Lipschitz Games

The Lipschitz constant of a finite normal-form game is the maximal change in some player's payoff when a single opponent changes his strategy. We prove that games with small Lipschitz constant admit pure ϵ-equilibria, and pinpoint the maximal Lipschitz constant that is sufficient to imply existence of a pure ϵ-equilibrium as a function of the number of players in the game and the number of strategies of each player. Our proofs use the probabilistic method.