Michèle Cohen

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 makers behavior is characterized by two functionsu andf. Our first result gives a condition under which the functionu characterizes the decision makers 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 is“sufficiently below” his functionu, hence if he is sufficiently“pessimistic.” 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.

Security level, potential level, expected utility: A three-criteria decision model under risk

Among the violations of expected utility (E.U.) theory which have been observed by experimenters, the violations of its independence axiom is, by far, the most common. It seems that, in many cases, these inconsistencies can be ascribed to the desire for security - called the security factor by L. Lopes (1986) - which “makes people attach special importance to the worst outcomes of risky decisions” as well as to the sole outcomes of riskless decisions (certainty effect). J.-Y. Jaffray (1988) has proposed a model which generalizes E.U. theory by taking into account this factor and is then able to account for certain violations. However, especially in experiments on choice involving prospective losses, violations of the von Neumann-Morgenstern independence axiom cannot be explained by the security factor alone and have to be partially ascribed to the potential factor (L. Lopes, 1986) which “reflects heightened attention to the best outcomes of decisions”, especially when the best outcome is the status quo. In this paper, we construct an axiomatic model for subjects taking into account simultaneously or alternatively the security factor and the potential factor. For this, as in Jaffrays model, it has been necessary to weaken not only the standard independence axiom but also the continuity axiom and, in the same time, to reinforce the dominance axiom. In the resulting model, choices are partially determined by the mere comparison of the (security level, potential level) (i.e. the (worst outcome, best outcome)) pairs offered, and completed by the maximization of an affine function of the expected utility, the coefficients of which depend on both the security level and potential level.In this model, a decision maker who (i) has constant marginal utility for money, (ii) is sensitive to the security factor alone in the domain of gains, (iii) is sensitive to the potential factor alone in the domain of losses, behaves as a risk averter for gains and a risk seeker for losses.

Four notions of mean preserving increase in risk, risk attitudes and applications to the Rank-Dependent Expected Utility model

This article presents various notions of risk generated by the intuitively appealing single-crossing operations between distribution functions. These stochastic orders, Bickel & Lehmann dispersion or (its equal-mean version) Quiggins monotone mean-preserving increase in risk and Jewitts location-independent risk, have proved to be useful in the study of Pareto allocations, ordering of insurance premia and other applications in the Expected Utility setup. These notions of risk are also relevant tothe Quiggin-Yaari Rank-dependent Expected Utility (RDEU) model of choice among lotteries. Risk aversion is modeled in the vNM Expected Utility model by Rothschild & Stiglitzs Mean Preserving Increase in Risk (MPIR). Realizing that in the broader rank-dependent set-up this order is too weak to classify choice, Quiggin developed the stronger monotone MPIR for this purpose. This paper reviews four notions of mean-preserving increase in risk - MPIR, monotoneMPIR and two versions of location-independent risk (renamed here left and right monotone MPIR) - and shows which choice questions are consistently modeled by each of these four orders.

Risk-Aversion Concepts in Expected- and Non-Expected-Utility Models

The non-expected-utility theories of decision under risk have favored the appearance of new notions of increasing risk like monotone increasing risk (based on the notion of comonotonic random variables) or new notions of risk aversion like aversion to monotone increasing risk, in better agreement with these new theories. After a survey of all the possible notions of increasing risk and of risk aversion and their intrinsic definitions, we show that contrary to expected-utility theory where all the notions of risk aversion have the same characterization ( u concave), in the framework of rank-dependent expected utility (one of the most well known of the non-expectedutility models), the characterizations of all these notions of risk aversion are different. Moreover, we show that, even in the expected-utility framework, the new notion of monotone increasing risk can give better answers to some problems of comparative statics such as in portfolio choice or in partial insurance. This new notion also can suggest more intuitive approaches to inequalities measurement.

An experimental study of updating ambiguous beliefs

‘Ambiguous beliefs’ are beliefs which are inconsistent with a unique, additive prior. The problem of their update in face of new information has been dealt with in the theoretical literature, and received several contradictory answers. In particular, the ‘maximum likelihood update’ and the ‘full Bayesian update’ have been axiomatized. This experimental study attempts to test the descriptive validity of these two theories by using the Ellsberg experiment framework.

Approximations of rational criteria under complete ignorance and the independence axiom

Rational decision making under complete ignorance, a limit case of uncertainty, is defined. Through a concept of approximation, a meaning is given to a criterion almost possessing a property. Rational criteria depend almost on the sole bounds of the outcome range of each decision, they are almost continuous, they can almost possess a transitive indifference relation; however, under Savages Independence Axiom, this last property restricts possible criteria to those which can be approximated, at least partly, by either the Maximin or the Maximax criterion.

Decision making in a case of mixed uncertainty: A normative model

Abstract We consider a case of uncertainty which is frequently met in various fields, e.g., in parametric statistics: Events {θ}, θ ∈ ∵, are members of family E on which the decision maker possesses no information at all; however, conditionally on the realization of {θ}, he is able to affix probabilities to all members of another family of events, F . We assume that the decision maker: (1) has a rational behavior under complete ignorance, for decisions whose results only depend on events of E ; (2) with {θ} known, maximizes his conditional expected utility for decisions whose results only depend on events of F ; (3) has (unconditional) preferences which are consistent with his conditional ones. These assumptions are shown to be sufficient to ensure an approximate representation of the decision makers preference by a real-valued function W which has the form W(f) = v[ Inf θ∈∵ E θ (u∘f), Sup θ∈∵ E θ (u∘f)] , where u and v, respectively, characterize the decision makers attitudes toward risk and toward complete ignorance.

Preponderence of the Certainty Effect Over Probability Distortion in Decision Making Under Risk

The appeal of expected utility (EU) theory — the standard model of decision analysis — as a normative model is not necessatily lessened by its poor quality as a descriptive model. As a matter of fact, observations of systematic violations of EU in Allais [1], Kahneman and Tversky [7], Mac Crimmon and Larsson [15], Kunreuther [10], Schoemaker [17] and elsewhere have had no apparent effect on its popularity.

Comonotonicity, Rank-Dependent Utilities and a Search Problem

Under the expected utility (EU) model, a decision maker (DM) is characterised by a utility function u, often assumed to be continuous and generally assumed to be non-decreasing. In this model, all the possible notions of risk aversion are merged and characterised by concavity of this utility function. Under the rank-dependent expected utility (RDEU) model (see [4, 12]), a DM is characterised by such a utility function (that plays the role of utility on certainty) in conjunction with a probability-perception function \(f: [0, 1] \to [0, 1]\) [0,1] that is non-decreasing and satisfies f(0) = 0, f(1) = 1. Such a DM prefers the random variable X to the random variable Y if and only if V(X) > V(Y), where the RDEU V (see [12, 16]) is given by