Approximately Solving Mean Field Games via Entropy-Regularized Deep Reinforcement Learning

TL;DR

This paper introduces an entropy-regularized deep reinforcement learning approach to approximately solve finite mean field games, overcoming convergence issues of traditional fixed point methods in high-dimensional settings.

Contribution

It proposes a novel entropy-regularized fixed point iteration method that guarantees convergence to approximate Nash equilibria in finite MFGs, including high-dimensional cases.

Findings

Proves non-contractiveness of fixed point operators in finite MFGs.

Demonstrates convergence of entropy-regularized methods where traditional methods fail.

Validates approach on high-dimensional problems using deep reinforcement learning.

Abstract

The recent mean field game (MFG) formalism facilitates otherwise intractable computation of approximate Nash equilibria in many-agent settings. In this paper, we consider discrete-time finite MFGs subject to finite-horizon objectives. We show that all discrete-time finite MFGs with non-constant fixed point operators fail to be contractive as typically assumed in existing MFG literature, barring convergence via fixed point iteration. Instead, we incorporate entropy-regularization and Boltzmann policies into the fixed point iteration. As a result, we obtain provable convergence to approximate fixed points where existing methods fail, and reach the original goal of approximate Nash equilibria. All proposed methods are evaluated with respect to their exploitability, on both instructive examples with tractable exact solutions and high-dimensional problems where exact methods become intractable. In high-dimensional scenarios, we apply established deep reinforcement learning methods and empirically combine fictitious play with our approximations.