Minimal conditions for parametric continuity of a utility representation

TL;DR

This paper characterizes when a utility representation depends continuously on parameters, identifying conditions on the parameter space and extending Berge's theorem to ensure joint continuity, with applications to economic and psychological models.

Contribution

It provides a characterization of parametric continuity for utility functions and identifies parameter spaces guaranteeing such representations, extending Berge's theorem for discrete alternatives.

Findings

Continuous utility representation exists under specific parameter space conditions.

Joint continuity is achieved for discrete alternatives via an extended Berge's theorem.

Applications include generalizing consumer choice models and reference-dependent preferences.

Abstract

Dependence on the parameter is continuous when perturbations of the parameter preserves strict preference for one alternative over another. We characterise this property via a utility function over alternatives that depends continuously on the parameter. The class of parameter spaces where such a representation is guaranteed to exist is also identified. When the parameter is the type or belief of a player, these results have implications for Bayesian and psychological games. When alternatives are discrete, the representation is jointly continuous and an extension of Berge's theorem of the maximum yields a continuous value function. We apply this result to generalise a standard consumer choice problem where parameters are price-wealth vectors. When the parameter space is lexicographically ordered, a novel application to reference-dependent preferences is possible.