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SUR DES MODELES NON-ADDITIFS EN THEORIE DES CHOIX INTERTEMPORELS ET DE LA DECISION DANS L'INCERTAIN

Abstract : Savage's (1954) model constitutes a main achievement in decision theory under uncertainty. It provides an axiomatization of subjective expected utility. A decision maker who fulfills the Savage's axioms chooses between acts according to their expected utility. Following this axiomatic method, additive representation can be obtained in different settings (Anscombe-Aumann (1963), Wakker (1990)). Despite the normative character of subjective expected utility models, empirical refutations arise quickly, for instance Ellsberg's paradox (1961). Choquet expected utility models can provide a response. Henceforth a decision maker does not have a subjective probability anymore but a subjective capacity (Choquet (1953)), a monotonic set function which is not necessarily additive. An integral theory with respect to capacities introduced by Choquet (1953), rediscovered and developed by Schmeidler (1986,1989) allows a generalization of the expected utility criterion. The axiomatic of Choquet expected utility models elaborated in an uncertainty framework can also be adapted in a temporal one. Hence the evaluation of a stream of incomes can be made in a non-additive way and embody variations between different successives periods (Gilboa (1989), De Waegenaere and Wakker (2001)). The first chapter deals with the integral representation of comonotonic additive and sequentially continuous from below or from above functionals. This representation through Choquet's (1953) integrals is based on sequential continuity, a natural condition in measure theory, and not on monotonicity as in Schmeidler (1986). Consequently games we consider are not necessarily monotonic but continuous from below or from above, properties which are equivalent to sigma-additivity for additive games. Finally, we provide some representation theorems for non-monotonic preferences but sequentially continuous from above or from below. The second chapter provides an axiomatization of some preferences in a temporal setting, which originates in Gilboa (1989) and carried on in Shalev (1997) in an Anscombe-Aumann's (1963) setting. We adopt here De Waegaenere and Wakker's (2001) method. Our aim is to take into account complementarities between different successive periods. For this we introduce a variation aversion axiom, that keeps additivity on income streams having the property of sequential comonotony. The extension to the infinite case is achieved through a behavioral axiom, myopia. Finally we present a generalization to the non-additive case of the discounted expected utility, axiomatized in Koopmans (1972). In the third chapter, we establish a Yosida-Hewitt(1952) decomposition theorem for totally monotone games on N, where any game is the sum of a sigma-continuous game and a pure game. This composition is obtained from an integral representation theorem on the set of belief functions, hence the Choquet (1953) integral of any bounded function, with respect to a totally monotone game admits an integral representation. Finally to every totally monotone sigma-continuous game is associated a unique M¨obius inverse on N; hence any Choquet integral of a bounded function on N with respect to a totally monotone sigma-continuous game obtains as the sum of an absolutely convergent series. The last chapter, deals with modelization of patience for countable streams of incomes. At first, we consider preferences that exhibits patience in the additive case. These preferences admit an integral representation with respect to pure probabilities, which coincide with Banach limits (Banach 1987). Then, we strenghten patience into time invariance. Lastly we consider naive patience, which leads to an impossibility theorem. Consequently, we give an extension of the preceeding results in a non-additive framework. We introduce a non-smooth additivity axiom which allows to represent preferences through a Choquet integral with convex capacity. In this case, patience translates into pure convex capacities. Likewise, time invariance expresses naturally in term of invariant convex capacities. Finally, naive patience admits for unique representation the inferior limit functional.
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Yann Rébillé. SUR DES MODELES NON-ADDITIFS EN THEORIE DES CHOIX INTERTEMPORELS ET DE LA DECISION DANS L'INCERTAIN. Economies et finances. Université Panthéon-Sorbonne - Paris I, 2002. Français. ⟨tel-00267890⟩

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