On the Interpretation of Sarin and Wakker's "A Simple Axiomatization of Nonadditive Expected Utility"

Abstract

IN AN INTERESTING RECENT PAPER, Sarin and Wakker (1992) (henceforth S-W) have provided a new axiomatization of expected utility maximization with (CEU). Its simplicity is attractive in that it provides a constructive interpretation of the role of capacities and Choquet integration in the main representation theorem. An important issue raised by the paper is the interpretation of the key Axiom P4 (Cumulative Dominance). S-W suggest an appreciation of the axiom as an adaptation of principles to nonadditive-probability contexts, most eloquently on page 1260. If viable, such an interpretation would supply the CEU model with a powerful intuitive foundation that has been lacking so far. This note argues that a interpretation can be maintained only at the price of either an arbitrary choice of specification which undermines its intuitive force or, alternatively, of an unintended restriction of the class of characterized preferences. Two logically independent arguments are presented. The first points out an arbitrariness in the of the more-likely-than relation in terms of preferences (Proposition 1). The second shows a similar arbitrariness in the of a stochastic dominance relation in terms of a more-likely-than relation (Proposition 2). In each case, the invoked symmetry conditions yield a characterization of CEU preferences with symmetric capacities, a nontrivial generalization of the SEU model that has received little attention in the literature. For notation and definitions, the reader is referred to S-W's paper. S-W's concern is to develop an intuitively convincing axiomatization of CEU-representable preference relations. Their key Axiom P4 is formulated in terms of a more-likelythan relation > on the algebra v of events that is defined in terms of the preference relation a on the set of acts Y To facilitate the subsequent discussion, their definition is introduced here formally as a condition on the pair of relations (a, >):