A Von Neumann-Morgenstern Representation Result Without Weak Continuity Assumption

Abstract

In the paradigm of von Neumann-Morgenstern, a representation of affine preferences in terms of an expected utility can be obtained under the assumption of weak continuity. Since the weak topology is coarse, this requirement is a priori far from being negligible. In this work, we replace the assumption of weak continuity by monotonicity. More precisely, on the space of lotteries on a real open interval, it is shown that any affine numerical representation of a preference order monotone with respect to the first stochastic order, admits a representation in terms of an expected utility for some nondecreasing utility function. As a consequence, any affine numerical representation on the subset of lotteries with compact support monotone with respect to the second stochastic order can be represented in terms of an expected utility for some nondecreasing concave utility function. We also provide such representations for affine preference order on the subset of those lotteries which fulfills some integrability conditions. The subtleties of the weak topology are also illustrated by some examples.

Keywords: von Neumann and Morgenstern representation, affine preference orders, automatic continuity, first stochastic order

JEL Classification: D08, D8, D81