Learning Graphon Mean Field Games and Approximate Nash Equilibria

TL;DR

This paper introduces a new discrete-time framework for graphon mean field games, providing theoretical guarantees and practical algorithms to approximate Nash equilibria in large dense graph games.

Contribution

It develops a novel discrete-time formulation for graphon mean field games, with rigorous existence and approximation results, and proposes learning schemes for equilibrium computation.

Findings

Finite-agent behavior converges to mean field predictions as system size increases.

The proposed algorithms successfully compute approximate Nash equilibria.

Empirical results validate the theoretical convergence and effectiveness of the methods.

Abstract

Recent advances at the intersection of dense large graph limits and mean field games have begun to enable the scalable analysis of a broad class of dynamical sequential games with large numbers of agents. So far, results have been largely limited to graphon mean field systems with continuous-time diffusive or jump dynamics, typically without control and with little focus on computational methods. We propose a novel discrete-time formulation for graphon mean field games as the limit of non-linear dense graph Markov games with weak interaction. On the theoretical side, we give extensive and rigorous existence and approximation properties of the graphon mean field solution in sufficiently large systems. On the practical side, we provide general learning schemes for graphon mean field equilibria by either introducing agent equivalence classes or reformulating the graphon mean field system as a classical mean field system. By repeatedly finding a regularized optimal control solution and its generated mean field, we successfully obtain plausible approximate Nash equilibria in otherwise infeasible large dense graph games with many agents. Empirically, we are able to demonstrate on a number of examples that the finite-agent behavior comes increasingly close to the mean field behavior for our computed equilibria as the graph or system size grows, verifying our theory. More generally, we successfully apply policy gradient reinforcement learning in conjunction with sequential Monte Carlo methods.