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 limitu * resulting from strategies approachingx * , some player i has a strategy yielding a payoff strictly above u i * 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.