On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games

A game is better-reply secure if for every nonequilibrium strategy x* and every payoff vector limit u* resulting from strategies approaching x*, some player i has a strategy yielding a payoff strictly above ui* even if the others deviate slightly from x*. If strategy spaces are compact and convex, payoffs are quasiconcave in the owner's strategy, and the game is better-reply secure, then a pure strategy Nash equilibrium exists. Better-reply security holds in many economic games. It also permits new results on the existence of symmetric and mixed strategy Nash equilibria.