An Axiomatization of Cumulative Prospect Theory

This paper presents a method for axiomatizing a variety of models for decision making under uncertainty, including Expected Utility and Cumulative Prospect Theory. This method identifies, for each model, the situations that permit consistent inferences about the ordering of value differences. Examples of rankdependent and sign-dependent preference patterns are used to motivate the models and the "tradeoff consistency" axioms that characterize them. The major properties of the value function in Cumulative Prospect Theory--diminishing sensitivity and loss aversion--are contrasted with the principle of diminishing marginal utility that is commonly assumed in Expected Utility. At tempts to explain decisions under uncertainty that violate Expected Utility have genera ted several new ideas. (For a survey, see Fishburn 1988.) One notion is that the objects of choice are prospects, defined in terms of gains and losses relative to neutral reference point, ra ther than acts defined in terms of final asset positions. This notion, first proposed by Markowitz (1952), is the cornerstone of K a h n e m a n and Tversky's (1979) prospect theory. The significance of the reference point stems f rom the observations that people are generally risk averse for gains, risk seeking for losses, and that losses loom larger than gains. Ano the r empirical generalization that is incorporated into prospect theory is the tendency to overweight small probabilities and to underweight high probabilities. It is natural to model this phenomenon by introducing decision weights obtained f rom a nonaddit ive t ransformation of the probability scale. The p rob lem then arises how to calculate expectation with respect to a nonaddit ive measure. Several authors (Weymark, 1981; Quiggin, 1982; Schmeidler, 1989; Yaari, 1987) proposed a solution, which reduces to the Choquet functional (Choquet, 1953-4). This model transforms cumulative rather than individual probabilities; we call it the cumulative model. 1 *The research of the first author has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. This work was supported by Grant No. 89-0064 from the Air Force Office of Scientific Research to the second author. 148 PETER WAKKER/AMOS TVERSKY It has been recognized by several authors (Starmer and Sugden, 1989; Luce and Fishburn, 1991; Tversky and Kahneman, 1992) that the basic elements of prospect theory are compatible with the cumulative functional, and that it is possible to construct a model that combines these approaches to produce what we call Cumulative Prospect Theory. This paper presents an axiomatization of Cumulative Prospect Theory, which extends and generalizes the formal development in Tversky and Kahneman (1992). The present treatment differs from Luce and Fishburn (1991) in that it does not involve an operation of joint receipt, does not require higher-order gambles, and does not restrict the shape of the value function. The axiomatization presented in this paper is based on the tradeoff consistency approach developed by Wakker (1989) and Tversky, Sattath, and Slovic (1988). It provides a unified procedure for axiomatizing Expected Utility and generalized expectation models, and is illustrated in Section 1. The key idea is that different models are characterized by the particular situations in which the ordering of value differences can be inferred from preferences. For example, an inference about the ordering of value differences that is valid in Expected Utility is not valid in the cumulative model unless the prospects in question are comonotonic, i.e., induce the same (desirability) ordering of states of nature. Likewise, an inference about value differences in Cumulative Prospect Theory requires, in addition, that the critical outcomes will have the same sign, i.e., are all gains or all losses. We show that a natural characterization of Cumulative Prospect Theory is obtained by accepting only these inferences. An aim of this paper has been to bring to the fore the tradeoff consistency approach to axiomatic preference theory. The natural way in which characterizations of earlier theories are extended to Cumulative Prospect Theory, as well as the simplicity of additional specifying conditions (for diminishing marginal utility and loss aversion), provide evidence for the fruitfulness of this approach. The paper is organized as follows. Section I gives examples of preferences that violate Expected Utility and exhibit rank-dependence and sign-dependency. Section 2 presents notations and definitions. The next four sections characterize Expected Utility, Cumulative Utility, Sign-Dependent Expected Utility, and Cumulative Prospect Theory. To demonstrate the unity and generality of the tradeoff-consistency approach, we axiomatize all four models. The first two have already been presented in Wakker (1989); they are used in the characterization of the other two. A schematic summary of the results appears in Table 7.1. The table shows that CPT concerns two independent generalizations of expected utility, one by rank-dependence, the other by sign-dependence. Rank-dependence permits, for instance, pessimism so that events associated with less favorable outcomes are weighted more heavily. Signdependence permits different decision weights for events associated with gains than for events associated with losses. The elicitation of value difference orderings is not disturbed by rank-dependence whenever the prospects used in the elicitation are comonotonic. The elicitation of value difference orderings is not disturbed by sign-dependence whenever the prospects used in the elicitation are "cosigned", i.e. the critical outcomes all have the same sign; this will be further explained below. Extensions and generalizations of the results are discussed in Section 8. For instance, it is demonstrated that double matching could have been omitted in Theorem 2.c of Tversky and Kahneman (1992), and the extension to infinite state spaces is given. Section AN AXIOMATIZATION OF CUMULATIVE PROSPECT THEORY 149 9 investigates the characteristics of the value function, and provides a method for comparing the value functions of different decision makers in Cumulative Prospect Theory, analogous to the Arrow-Pratt results for Expected Utility. 2 Loss aversion is characterized ("losses loom larger than gains"). Proofs are presented in the Appendix.