Review of: Generalized expected utility theory: The rank-dependent model

Machina,-Mark-J. (Reviewer)

U of CA, San Diego

Journal-of-Economic-Literature; 32(3), September 1994, pages 1237-38.

The last decade or so has seen a profusion of new models of choice under uncertainty, all offered as alternatives to the classical "expected utility" model of von Neumann-Morgenstern, and all designed to explain--or at least to accommodate--a growing body of observed violations of the assumptions and predictions of the expected utility model. This volume presents a self-contained survey of these developments, with special emphasis on what is probably the most successful of these alternatives: the author's own "rank-dependent expected utility" model.

The rank-dependent model of preferences over lotteries consists of a very specific modification of the classical expected utility model, which drops the "linearity in the probabilities" feature of the classical model. Recall that the expected utility formula for evaluating a lottery consists of weighting the utility of each outcome by its associated probability, then summing over all such terms. Although the rank-dependent model is not the first attempt to allow probabilities to enter nonlinearly into preferences, it is the first successful attempt. An earlier proposal by Edwards in the 1950s (revived by Kahneman and Tversky in the 1970s) consists of separately transforming each probability by a nonlinear "decision weight" function, then multiplying the utility of each outcome by its transformed probability, and then summing. However, such models were found to violate a preference for first order stochastically dominating lotteries, which is the stochastic analogue of positive marginal utility of wealth. Quiggin's rank-dependent model solves this problem by adopting a decision weight function which still depends upon an outcome's probability, but also incorporates the "relative ranking" of the outcome, as measured by the likelihood of receiving any less-preferred outcome. Such a model may be termed "expected utility with `rank-dependent' transformed probabilities," or the more common abbreviation "rank-dependent expected utility."

The publication history of the rank-dependent expected utility model attests to its role as the most natural and useful modification of the classical expected utility formula. First developed in Quiggin's 1979 undergraduate thesis, it was published by him in the 1982 Journal of Economic Behavior and Organization. Unfortunately, this paper did not receive the attention it deserved, and the rank-dependent model was independently discovered and published at least three more times: in a special case by Yaari in 1987, by Allais in 1988, and in a context of social welfare measurement by Weymark in 1981. Since these publications, and the eventual recognition of Quiggin's original contribution, the model has received the attention of noted researchers such as Chew, Green, Karni, Roell, Safra, Segal, Tversky, and Wakker.

The present volume provides an ideal introduction and reference to both the rank-dependent model and the more general context in which it lies, namely the current theoretical and empirical challenge to the expected utility approach. The book is organized into four sections. The first prepares the reader by presenting a self-contained development of the expected utility model, including its history, intuition, axiomatic development, most important applications such as stochastic dominance and risk aversion, and the most general comparative statics properties of the model. This section concludes with a survey of the many types of systematic violations of the model, ranging from the well-known "Allais Paradox" and similar effects, through the "ambiguity effects" discovered by Ellsberg, the "Preference Reversal Phenomenon" uncovered by Slovic and Lichtenstein, and other issues involving simultaneous gambling and insurance.

The second section consists of a development and analysis of the rank-dependent model. It begins with the intuition behind the original construction of the model, and an intuitive discussion of various methods of viewing the model. Recognizing that any proposed alternative to expected utility must be expected to match its extensive theoretical power, the author shows how the rank-dependent model can be used to provide analyses of risk aversion and general comparative statics that are essentially as powerful as those provided by the classical model. Arguing that the role of such alternatives is also to provide answers to questions outside the scope of the classical model, the author provides a rank-dependent analysis of the theory of optimal lottery design, which goes quite beyond the original 1948 Friedman and Savage expected utility-based analysis of this issue.

The third section extends the analysis of the rank-dependent model in three directions. The first direction encompasses normative issues such as the model's implications concerning portfolio diversification, attitudes toward probabilistic randomization, dynamic consistency of choice, and attitudes toward the revelation of information. Each of these issues is important in the study of non-expected utility models, and each is effectively discussed by the author. The second direction concerns the experimental evidence: here the author shows how the rank dependent model is capable of accommodating the types of empirical phenomena listed in the paragraph before last. The third direction concerns the axiomatic construction of the rank-dependent model, and summarizes the work of both the author and others along these lines.

The final section examines the rank-dependent model from some more general points of view. One of these points of view is the analytical approach of "generalized expected utility analysis" I developed in 1982, in which the properties of a non-expected utility model are determined via an analysis of its "local utility functions" (i.e., local expected utility approximations). A second point of view consists of a general categorization of the rank-dependent and related models in terms of the nature of their respective relaxations of the expected utility "Independence Axiom." The final chapter in the volume offers some ideas toward extending the model to the topics of social welfare functions, intertemporal choice, intertemporal choice under uncertainty, and the psychological phenomenon of "framing."

This is an extremely well written book. The author simultaneously combines the tasks of intuitively presenting technical models and results with that of constantly relating these models and results to the "big picture"--that is, to the evolving literature on choice under uncertainty. This book will be found to be valuable by both novices to the literature and current experts.