ZOR - Methods and Models of Operations Research (1993) 37:231-256 V~. I ] ~4 - v Stochastic Dominance with Nonadditive Probabilities RAINER DYCKERHOFF AND KARL MOSLER Universit~it der Bundeswehr Hamburg, Holstenhofweg 85, 2000 Hamburg 70, Germany Abstract: Choquet expected utility which uses capacities (i.e. nonadditive probability measures) in place of a-additive probability measures has been introduced to decision making under uncertainty to cope with observed effects of ambiguity aversion like the Ellsberg paradox. In this paper we present necessary and sufficient conditions for 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 Sf dC = ~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. Key words and Phrases: expected utility, Choquet integral, anticipated utility, rank dependent ex- pected utility, n-th degree stochastic dominance, set dominance. 1 Introduction A nonadditive probability measure (also called capacity) is a set function which is defined on a set algebra, ranges from 0 to 1, and is increasing. Every probabil- ity measure is a capacity. Integration with respect to a capacity has been investi- gated by Choquet (1953/54). In recent years nonadditive probability measures have been used in decision theory to cope with observed violations of expected utility (EU). Schmeidler (1989), Gilboa (1987, 1989), Wakker (1989), and Nakamura (1990) provide sets of axioms under which a given preference between uncertain prospects can be represented by the Choquet integral of a utility function with respect to a capacity. These models were developed to explain observed effects of ambiguity aversion like the Ellsberg paradox. Closely related to this are the nonlinear expected utility models by Quiggin (1982), Segal (1984), and Chewl Karni, and 0340 9422/93/3/231-25652.50 9 1993 Physica-Verlag, Heidelberg