Reinforcement Nash Equilibrium Solver

TL;DR

This paper introduces RENES, a novel reinforcement learning approach that trains a single policy to modify games, enabling existing solvers to better approximate Nash Equilibria across various game sizes and types.

Contribution

We propose a reinforcement learning framework using graph neural networks and tensor decomposition to enhance NE approximation in general-sum games, generalizing across game sizes and types.

Findings

RENES improves NE approximation for multiple solvers.

It generalizes well to unseen large-scale games.

It outperforms traditional inexact solvers in experiments.

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 $\alpha$-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., $\alpha$-rank, CE, FP and PRD, and can be generalized to unseen games.