Equilibrium existence and approximation of regular discontinuous games

Many conditions have been introduced to ensure equilibrium existence in games with discontinuous payoff functions. This paper introduces a new condition, called regularity, that is simple and easy to verify. Regularity requires that if there is a sequence of strategies converging to s* such that the players' payoffs along the sequence converge to the best-reply payoffs at s*, then s* is an equilibrium. We show that regularity is implied both by Reny's better-reply security and Simon and Zame's endogenous sharing rule approach. This allows us to explore a link between these two distinct methods. Although regularity implies that the limits of epsilon-equilibria are equilibria, it is in general too weak for implying equilibrium existence. However, we are able to identify extra conditions that, together with regularity, are sufficient for equilibrium existence. In particular, we show how regularity allows the technique of approximating games both by payoff functions and space of strategies.