Deep Fictitious Play for Finding Markovian Nash Equilibrium in Multi-Agent Games

TL;DR

This paper introduces a deep neural network algorithm based on fictitious play to efficiently find Markovian Nash equilibria in large N-player stochastic differential games, overcoming the curse of dimensionality.

Contribution

It develops a novel deep learning approach combining fictitious play and deep BSDE methods to solve high-dimensional stochastic games for the first time.

Findings

Successfully computes equilibria in 50-player games with common noise.

Accurately solves large N-player games where traditional methods fail.

Validates approach with multiple numerical examples involving diverse agents.

Abstract

We propose a deep neural network-based algorithm to identify the Markovian Nash equilibrium of general large $N$-player stochastic differential games. Following the idea of fictitious play, we recast the $N$-player game into $N$ decoupled decision problems (one for each player) and solve them iteratively. The individual decision problem is characterized by a semilinear Hamilton-Jacobi-Bellman equation, to solve which we employ the recently developed deep BSDE method. The resulted algorithm can solve large $N$-player games for which conventional numerical methods would suffer from the curse of dimensionality. Multiple numerical examples involving identical or heterogeneous agents, with risk-neutral or risk-sensitive objectives, are tested to validate the accuracy of the proposed algorithm in large group games. Even for a fifty-player game with the presence of common noise, the proposed algorithm still finds the approximate Nash equilibrium accurately, which, to our best knowledge, is difficult to achieve by other numerical algorithms.