Lipschitz Bernoulli Utility Functions.

The article focuses on characterizing Lipschitz Bernoulli utility functions as representations of preference preorders on spaces of lotteries, extending classical expected utility theory. It introduces a novel behavioral axiom—termed the Lipschitz property—on preorders defined over the set Δ₁(X) of Borel probability measures with finite first moment on a separable metric space X, equipped with the Wasserstein 1-metric. The main result establishes that a preorder on Δ₁(X) is affine and satisfies this Lipschitz condition if and only if it admits a representation by a family of Lipschitz Bernoulli utility functions, which may be unbounded. This framework generalizes the von Neumann–Morgenstern expected utility theorem by allowing incomplete preferences and unbounded utilities, and it encompasses virtually all stochastic orders as Lipschitz affine preorders. The paper further explores structural properties, uniqueness, and properness of such multiutilities, relates the Lipschitz condition to a prior definition by Levin, and applies the results to the affine core of preorders and to state-dependent acts in the Anscombe–Aumann framework.