Lipschitz Bernoulli utility functions

Abstract

We obtain variants of the classical 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 may 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 completion of the Kantorovich-Rubinstein space is the canonical predual of the Banach space of Lipschitz functions that vanish at a fixed point. Two applications are given, one to the theory of non-expected utility theory, and the other to the theory of decision-making under uncertainty.