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Stochastic dominance with nonadditive probabilities. (English) Zbl 0780.90009

Summary: Choquet expected utility which uses capacities (i.e. nonadditive probability measures) in place of \(\sigma\)-additive probability measures has been introduced to decision making under uncertainty to cope with observed effects of ambiguity aversion like the Ellsberg paradox. We present necessary and sufficient conditions of stochastic dominance between capacities (i.e. the expected utility with respect to one capacity exceeds that with respect to the other one for a given class of utility functions). One wide class of conditions refers to probability inequalities on certain families of sets. To yield another general class of conditions we present sufficient conditions for the existence of a probability measure \(P\) with \(\int f dC=\int f dP\) for all increasing functions \(f\) when \(C\) is a given capacity. Examples include \(n\)-th degree stochastic dominance on the reals and many cases of so-called set dominance. Finally, applications to decision making are given including anticipated utility with unknown distortion function.

MSC:

91B16 Utility theory
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