Lipschitz Bernoulli utility functions

TL;DR

This paper extends the von Neumann-Morgenstern expected utility theorem to include Lipschitz Bernoulli utility functions over arbitrary separable metric prize spaces, accommodating unbounded utilities and broad preference axioms.

Contribution

It introduces a novel behavioral axiom and proves a representation theorem for Lipschitz utility functions in a general metric setting, expanding classical utility theory.

Findings

Utility functions can be Lipschitz continuous in a general metric space.

The results apply to non-expected utility and decision-making under uncertainty.

The theorem accommodates unbounded utilities and broad preference relations.

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.