Among the most popular models for decision under risk and uncertainty are the rank-dependent models, introduced by Quiggin and Schmeidler. Central concepts in these models are rank-dependence and comonotonicity. It has been suggested in the literature that these concepts are technical tools that have no intuitive or empirical content. This paper describes such contents. As

Under prospect theory, three components influence the risk attitude of a decision maker: the utility function, the probability weighting function, and loss aversion. Loss aversion reflects the observed behavior of decision makers' being more sensitive to losses than to gains, resulting in a utility function that is steeper for losses than for gains. Much of the empirically observed risk aversion is due to loss aversion. This paper proposes an index of loss aversion. It also demonstrates how the degree of loss aversion of two decision makers can be compared and how its influences on comparative risk aversion can be examined. The main result characterizes comparative loss aversion in terms of preferences.

Quality-adjusted life years (QALY) utility models are multiattribute utility models of survival duration and health quality. This paper formulates six classes of QALY utility models and axiomatizes these models under expected utility (EU) and rank-dependent utility (RDU) assumptions. The QALY models investigated in this paper include the standard linear QALY model, the power and exponential multiplicative models, and the general multiplicative model. Emphasis is placed on a preference assumption, the zero condition, that greatly simplifies the axiomatizations under EU and RDU assumptions. The RDU axiomatizations of QALY models are generally similar to their EU counterparts, but in some cases, they require modification because linearity in probability is no longer assumed, and rank dependence introduces asymmetries between the domains of better-than-death health states and worsethan-death health states. 1999 Academic Press This paper concerns the foundations of quality-adjusted life years (QALY) utility models. QALY utility models are widely used in the expected utility analysis of health decisions because they provide an outcome measure that integrates the duration and quality of survival. Before discussing the specifics of these models, it will be helpful to motivate the discussion by describing the role played by QALY utility models in health decision analysis (Weinstein et al., 1980; Sox, Blatt,

Peter P. Wakker has forcefully shown the importance for decision theory of a condition that he called "Cardinal Coordinate Independence". Indeed, when the outcome space is rich, he proved that, for continuous weak orders, this condition fully characterizes the Subjective Expected Utility model with a finite number of states. He has furthermore explored in depth how this condition can be weakened in order to arrive at characterizations of Choquet Expected Utility and Cumulative Prospect Theory. This note studies the consequences of this condition in the absence of any transitivity assumption. Complete preference relations satisfying Cardinal Coordinate Independence are shown to be already rather well-behaved. Under a suitable necessary order denseness assumption, they may always be represented using a simple numerical model.

During the last 25 years, prospect theory and its successor, cumulative prospect theory, replaced expected utility as the dominant descriptive theories of risky decision making. Although these models account for the original Allais paradoxes, 11 new paradoxes show where prospect theories lead to self-contradiction or systematic false predictions. The new findings are consistent with and, in several cases, were predicted in advance by simple “configural weight ” models in which probability-consequence branches are weighted by a function that depends on branch probability and ranks of consequences on discrete branches. Although they have some similarities to later models called “rank-dependent utility, ” configural weight models do not satisfy coalescing, the assumption that branches leading to the same consequence can be combined by adding their probabilities. Nor do they satisfy cancellation, the “independence ” assumption that branches common to both alternatives can be removed. The transfer of attention exchange model, with parameters estimated from previous data, correctly predicts results with all 11 new paradoxes. Apparently, people do not frame choices as prospects but, instead, as trees with branches.

If an agent (weakly) prefers early resolution of uncertainty then the recursive forms of both the most commonly used non-expected utility models, betweenness and rank dependence, almost reduce to Kreps & Porteus's (1978) recursive expected utility. Simon Grant Department of Economics, Faculties Australian National University Atsushi Kajii Institute of Policy and Planning Sciences University of Tsukuba Ben Polak Economics Department Yale University We thank Eddie Dekel and Boaz Moselle. 1 Introduction Kreps & Porteus (1978) recursive expected utility model allows an agent to care about the timing of the resolution of uncertainty. For example, an anxious agent may prefer early resolution while a hopeful agent may prefer late. Recursive expected utility achieves this #exibility by relaxing the reduction of compound lottery axiom for temporal lotteries. The model remains tractable thanks to recursivity: preferences today are built up from preferences tomorrow that do not themselves dep...

. This paper proposes a decomposition of nonadditive decision weights into a component reflecting risk attitude and a component depending on belief. The decomposition is based solely on observable preference and does not invoke other empirical primitives such as statements of judged probabilities. The characterizing preference condition (less sensitivity towards uncertainty than towards risk) deviates somewhat from the often-studied ambiguity aversion but is confirmed in the empirical data. The decomposition only invokes one-nonzero-outcome prospects and is valid under all theories with a nonlinear weighting of uncertainty. * This paper started as a joint project with Amos Tversky. Due to his untimely death, I had to complete it on my own and am alone responsible for any errors. In agreement with the judgment of people close to Amos (Kahneman 2000, B. Tversky 2000), I have become the sole author of this paper. Four referees made helpful comments. Special thanks are due to Craig Fox fo...

The dominance principle states that one should prefer the option with consequences that are at least as good as those of other options for any state of the world. When applied to judged prices of gambles, the dominance principle requires that increasing one or more outcomes of a gamble should increase the judged price of the gamble, with everything else held constant. Previous research has uncovered systematic violations of the dominance principle: people assign higher prices to a gamble with a large probability of winning an amount, Y, otherwise zero, than they do to a superior gamble with the same chance of winning Y, otherwise winning a small amount, X! These violations can be explained by a configural-weight theory in which two-outcome gambles are represented with two sets of decision weights; one set for outcomes having values of zero and another set for lower-valued outcomes that have nonzero values. The present paper investigates whether dominance violations are limited to two-outcome gambles. Results show that people violate the dominance principle with three-outcome gambles even with financial incentives. Furthermore, results could be predicted from the configural-weight theory. The data do not support the view that configural weighting is caused by a shift in strategy that would apply only to two-outcome gambles. KEY WORDS dominance principle; two-outcome gambles; three-outcome gambles; configural-weight theory

This paper introduces a new preference condition that can be used to justify (or criticize) expected utility. It is based on a method for deriving "comparisons of tradeoffs" from ordinal preferences. Our condition simplifies earlier conditions thus making them more accessible to nonspecialists, and at the same time provides more general and powerful tools to specialists. It is more closely related to empirical methods for measuring utility than conditions published before. The condition thus provides a unifying tool for quantitatively measuring, qualitatively testing, and normatively justifying expected utility. Although, in a formal sense, our method reveals risky utility, we hope that it can nevertheless appeal to a concept of cardinal utility prior t...

making under risk and uncertainty can you think of? Readers of this article will no doubt be familiar with Expected