Existence of equilibria in constrained discontinuous games

We establish the existence of pure strategy equilibria in games with discontinuous payoffs where the set of feasible actions of each player varies, also in a discontinuous fashion, as a function of the actions of the other players. Such games are used in modeling abstract economies and other games where players share common constraints. Our approach circumvents the difficulties that arise from the presence of discontinuities by modifying the original problem and allowing the players to use strategies that possibly lie outside their feasible sets. We then show that each modified game has ε-equilibria points. Under certain conditions, and as the extent of modification becomes smaller and smaller and ε approaches zero, the ε-equilibria points of the modified games will converge to a strategy profile that is an equilibrium of the original game. Hence, we obtain a set of sufficient conditions for the existence of pure equilibria of the original game. We apply our results to a number of classic games that have discontinuous payoffs and discontinuous constraint correspondences.