GENERALIZED EXPECTED UTILITY THEORY THE RANK-DEPENDENT MODEL

Transcription

1 GENERALIZED EXPECTED UTILITY THEORY THE RANK-DEPENDENT MODEL

2 GENERALIZED EXPECTED UTILITY THEORY THE RANK-DEPENDENT MODEL John Quiggin Australian National University... " SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

3 Library of Congrcss Cataloging-In-Publlcatlon Data Quiggin. John. Gcneralized expeeted utility theory the rank-depcndent modei I John Quiggin. p. em. Include. bibiiographieal referenees and index. ISBN S ISBN (ebook) DOI / Utility theory-.mathematicai model.. 1. Title. HB20l.Q '03 dc CIP Copyright 1993 by Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1993 Softcover reprint of the hardcover 1 st edition 1993 AII rights reserved. No part ofthis publieation may be reproduced. stored in a retricval systemor transmitted in any farm orby any means, mechanical. photo copying. record ing. or olhcrwise, withoul the prior written permission of the publisher, Springer-Science+Business Media, B.V.. Printed on acid-free pa per.

4

5 vii CONTENTS PART 1 THE EXPECTED UTILITY MODEL CHAPTER 1 Uncertainty The Nature of Uncertainty Descriptive and Normative Models 1.3 Scientific Research Programs 4 6 CHAPTER 2 Background EU Theory.. History EU Theory - An Informal Discussion Notation.. The Discrete Case 2.4 Notation.. The General Case Preferences 15 CHAPTER 3 EU Theory The EU Functional EU Axioms Risk Aversion in EU Theory Stochastic Dominance Risk Preference in EU Theory Applications of EU Theory The General Control Problem 28 Concluding Comments 31 Appendix: 33 CHAPTER 4 The Challenge to EU Theory The Allais Prohlem The Common Ratio Effect Ambiguity Preference Reversal 4.5 Gambling Insurance Construction of Utility Functions Probability Weighting Prospect Theory and Subjectively Weighted Utility 50

6 viii PART 2 THE RANK-DEPENDENT MODEL CHAPTER 5 Rank-Dependent Expected Utility - An Outline The Development of RDEU Theory The RDEU Model The Transfonnation Function RDEU - Probability Weighting Interpretation The Dual Approach The Allais Model RDEU and Ambiguity RDEU as a Natural Extension 72 Concluding Comment. 74 CHAPTER 6 Risk Aversion in RDEU Theory The Correspondence Principle Pessimism and RDEU Risk Aversion First and Second Order Risk Aversion Risk Premiums and Coefficients of Risk Aversion An Alternative Definition of Increasing Risk The Rothschild-Stiglitz Theory of Increasing Risk Risk Seeking Behavior 88 Concluding Comments 89 CHAPTER 7 Comparative Statics for RDEU Theory Comparative Static Problems Extension Methods The Extension to RDEU Risk Aversion, Risk Seeking and Comparative Statics Monotone Spreads and Comparative Risk Aversion Comparative Risk Aversion in RDEU 99 Concluding Comments 100 CHAPTER 8 Risk Seeking and Lottery Design Lotteries with a Single Prize The General Case Evidence from observed lottery designs III 8.4 Racetrack Betting 112 Concluding Comments 113

7 Lx PART 3 FURTHER PROPERTIES OF THE MODEL CHAPTER 9 Some Normative Properties of RDEU Diversification Quasiconcavity, Quasiconvexity and Betweenness Dynamic Consistency RDEU and Information 125 Concluding Comments 126 CHAPTER 10 RDEU and Experimental Evidence The Experimental Approach The Allais Problem The Common Ratio Effect Ambiguity and Reduction ofcompound Lotteries Preference Reversal The Recent Empirical Evidence on RDEU Estimating a Functional Form 140 CHAPTER 11 Axiomatic Approaches to RDEU 145 Il.l The Probability Weighting Approach The Dual Approach Ordinal Independence and Measure Representation Trade-OffConsistency The Reduction ofcompound Lotteries Axiom Ordinally Independent Generalhations 155 1l.7 Cumulative Prospect Theory The Space ofoutcomes 158 Appendix - Notes on the Original Axiomatization of ROEU 159 PART 4 GENERALIZED EXPECTED UTILITY THEORY CHAPTER 12 Generalized Smooth Utility and RDEU The Model Comparative Statics RDEU and Generalized Smooth Preferences Risk Seeking 169

8 x CHAPTER 13 Stochastic Dominance and Independence Rules Notation Dominance Rules Independence Independence in RDEU Betweenness Rules 180 Concluding Comments 182 CHAPTER 14 Extensions Social Welfare Functions Constitutional Choice Choice over Time Time and Uncertainty Framing and Coding 191 References Index of Names Topic Index

9 Preface xi Since the publication of von Neumann and Morgenstern's classic Theory of Games and Economic Behavior in 1944, economic analysis of choice under uncertainty has been dominated by the expected utility (EU) model. The EU model has been one of the great success stories of modern economic analysis. A wide range of phenomena that previously lay outside the domain of formal economic analysis have been successful modelled. The EU framework has been a fruitful source of new concepts and methods of analysis. Most importantly, the EU model permits the application of standard methods of comparative statics and dynamics to the analysis ofchoice under uncertainty. Yet the EU model has never been without its critics. Psychologists such as Edwards and EUsberg accumulated evidence that individual choices under uncertainty were inconsistent with the predictions of the EU model. Applied work in areas such as finance was dominated by the simpler, and apparently more tractable, mean-variance analysis populari7~d by Markowitz. The most fundamental criticisms were made in the early 1950s by A1lais, who focused attention on the EU independence axiom. Allais proposed a striking counterexample, the so-called 'Allais paradox', which induced even staunch advocates of EU, such as Savage, to make choices inconsistent with the independence axiom. Despite this apparent refutation of EU predictions, AUais' criticisms were not taken very seriously. A revival of interest in these issues among economists began in the late 1970s with the publication of a number of papers presenting models based on the idea of probability weighting. The most influential was the prospect theory of Kahneman and Tversky, which permitted the analysis of behavior inconsistent with EU theory, at the price of introducing violations of dominance. Other models presented around the same time by Handa and Karmarkar also involved violations of dominance, leading to some scepticism as to the possibility of developing any normatively appealing alternative to EU. In the 1980s, this scepticism was dispelled. A numher of generalizations of EU were proposed, most of which were capahle of explaining the Allais 'paradox' and other evidence inconsistent with EU, while preserving transitivity and dominance. Along with numerous specific models came the highly influential work of Machina who showed that preference functionals inconsistent with the independent axioms could be modelled in terms of local utility functions and that properties of the local utility functions such as monotonicity and risk-aversion carried over to the global preference functional. Generalized expected utility analysis is now a flourishing sub-field of economics, with a large and active research community and the usual appurtenances, such as a biennial conference (the Foundations of Utility Research conference) and a specialist journal (the Journal ofrisk and Uncertainty). There are now dozens ofcompeting models and a considerable literature exploring thcir theoretical properties and comparing their empirical performance. Outside this community, however, the new ideas have had only a limited impact. The EU model remains the principal tool for the analysis of choice under uncertainty. There is a widespread view that generalized models are too difficult to handle or incapable ofgenerating sharp results. There is, then, a need to show that the new models can be used in the kinds of economic analysis for which EU has proved so powerful, and that they can yield new and interesting results. This book is an attempt to meet this need by describing one of the most popular of the generalized models - the rank-dependent expected utility model (RDEU), also known as anticipated utility (AU), expected utility with rank-dependent preferences (EURDP), the

10 xii Jl-e model, the dual theory of choice under uncertainty and simply as rank-dependent utility (ROU). As the profusion of names indicates, the model has been approached in many different ways by many different authors. For this reason, consideration of a single model throws light on many of the concerns that have motivated the development of generalized expect utility theories. The popularity of the ROEU model rests on its simplicity and tractability. The model has a simple symmetrical representation in which outcomes are transformed by a von Neumann-Morgenstern utility function and (cumulative) probabilities are transformed by a weighting function. This makes it easy to derive specific functional forms. The fact that the utility function is essentiauy the sameas that ofeu theory means that ideas developed in the EU context, such as coefficients ofabsolute and relative riskaversion, have a direct interpretation in the RDEU framework. The standard tools of analysis developed for EU theory may be applied, with only minor modifications, to the ROEU model. However, because ROEU admits behavior inconsistent with EU, such as simultaneous gambling and insurance, the field ofpotential applications is widened. The ROEU model is not so much a competitor to EU as an extension based on less restrictive assumptions. In order to reach as wide an audience as possible, I have attempted to present the key ideas using only simple mathematical tools, at the expense ofrigor and elegance ifnecessary. In particular, most ofthe main results can be understood purely in terms ofdiscrete probability distributions. This book draws on the work of a great many of my colleagues, too many to name individually. However, the inspiration provided by the work of Maurice Allais cannot go unmentioned. Mark Machina gave me the encouragement to persevere at a time when my efforts in this field seemed entirely fruitless. Zachary Rolnik encouraged me to write this book, and has been a supportive editor. Jock Anderson, Uzi Segal and Peter Wakker read earlierdrafts ofthe book, and forced me to undertake the major revisions that were necessary. Without their helpful and critical comments, this would have been a far less satisfactory work. Shirley Halton read the proofs and detected numerous errors. For all remaining errors, logical and typographical, I am entirely responsible. This book was produced and set in camera-ready form, using an Apple Macintosh IIci with Nisus word processing software and Aldus Freehand graphics.