Mixture Independence Foundations for Expected Utility

An alternative preference foundation for expected utility is provided. Our segregated approach considers four implications of the von Neumann-Morgenstern independence axiom. The monotonicity principle is, for a transitive relation, equivalent to monotonicity with respect to first-order stochastic dominance. Rank-dependent separability is similar to the comonotonic sure-thing principle used in the ambiguity literature. The remaining two properties are weak formulations of the independence principle which invoke the latter only for probability mixtures with the extreme, that is the best, respectively, the worst outcome. Worst-outcome mixture independence speaks directly to the common ratio example of Allais. The dual of it is called best-outcome mixture independence. These four implications of independence, together with completeness, transitivity and continuity of a preference relation, characterize expected utility. Furthermore, if rank-dependent separability is dropped, expected utility still holds on each subset of three-outcomes lotteries that give positive probability to both the best and the worst outcomes