Correlation-Based Algorithm for Team-Maxmin Equilibrium in Multiplayer Extensive-Form Games

Efficient algorithms computing a Nash equilibrium have been successfully applied to large zero-sum two-player extensive-form games (e.g., poker). However, in multiplayer games, computing a Nash equilibrium is generally hard, and the equilibria are not exchangeable, which makes players face the problem of selecting one of many different Nash equilibria. In this paper, we focus on an alternative solution concept in zero-sum multiplayer extensive-form games called Team-Maxmin Equilibrium (TME). It is a Nash equilibrium that maximizes each team member's utility. As TME is unique in general, it avoids the equilibrium selection problem. However, it is still difficult (FNP-hard) to find a TME. Computing it can be formulated as a non-convex program, but existing algorithms are capable of solving this program for only very small games. In this paper, we first refine the complexity result for computing a TME by using a correlation plan to show that a TME can be found in polynomial time in a specific class of games according to our boundary for complexity. Second, we propose an efficient correlation-based algorithm to solve the non-convex program for TME in games not belonging to this class. The algorithm combines two special correlation plans based on McCormick envelopes for convex relaxation and von Stengel-Forges polytope for correlated equilibria. We show that restricting the feasible solution space to von Stengel-Forges polytope will strictly reduce the feasible solution space after convex relaxation of nonlinear terms. Finally, experiments show that our algorithm is about four orders of magnitude faster than the prior state of the art and can solve many previously unsolvable games.