Existence of Equilibria in Discontinuous and Nonconvex Games

This paper investigates the existence of pure strategy, dominant strategy, and mixed strategy Nash equilibria in discontinuous and/or nonconvex games. We introduce a new notion of very weak continuity, called weak transfer quasi-continuity, which is weaker than the most known weak notions of continuity, including diagonal transfer continuity in Baye et al. [1993] and better-reply security in Reny [1999], and holds in a large class of discontinuous games. We show that weak transfer quasi-continuity, together with the compactness of strategy space and the quasiconcavity or (strong/weak) diagonal transfer quasiconcavity of payoffs, permits the existence of a pure strategy Nash equilibrium. We provide sufficient conditions for weak transfer quasi-continuity by introducing notions of weak transfer continuity, weak transfer upper continuity, and weak transfer lower continuity. Moreover, an analogous analysis is applied to show the existence of dominant strategy and mixed strategy Nash equilibria in discontinuous games.