A Simple Axiomatization of Nonadditive Expected Utility

Abstract

This paper provides an extension of Savage's subjective expected utility theory for decisions under uncertainty. It includes in the set of events both events for which probabilities are additive and ambiguous events for which probabilities are permit- ted to be nonadditive. The main axiom is cumulative dominance, which adapts stochastic dominance to decision making under uncertainty. We derive a Choquet expected utility representation and show that a modification of cumulative dominance leads to the classical expected utility representation. The relationship of our approach with that of Schmeidler, who uses a two-stage formulation to derive Choquet expected utility, is also explored. Our work may be viewed as a unification of Schmeidler (1989) and Gilboa (1987). SAVAGE'S (1954) SUBJECTIVE EXPECTED UTILITY (SEU) theory has been widely adopted as the guide for rational decision making in the face of uncertainty. In SEU theory both the probabilities and the utilities are derived from preferences (see also Ramsey (1931)). This represents a hallmark contribution, as it avoids the reliance on introspection for quantifying tastes and beliefs. We continue in Savage's vein and extend his theory to derive a more general nonaddi- tive expected utility representation, called Choquet expected utility (CEU). Schmeidler (1989, first version 1982) made the first contribution in providing a CEU representation and Gilboa (1987) extended this work. We develop this line of research further by providing an intuitive axiomatization of CEU. The key distinction between our work and that of Savage is that we identify two types of events-unambiguous and ambiguous. People feel relatively sure about the probabilities of events. An example of an event could be the outcome of a toss of a fair coin (heads or tails). We assume that Savage's axioms hold for a sufficiently rich set of unambiguous acts, i.e., acts measurable with respect to the events. The probabilities of ambiguous events, however, are not known with precision. An example of such an event could be next week's weather conditions (rain or sunshine). Ambiguity in the probability of such events may be caused, for example, by a lack of available information relative to the amount of conceivable information (Keynes (1921)). Most people exhibit a reluctance to bet on events with ambiguous