Alain Chateauneuf

Some Characterizations of Lower Probabilities and Other Monotone Capacities through the Use of Mobius Inversion

Monotone capacities (on finite sets) of finite or infinite order (lower probabilities) are characterized by properties of their Mobius inverses. A necessary property of probabilities dominating a given capacity is demonstrated through the use of Gale’s theorem for the transshipment problem. This property is shown to be also sufficient if and only if the capacity is monotone of infinite order. A characterization of dominating probabilities specific to capacities of order 2 is also proved.

Choice under uncertainty with the best and worst in mind: Neo-additive capacities

We develop the simplest generalization of subjective expected utility that can accommodate both optimistic and pessimistic attitudes towards uncertainty-Choquet expected utility with non-extreme-outcome-additive (neo-additive) capacities. A neo-additive capacity can be expressed as the convex combination of a probability and a special capacity, we refer to as a Hurwicz capacity, that only distinguishes between whether an event is impossible, possible or certain. We show that neo-additive capacities can be readily applied in economic problems, and we provide an axiomatization in a framework of purely subjective uncertainty.

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.

On the use of capacities in modeling uncertainty aversion and risk aversion

Abstract Schmeidler (1989) and Yaari (1987) have proposed models where attitudes towards uncertainty (or risk) are characterized by not necessarily additive probabilities, and are therefore kept separate from attitudes towards wealth. The axiomatics of both models include a comonotonicity axiom the interpretation of which is delicate. Here, assuming as Yaari that the decision maker displays a constant marginal utility of wealth, we show that the comonotonicity axiom can be significantly weakened while being intuitively meaningful, and that it is dispensable when characterizing some forms of strong uncertainty (or risk) aversion. This investigation is performed within a simpler but possibly more intuitive framework than Schmeidlers and Yaaris.

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.

Modeling Attitudes Towards Uncertainty and Risk Through the Use of Choquet Integral

The aim of this paper is to present in a unified framework a survey of some results related to Choquet Expected Utility (CEU) models, a promising class of models introduced separately by Quiggin [35], Yaari [48] and Schmeidler [40, 41] which allow to separate attitudes towards uncertainty (or risk) from attitudes towards wealth, while respecting the first order stochastic dominance axiom.

Continuous representation of a preference relation on a connected topological space

Abstract Necessary and sufficient conditions are given for the existence of two continuous real valued functions u and v on a connected topological space X endowed with a preference relation ≺ (i.e., an asymmetric binary relation) such that y is preferred to x if and only if v(y)>u(x). It is shown that these conditions - a slight generalization of the usual ones encountered in classical utility theory - entail the existence of such a continuous representation u, v with u and v continuous utility functions for two complete preorders intimately connected with the preference relation ≺.

Comonotonicity axioms and rank-dependent expected utility theory for arbitrary consequences

Abstract This paper presents a new axiomatization of the rank-dependent expected utility (RDEU) model in the general framework of simple distributions over a connected compact metric space. The result is mainly achieved through two comonotonicity axioms: the comonotonic sure-thing principle (C.S.T.P.) and the comonotonic mixture independence axiom (C.M.I.A.) an adaptation of mixture independence to RDEU. In a first step a characterization of the rank-dependent utility (R.D.U.) model through the C.S.T.P. is given. The paper ends for RDEU theory with a characterization of attraction for certainty, in terms of dual subadditivity of the probability transformation.

From local to global additive representation

This paper studies continuous additive representations of transitive preferences on connected subdomains of product sets. Contrary to what has sometimes been thought, local additive representability does not imply global additive representability. It is shown that the result can nevertheless be established under some additional connectedness conditions. This generalizes previous results on additive representations on (subsets of) product sets.

Decomposable capacities, distorted probabilities and concave capacities

Abstract During the last few years, capacities have been used extensively to model attitudes towards uncertainty. We describe the links between some classes of capacities, namely between decomposable capacities introduced by Dubois and Prade and other capacities, such as concave or convex capacities, and distorted probabilities that appeared in two new models of non-additive expected utility theory (Schmeidler, Econometrica , 1989, 57, 571–587; Yaari, Econometrica , 1987, 55, 95–115). It is shown that the most well-known decomposable capacities prove to be distorted probabilities, and that any concave distortion of a probability is decomposable. The paper ends by successively characterizing decomposable capacities that are concave distortions of probabilities, and ⊥-decomposable capacities (for triangular conorms ⊥) that are concave, since decomposable capacities prove to be much more related to concavity than convexity.