corresponding consequences in the grand world.

Extension ofthe Small World Concept Single Contingent Decisions Decisions can be made regardless of whether additional information has been acquired. Contingent decisions are decision rules that prescribe how decisions should be made in light of new infonnation. A plan consists of a

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8. Information Structure Exploration as Planning for a Unitary Organization

set of contingent decisions. Planning is equivalent to gathering infonnation to reduce uncertainty in making contingent decisions. In this section, we provide a theoretical basis prescribing how such a contingent decision should be made. Following Marschak (1974), assume, with slight modifications, tbe planner of the unitary organization perfonns two kinds of activities: 1) To make an observation (gather information) on the external world (grand world), and 2) To perform an action upon the external world based on the information gatbered.

The above activities can be refonnulated to explicitly take into account planning behavior and how small worlds are constructed. That is, in addition to the above two kinds of activities, the decision maker specifies a desired small world through changing tbe probability distribution of tbe states in tbat world. This planning under goal setting will be discussed following a simpler situation in which the plallller gathers more valuable information without anticipating changes in the probability distribution of the states. Though the following axiomatic construct and notations are closely based on Marschak and Radner's formulation (1978), I aimhere to interpret tbe nonnative infonnation gathering process in planning through more rigorous tenns in combination of Savage's work as depicted earlier. Most of the following equations, e.g., Equations (1) through (7) and theorems, e.g., Lemma 1, are derived directly from Marschak and Radner's fonnulation with slight modification to incorporate multiple, independent

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random variables to provide a theoretical context for my illustration of planning behavior. More specifically, the following argument follows exactly the argument of Marschak and Radner, but generalizes it to the case of multiple, independent random variables. Thus I have modified their equations by adding subscripts to account for multiple instances. The important point is to note that as long as the multiple variables have independent distributions, the argument of Marschak and Radner for a single variable still holds, e.g., Theorem 1. The following elaboration can therefore be considered as an extension of their original work. Letxk be a state in the grand world for a particular random variable Xi, for

k = 1,2, ... ,po Each player gathers infOlmation through an infOlmation gathering process, or infOlmation structure (Marschak and Radner, 1978),

'b, which yields

a set of signals

yJ,j

= 1,2, ... ,n, fOlming n partitions for

the outcomes of the random variable of the grand world. Each grand world state xk corresponds to a signal

yJ. The states in a small world are thus the

partitions in fonn of the signals. Following Marschak and Radner, the action performed by a player is detennined by a decision rule a-;, and the structure ('b,a establishes the l)

organizational form of the player's decision situation, which is equivalent to a small world according to my definition. Given the infonnation structure 'b and the true state of the grand world

yj

is determined by

yj

xk, the information signal

~ 7],(xk); the payoff function for the player can be

denoted as (1)

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8. Information Structure Exploration as Planning for a Unitary Organization

where a! is an action taken in light of the signal yJ and the decision rule al. Assume the probability density function on

xk

to be if,. The expected

payoff for the player considering all the possible grand world states becomes (2) The problem is then to determine the uncontrolled organizational fmm (7)" a,) such that U is maximized. Note that

0 ,

and if, are non-controllable

and that a! is a contingent decision in that the decision will be made based on the infonnation gathered 'b about the true state xk , i.e., lb[xkJ. Solving this problem for all random variables in one time period is equivalent to making plans tlirough gathering information (7),) that are contingent on tlie resulting signals and the true states of the grand world tliat evoke indirectly different decisions based on the decision rule a !. Plans consist of contingent decisions that can be evaluated based either on the states of the environment or on the signals or information. If the relation between the signals and the states, that is 1]1, is knO\vn, selection of decisions based on anyone of the two particular sets of arguments is sufficient as sho\Vll above. One way to explore the characteristics of the best decision rule a! is to fix 1]!, thus leaving only thea! to the individual's choice. The expected payoff yielded by the best decision function, given the information structure 1]!, is denoted by (3)

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Let l]i[xkJ = yj. The consequences of an action, a ) (Yj), are unknown to the decision maker because there is more than one state corresponding to the signal

yJ. In this situation, the best decision rule must be selected so as

to maximize the conditional expectation of the payoff, given that the true state xk is in yjbecause a signal is a subset of states. That is, to maximize the expected payoff for a given decision function (4) We can group the state xk according to the corresponding signals yJ =

I),[xkl and the expected payoff above can be rewritten as (5) Choosing a decision function a ) that maximizes U above is equivalent to choosing, for each signal

yJ, an action a ) (yJ) that maximizes the telTIl

L w,[xk, a(yJ)]