Existence of Equilibria in Discontinuous Games

This paper investigates the existence of pure strategy, dominant strategy, and mixed strategy Nash equilibria in discontinuous games. We introduce a new notion of weak continuity, called weak transfer quasi-continuity, which is weaker than 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 it, together with strong diagonal transfer quasiconcavity introduced in the paper, is enough to guarantee the existence of Nash equilibria in compact and convex normal form games. 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.