Empirical Distribution of Equilibrium Play and Its Testing Application

AbstractWe show that in any n-player, m-action normal-form game, we can obtain an approximate equilibrium by sampling any mixed-action equilibrium a small number of times. We study three types of equilibria: Nash, correlated, and coarse-correlated. For each one we obtain upper and lower bounds on the number of samples required for the empirical distribution over the sampled action profiles to form an approximate equilibrium with probability close to 1.These bounds imply that using a small number of samples we can test whether or not players are playing according to an approximate equilibrium, even in games where n and m are large. In addition, our results substantially improve previously known upper bounds on the support size of approximate equilibria in games with many players. In particular, for the three types of equilibria we show the existence of approximate equilibrium with support-size polylogarithmic in n and m, whereas the previously best-known upper bounds were polynomial in n Hémon et al. [Hémon S, de Rougemont M, Santha M 2008 Approximate nash equilibria for multi-player games. Algorithmic Game Theory Springer, 267-278], Germano and Lugosi [Germano F, Lugosi G 2007 Existence of sparsely supported correlated equilibria. Econom. Theory 323:575-578], Hart et al. [Hart S, Mas-Colell A, Babichenko Y 2013 Simple Adaptive Strategies: From Regret-Matching to Uncoupled Dynamics, Vol. 4 World Scientific Publishing Company].