Pure strategy dominance with quasiconcave utility functions

Abstract

By a result of Pearce (1984), in a finite strategic form game, the set of a player's serially undominated strategies coincides with her set of rationalizable strategies. In this note we consider an extension of this result that applies to games with continuous utility functions that are quasiconcave in own action. We prove that in such games, when the players are endowed with compact, metrizable, and convex action spaces, a strategy of some player is dominated by some other pure strategy if and only if it is not a best reply to any belief over the strategies adopted by her opponents. For own-quasiconcave games, this can be used to give a characterization of the set of rationalizable strategies, different from the one given by Pearce. Moreover, expected utility functions defined on the mixed extension of a game are always own-quasiconcave, and therefore the result in this note generalizes Pearce''s characterization to infinite games, by a simple shift of perspective.