Reinforcement Nash Equilibrium Solver
Xinrun Wang, Chang Yang, Shuxin Li, Pengdeng Li, Xiao Huang, Hau Chan, Bo An
TL;DR
This work tackles the challenge of computing Nash Equilibria in general-sum games, where exact NE computation is PPAD-Complete and traditional inexact solvers may diverge from NE. It introduces RENES, a reinforcement-learning-based framework that learns a single policy to modify game instances of varying size and then applies standard solvers on the modified games, using α-rank graphs and CP tensor decomposition to enable size-agnostic modification and a PPO-based training loop to optimize over both the game modifications and solver performance. The approach yields consistent improvements across multiple solvers ($α$-rank, CE, FP, PRD) on large-scale normal-form games and generalizes to unseen games, demonstrating the potential of pre-training a solver-agnostic modification policy. The work highlights a new direction in which game modification serves as a pre-training task to enhance equilibrium approximations, with implications for scalability, generalization, and potential extensions to other solution concepts and game types.
Abstract
Nash Equilibrium (NE) is the canonical solution concept of game theory, which provides an elegant tool to understand the rationalities. Though mixed strategy NE exists in any game with finite players and actions, computing NE in two- or multi-player general-sum games is PPAD-Complete. Various alternative solutions, e.g., Correlated Equilibrium (CE), and learning methods, e.g., fictitious play (FP), are proposed to approximate NE. For convenience, we call these methods as "inexact solvers", or "solvers" for short. However, the alternative solutions differ from NE and the learning methods generally fail to converge to NE. Therefore, in this work, we propose REinforcement Nash Equilibrium Solver (RENES), which trains a single policy to modify the games with different sizes and applies the solvers on the modified games where the obtained solution is evaluated on the original games. Specifically, our contributions are threefold. i) We represent the games as $α$-rank response graphs and leverage graph neural network (GNN) to handle the games with different sizes as inputs; ii) We use tensor decomposition, e.g., canonical polyadic (CP), to make the dimension of modifying actions fixed for games with different sizes; iii) We train the modifying strategy for games with the widely-used proximal policy optimization (PPO) and apply the solvers to solve the modified games, where the obtained solution is evaluated on original games. Extensive experiments on large-scale normal-form games show that our method can further improve the approximation of NE of different solvers, i.e., $α$-rank, CE, FP and PRD, and can be generalized to unseen games.