Peter P. Wakker

An index of loss aversion

To a considerable extent, risk aversion as it is commonly observed is caused by loss aversion. Several indexes of loss aversion have been proposed in the literature. The one proposed in this paper leads to a clear decomposition of risk attitude into three distinct components: basic utility, probability weighting, and loss aversion. The index is independent of the unit of payment. The main theorem shows how the indexes of different decision makers can be compared through observed choices.

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.

Separating Marginal Utility and Probabilistic Risk Aversion

This paper is motivated by the search for one cardinal utility for decisions under risk, welfare evaluations, and other contexts. This cardinal utility should have meaningprior to risk, with risk depending on cardinal utility, not the other way around. The rank-dependent utility model can reconcile such a view on utility with the position that risk attitude consists of more than marginal utility, by providing a separate risk component: a ‘probabilistic risk attitude’ towards probability mixtures of lotteries, modeled through a transformation for cumulative probabilities. While this separation of risk attitude into two independent components is the characteristic feature of rank-dependent utility, it had not yet been axiomatized. Doing that is the purpose of this paper. Therefore, in the second part, the paper extends Yaaris axiomatization to nonlinear utility, and provides separate axiomatizations for increasing/decreasing marginal utility and for optimistic/pessimistic probability transformations. This is generalized to interpersonal comparability. It is also shown that two elementary and often-discussed properties — quasi-convexity (‘aversion’) of preferences with respect to probability mixtures, and convexity (‘pessimism’) of the probability transformation — are equivalent.

A Simple Axiomatization of Nonadditive Expected Utility

This paper provides an extension of L. J. Savages subjective expected utility theory for decisions under uncertainty. It includes in the set of events both unambiguous events for which probabilities are additive and ambiguous events for which probabilities are permitted to be nonadditive. The main axiom is cumulative dominance, which adapts stochastic dominance to decision-making under uncertainty. The authors derive a Choquet expected utility representation and show that a modification of cumulative dominance leads to the classical expected utility representation. The relationship of their approach with that of D. Schmeidler, who uses a two-stage formulation to derive Choquet expected utility, is also explored. Copyright 1992 by The Econometric Society.

The utility of health states after stroke : A systematic review of the literature

Background— To perform decision analyses that include stroke as one of the possible health states, the utilities of stroke states must be determined. We reviewed the literature to obtain estimates of the utility of stroke and explored the impact of the study population and the elicitation method. Summary of Review— We searched various databases for articles reporting empirical assessment of utilities. Mean utilities of major stroke (Rankin Scale 4 to 5) and minor stroke (Rankin Scale 2 to 3) were calculated, stratified by study population and elicitation method. Additionally, the modified Rankin Scale was mapped onto the EuroQol classification system. Utilities were obtained from 23 articles. Patients at risk for stroke assigned utilities of 0.26 and 0.55 to major and minor stroke, respectively. Stroke survivors assigned higher utilities to both major (0.41) and minor stroke (0.72). The EuroQol completed by stroke survivors revealed a utility of 0.32 and 0.71 for major and minor stroke, respectively. Utilities elicited by the Standard Gamble were generally higher, while those obtained by the Visual Analogue Scale were lower than the Time Trade Off values. Remaining variation between utilities may be caused by differences in definitions of the health states. The mapped EuroQol indicated a utility of 0.64 for minor stroke and a value just below zero for major stroke. Conclusions— For minor stroke, a utility between 0.50 and 0.70 seems to be reasonable for both decision analyses and cost-effectiveness studies. The utility of major stroke may range between 0 and 0.30 and may possibly be negative.

Explaining the characteristics of the power (CRRA) utility family

The power family, also known as the family of constant relative risk aversion (CRRA), is the most widely used parametric family for fitting utility functions to data. Its characteristics have, however, been little understood, and have led to numerous misunderstandings. This paper explains these characteristics in a manner accessible to a wide audience.

Characterizing QALYs by Risk Neutrality

This paper shows that QALYs can be derived from more elementary conditions than thought hitherto in the literature: it suffices to impose risk neutrality for life years in every health state. This derivation of QALYs is appealing because it does not require knowledge of concepts from utility theory such as utility independence. Therefore our axiomatization greatly facilitates the assessment of the normative (non)validity of QALYs in medical decision making. Moreover, risk neutrality can easily be tested in experimental designs, which makes it straightforward to assess the descriptive (non)validity of QALYs.

Comonotonic Independence: The Critical Test between Classical and Rank-Dependent Utility Theories

This article compares classical expected utility (EU) with the more general rank-dependent utility (RDU) models. The difference between the independence condition for preferences of EU and its comonotonic generalization in RDU provides the exact demarcation between EU and rank-dependent models. Other axiomatic differences are not essential. An experimental design is described that tests this difference between independence and comonotonic independence in its most basic form and is robust against violations of other assumptions that may confound the results, in particular the reduction principle and transitivity. It is well known that in the classical counterexamples to EU, comonotonic independence performs better than full-force independence. For our more general choice pairs, however, we find that comonotonic independence does not perform better. This is contrary to our prior expectation and suggests that rank-dependent models, in full generality, do not provide a descriptive improvement over EU. For rank-dependent models to have a future, submodels and choice situations need to be identified for which rank-dependence does contribute descriptively.

CONTINUOUS SUBJECTIVE EXPECTED UTILITY WITH NON-ADDITIVE PROBABILITIES*

A well-known theorem of Debreu about additive representations of preferences is applied in a non-additive context, to characterize continuous subjective expected utility maximization for the case where the probability measures may be non-additive. The approach of this paper does not need the assumption that lotteries with known (objective) probability distributions over consequences are available.

Original and cumulative prospect theory: a discussion of empirical differences

This note discusses differences between prospect theory and cumulative prospect theory. It shows that cumulative prospect theory is not merely a formal correction of some theoretical problems in prospect theory, but it also gives different predictions. Experiments are described that favor cumulative prospect theory.