Existence of a pure strategy equilibrium in finite symmetric games where payoff functions are integrally concave

In this paper we show that a finite symmetric game has a pure strategy equilibrium if the payoff functions of players are integrally concave (the negative of the integrally convex functions due to Favati and Tardella [Convexity in nonlinear integer programming, Ricerca Operativa, 1990, 53:3–44]). Since the payoff functions of any two-strategy game are integrally concave, this generalizes the result of Cheng et al. [Notes on equilibria in symmetric games, Proceedings of the 6th Workshop On Game Theoretic And Decision Theoretic Agents, 2004, 23–28]. A simple algorithm to find an equilibrium is also provided.