Lipschitz Bernoulli Utility Functions

Abstract

We obtain several variants of the classic von Neumann–Morgenstern expected utility theorem with and without the completeness axiom in which the derived Bernoulli utility functions are Lipschitz. The prize space in these results is an arbitrary separable metric space, and the utility functions are allowed to be unbounded. The main ingredient of our results is a novel (behavioral) axiom on the underlying preference relations, which is satisfied by virtually all stochastic orders. The proof of the main representation theorem is built on the fact that the dual of the Kantorovich–Rubinstein space is (isometrically isomorphic to) the Banach space of Lipschitz functions that vanish at a fixed point. An application to the theory of nonexpected utility is also provided.