Approximating N-Player Nash Equilibrium through Gradient Descent

TL;DR

This paper introduces a gradient descent method to efficiently approximate Nash equilibria in N-player general-sum games, addressing computational challenges and outperforming existing algorithms in diverse game settings.

Contribution

It proposes a novel gradient-based approach that transforms NE computation into a global minimum search, with proven convergence for convex utility functions.

Findings

Outperforms Tsaknakis-Spirakis algorithm, fictitious play, and regret matching.

Demonstrates robustness with increasing players and actions.

Achieves convergence rate of O(1/T) for convex utility functions.

Abstract

Decoding how rational agents should behave in shared systems remains a critical challenge within theoretical computer science, artificial intelligence and economics studies. Central to this challenge is the task of computing the solution concept of games, which is Nash equilibrium (NE). Although computing NE in even two-player cases are known to be PPAD-hard, approximation solutions are of intensive interest in the machine learning domain. In this paper, we present a gradient-based approach to obtain approximate NE in N-player general-sum games. Specifically, we define a distance measure to an NE based on pure strategy best response, thereby computing an NE can be effectively transformed into finding the global minimum of this distance function through gradient descent. We prove that the proposed procedure converges to NE with rate $O(1/T)$ ($T$ is the number of iterations) when the utility function is convex. Experimental results suggest our method outperforms Tsaknakis-Spirakis algorithm, fictitious play and regret matching on various types of N-player normal-form games in GAMUT. In addition, our method demonstrates robust performance with increasing number of players and number of actions.