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

Philip J. Reny

Abstract

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.