Numerical Representation of Convex Preferences Over Anscombe-Aumann Acts

Abstract

In this paper we derive a numerical representation for general complete preferences respecting the following two principles: a) more is better than less, b) averages are better than extremes. To be able to distinguish between risk and ambiguity we work in an Anscombe-Aumann framework. Our main result is a quasi-concave numerical representation for a class of preferences wide enough to accommodate Ellsberg as well as Allais-type behavior. Instead of assuming the usual monotonicity we suppose that our preferences are monotone with respect to first order stochastic dominance. Preference for averages is expressed through convexity. No independence assumptions of any form are made. In general, our preferences intertwine attitudes towards risk and ambiguity. But if one assumes a weak form of Savage’s sure thing principle, there is separation between risk and ambiguity attitudes, and the representation decomposes into state dependent preference functionals over the consequences and a quasi-concave functional aggregating the preferences of the decision maker in different states of the world.