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 Formula: see text-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 decomposition-based 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. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms. Funding: This work was supported by the National Institute of Dental and Craniofacial Research Grant R01DE028283. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. The funding agreements ensured the authors’ independence in designing the study, interpreting the data, writing, and publishing the report. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0072 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0072 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .