Minimal conditions for parametric continuity of a utility representation

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. May 13, 2015. I thank an anonymous referee, Sander Heinsalu, Atsushi Kajii, Jeff Kline, Andrew McLennan, Aliandra Nasif, John Quiggin and Maxwell Stinchcombe for their useful feedback. Some of the present results appeared in my Ph.D dissertation. Department of Economics, University of Queensland. Email [email protected]