Logarithmic Query Complexity for Approximate Nash Computation in Large Games

Paul W. Goldberg, Francisco J. Marmolejo Cossío, Zhiwei Steven Wu

Abstract: We investigate the problem of equilibrium computation for “large” n-player games where each player has two pure strategies. Large games have a Lipschitz-type property that no single player’s utility is greatly affected by any other individual player’s actions. In this paper, we assume that a player can change another player’s payoff by at most $$\frac{1}{n}$$ by changing her strategy. We study algorithms having query access to the game’s payoff function, aiming to find $$\varepsilon $$ -Nash equilibria. We seek algorithms that obtain $$\varepsilon $$ as small as possible, in time polynomial in n. Our main result is a randomised algorithm that achieves $$\varepsilon $$ approaching $$\frac{1}{8}$$ in a completely uncoupled setting, where each player observes her own payoff to a query, and adjusts her behaviour independently of other players’ payoffs/actions. $$O(\log n)$$ rounds/queries are required. We also show how to obtain a slight improvement over $$\frac{1}{8}$$ , by introducing a small amount of communication between the players.