The Zero Regrets Algorithm: Optimizing over Pure Nash Equilibria via Integer Programming

Designing efficient algorithms to compute Nash equilibria poses considerable challenges in algorithmic game theory and optimization. In this work, we employ integer programming techniques to compute Nash equilibria in integer programming games, a class of simultaneous and noncooperative games in which each player solves a parameterized integer program. We introduce zero regrets, a general and efficient cutting-plane algorithm to compute, enumerate, and select Nash equilibria. Our framework leverages the concept of equilibrium inequality, an inequality valid for any Nash equilibrium, and the associated equilibrium separation oracle. We evaluate our algorithmic framework on a wide range of practical and methodological problems from the literature, providing a solid benchmark against the existing approaches.