Identifying Socially Optimal Equilibria Using Combinatorial Properties of Nash Equilibria in Bimatrix Games

Nash equilibrium is arguably the most fundamental concept in game theory, which is used to analyze and predict the behavior of the players. In many games, there exist multiple equilibria, with different expected payoffs for the players, which in turn raises the question of equilibrium selection. In this paper, we study the NP-hard problem of identifying a socially optimal Nash equilibrium in two-player normal-form games (called bimatrix games), which may be represented by a mixed integer linear program (MILP). We characterize the properties of the equilibria and develop several classes of valid inequalities accordingly. We use these theoretical results to provide a decompositionbased reformulation of the MILP, which we solve by a branch-and-cut algorithm. Our extensive computational experiments demonstrate superiority of our approach over solving the MILP formulation through feeding it into a commercial solver or through the "traditional" Benders' decomposition. Of note, our proposed approach can find provably optimal solutions for many instances.