A Direct Second-Order Method for Solving Two-Player Zero-Sum Games
David Yang, Yuan Gao, Tianyi Lin, Christian Kroer
TL;DR
The paper addresses efficient computation of Nash equilibria in two-player zero-sum games by introducing a direct second-order method based on Douglas–Rachford splitting solved with a regularized semi-smooth Newton method, complemented by a global progression phase using PRM+. It establishes a lifted fixed-point framework linking equilibria to DRS fixed points, provides convergence guarantees including local quadratic convergence, and develops a hybrid algorithm that combines global first-order progress with rapid local convergence. Theoretical results are supported by extensive numerical experiments on matrix games, showing large speedups in high-precision equilibrium computation while preserving problem structure. Overall, the work presents a principled, scalable approach for high-accuracy equilibrium computation in large-scale zero-sum games.
Abstract
We introduce, to our knowledge, the first direct second-order method for computing Nash equilibria in two-player zero-sum games. To do so, we construct a Douglas-Rachford-style splitting formulation, which we then solve with a semi-smooth Newton (SSN) method. We show that our algorithm enjoys local superlinear convergence. In order to augment the fast local behavior of our SSN method with global efficiency guarantees, we develop a hybrid method that combines our SSN method with the state-of-the-art first-order method for game solving, Predictive Regret Matching$^+$ (PRM$^+$). Our hybrid algorithm leverages the global progress provided by PRM$^+$, while achieving a local superlinear convergence rate once it switches to SSN near a Nash equilibrium. Numerical experiments on matrix games demonstrate order-of-magnitude speedups over PRM$^+$ for high-precision solutions.