Risk seeking with diminishing marginal utility in a non-expected utility model

The present work takes place in the framework of a non-expected utility model under risk: the RDEU theory (Rank Dependent Expected Utility, first initiated by Quiggin under the denomination of Anticipated Utility), where the decision maker's behavior is characterized by two functionsu andf. Our first result gives a condition under which the functionu characterizes the decision maker's attitude towards wealth. Then, defining a decision maker as risk averter (respectively risk seeker) when he always prefers to any random variable its expected value (weak definition of risk aversion), the second result states that a decision maker who has an increasing marginal utility of wealth (a convex functionu) can be risk averse, if his functionf issufficiently below his functionu, hence if he is sufficientlypessimistic. Obviously, he can also be risk seeking with a diminishing marginal utility of wealth. This result is noteworthy because with a stronger definition of risk aversion/risk seeking, based on mean-preserving spreads, Chew, Karni, and Safra have shown that the only way to be risk averse (in their sense) in RDEU theory is to have, simultaneously, a concave functionu and a convex functionf.     

A theoretically-consistent empirical model of non-expected utility: An application to nuclear-waste transport

Few theoretically-consistent empirical models addressing the relationship between ambiguity, risk, and preferences for health and safety exist. To fill this gap, we propose a theoretical non-expected-utility model (NEUM) that is relatively easy to estimate using an interval-data model. The NEUM we develop hinges upon two sources of variability, one over risk and the other over ambiguity about the risk. Using data from a survey of Nevada residents concerning risks from nuclear-waste transport, we provide individual-specific welfare estimate for a risk increase. Our findings suggest that negative externalities from nuclear-waste transport perceived risks and ambiguity may be substantial. JEL Codes D81 · D62 · Q53 · Q51     

The arguments of utility: Preference reversals in expected utility of income models

There is a debate in the literature about the arguments of utility in expected utility theory. Some implicitly assume utility is defined on final wealth whereas others argue it may be defined on initial wealth and income separately. I argue that making income and wealth separate arguments of utility has important implications that may not be widely recognized. A framework is presented that allows the unified treatment of expected utility models and anomalies. I show that expected utility of income models can predict framing induced preference reversals, a willingness to pay-willingness to accept gap for lotteries, and choice-value preference reversals. The main contribution is a theorem. It is proved that for all utility functions where initial wealth and income enter separately, either there will be preference reversals or preferences can be represented by a utility function defined on final wealth alone.     

Mean utility preserving increases in risk for state dependent utility functions

This paper defines the concept of a mean utility preserving spread across states (MUPSAS) for state dependent utility functions and analyzes the behavioural impact of shifts in the probability distribution of wealth across states such that overall mean utility is preserved. The main result provides an alternative way of ranking state dependent utility functions according to their degree of risk aversion (thus extending Kami's theorem of comparative risk aversion) and establishes a link between increases in risk and risk aversion for state dependent preferences. In a portfolio problem where preferences and the rate of return of the risky venture are state dependent, we find sufficient conditions to determine the impact of a MUPSAS on the optimal share of the portfolio invested in the risky asset.

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Pareto utility

In searching for an appropriate utility function in the expected utility framework, we formulate four properties that we want the utility function to satisfy. We conduct a search for such a function, and we identify Pareto utility as a function satisfying all four desired properties. Pareto utility is a flexible yet simple and parsimonious two-parameter family. It exhibits decreasing absolute risk aversion and increasing but bounded relative risk aversion. It is applicable irrespective of the probability distribution relevant to the prospect to be evaluated. Pareto utility is therefore particularly suited for catastrophic risk analysis. A new and related class of generalized exponential (gexpo) utility functions is also studied. This class is particularly relevant in situations where absolute risk tolerance is thought to be concave rather than linear.     

Advances in multiattribute utility theory

Several advances in multiattribute expected utility theory have emerged recently. Much of the existing theory deals with independence axioms on whole attributes and the corresponding utility decompositions. This paper reviews three alternate approaches for obtaining representations of multiattribute utility functions: (1) multi-valent preference analysis, (2) approximation methods, and (3) indifference spanning analysis. Unlike some utility decompositions, these approaches require the assessment of only single-attribute functions which makes implementation relatively simple. Only multivalent preference analysis and indifference spanning analysis, however, provide axioms that can be empirically tested to justify a particular utility representation.This research was supported in part by the Office of Naval Research under Contract No. N00014-78-C-0638, Task No. NR-277-258.     

The utility of gambling

A tiny utility of gambling is appended to an expected utility model for a risk-averse individual. It is shown that the model can explain small payoff gambles, large prize lotteries, and patterns of risk-seeking in the experimental evidence that are puzzling from the viewpoint of standard theory. At the same time, the model maintains expected utility theory's ability to explain insurance purchase, portfolio diversification, and other risk-averting behavior. The tiny utility of gambling could equally well be appended to models of risky choice other than the expected utility model.     

Conditional expected utility

Let \({\mathcal {E}}\) be a class of events. Conditionally Expected Utility decision makers are decision makers whose conditional preferences \(\succsim _{E}\), \(E\in {\mathcal {E}}\), satisfy the axioms of Subjective Expected Utility (SEU) theory. We extend the notion of unconditional preference that is conditionally EU to unconditional preferences that are not necessarily SEU. We study a subclass of these preferences, namely those that satisfy dynamic consistency. We give a representation theorem, and show that these preferences are Invariant Bi-separable in the sense of Ghirardato et al. (Journal of Economic Theory 118:133–173, 2004). We also show that these preferences have only a trivial overlap with the class of Choquet Expected Utility preferences, but there are plenty of preferences of the \(\alpha \)-Maxmin Expected Utility type that satisfy our assumptions. We identify several concrete settings where our results could be applied. Finally, we consider the special case where the unconditional preference is itself SEU, and compare our results with those of Fishburn (Econometrica 41:1–25, 1973).     

Local utility functions

In Machina's approach to generalised expected utility theory, decision makers maximise a choice functional which is smooth but not linear in the probabilities. When evaluating small changes, the choice functional can be approximated by the expectation of a local utility function. This local utility function is not however invariant under large changes in risk. This paper gives a simple explicit formula which can be used to write down the local utility functions of some common decision rules.     

Ordinal independence in nonlinear utility theory

Individual behavior under uncertainty is characterized using a new axiom, ordinal independence, which is a weakened form of the von Neumann-Morgenstern independence axiom It states that if two distributions share a tail in common, then this tail can be modified without altering the individual's preference between these distributions. Preference is determined by the tail on which the distributions differ. This axiom implies an appealing and simple functional form for a numerical representation of preferences. It generalizes the form of anticipated utility, and it explains some well-known forms of behavior, such as the Friedman-Savage paradox, that anticipated utility cannot.     

Interval scalability of rank-dependent utility

Luce and Narens (Journal of Mathematical Psychology, 29:1–72, 1985) showed that rank-dependent utility (RDU) is the most general interval scale utility model for binary lotteries. It can be easily established that this result cannot be generalized to lotteries with more than two outcomes. This article suggests several additional conditions to ensure RDU as the only utility model with the desired property of interval scalability in the general case. The related axiomatizations of some special cases of RDU of independent interest (the quantile utility, expected utility, and Yaari’s dual expected utility) are also given.     

The value of information in anticipated utility theory

A well-known property of expected utility theory is that the value of information is nonnegative. Given the widespread dissatisfaction with the expected utility hypothesis, a natural question to ask is whether competing theories of choice preserve this property. This article considers one widely discussed alternative to expected utility, anticipated utility theory. We show that, like expected utility, the anticipated value of perfect information is always nonnegative. The value of imperfect information, however, may be negative, though the precise valuation of information depends upon whether the reduction of compound lotteries axiom is used to derive the anticipated utility functional.I am indebted to Edi Karni, Peter Wakker, and an anonymous referee for helpful comments on earlier versions of this article. They are of course in no way responsible for errors or obscurities in the present version.     

Stochastic expected utility theory

This paper proposes a new decision theory of how individuals make random errors when they compute the expected utility of risky lotteries. When distorted by errors, the expected utility of a lottery never exceeds (falls below) the utility of the highest (lowest) outcome. This assumption implies that errors are likely to overvalue (undervalue) lotteries with expected utility close to the utility of the lowest (highest) outcome. Proposed theory explains many stylized empirical facts such as the fourfold pattern of risk attitudes, common consequence effect (Allais paradox), common ratio effect and violations of betweenness. Theory fits the data from ten well-known experimental studies at least as well as cumulative prospect theory.

Approaches to cardinal utility

In a previous article (see [3]) a system of axioms is proposed stating conditions which are necessary and sufficient to determine a cardinal utility function on any set, finite or infinite, of outcomes X. The present paper discusses and interprets the meaning of those axioms, and compares this new approach to cardinal utility with the utility differences approach proposed by Alt and Frisch, among others, and with the expected utility approach of von-Neuman and Morgenstern. The notion of repetition of the same choice situation is presented and its interpretation discussed. It is then argued that this notion leads naturally to the system of axioms presented in On Cardinal Utility. It is also argued that this notion must be used if we want to have a more clear understanding of the meaning of the axioms proposed by Alt and Frisch. Finally, it is remarked that since uncertainty is not present in the new approach, it is free of the paradoxes that have plagued the expected utility hypothesis.     

Economic choice in generalized expected utility theory

Generalized expected utility models have enjoyed considerable success in explaining observed choices under uncertainty. However, there has been only limited progress in deriving comparative static results. This paper presents a general framework which permits the incorporation of a wide range of generalized expected utility models, but is sufficiently powerful to permit the derivation of comparative static results. The central idea is to represent preferences by the expected utility of a transformed probability distribution.     

A comparison of non-transferable utility values

Three values for non-transferable utility games -- the Harsanyi NTU-value, the Shapley NTU-value, and the Maschler--Owen consistent NTU-value -- are compared in a simple example.     

The structure of random utility models

I am grateful to Joseph B. Kadane for numerous constructive suggestions offered during discussions of this research. The financial sponsorship of the U.S. Department of Transportation through grant DOT-OS-4006 is also acknowledged. The opinions and conclusions expressed herein are solely those of the author.Assistant Professor of Economics, School of Urban and Public Affairs, Carnegie-Mellon University.     

The value of perfect information in nonlinear utility theory

Wakker (1988) has recently shown that, in contrast to an expected utility maximizer, the value of information will sometimes be negative for an agent who violates the independence axiom of expected utility theory. We demonstrate, however, that the value ofperfect information will always be nonnegative if the agent satisfies a weak dominance axiom. This result thus mitigates to some degree the normative objection to nonlinear utility theory implicit in Wakker's finding.