Computing Optimal Nash Equilibria in Multiplayer Games

Designing efficient algorithms to compute a Nash Equilibrium (NE) in multiplayer games is still an open challenge. In this paper, we focus on computing an NE that optimizes a given objective function. For example, when there is a team of players independently playing against an adversary in a game (e.g., several groups in a forest trying to interdict illegal loggers in green security games), these team members may need to find an NE minimizing the adversary's utility. Finding an optimal NE in multiplayer games can be formulated as a mixed-integer bilinear program by introducing auxiliary variables to represent bilinear terms, leading to a huge number of bilinear terms, making it hard to solve. To overcome this challenge, we first propose a general framework for this formulation based on a set of correlation plans. We then develop a novel algorithm called CRM based on this framework, which uses Correlation plans with their Relations to restrict the feasible solution space after the convex relaxation of bilinear terms while Minimizing the number of correlation plans to reduce the number of bilinear terms. We show that our techniques can significantly reduce the time complexity, and CRM can be several orders of magnitude faster than the state-of-the-art baseline.