Numerical Methods for Mean Field Games and Mean Field Type Control

TL;DR

This paper reviews various numerical methods for solving Mean Field Games and Mean Field Control problems, emphasizing PDE schemes, optimization techniques, monotone operator approaches, and machine learning-based stochastic methods.

Contribution

It provides a comprehensive overview of recent numerical approaches for MFG and MFC, including heuristics, PDE schemes, variational methods, and machine learning techniques.

Findings

Numerical schemes for forward-backward PDE systems are discussed.

Optimization techniques for variational problems are reviewed.

Machine learning methods for stochastic MFG and MFC are introduced.

Abstract

Mean Field Games (MFG) have been introduced to tackle games with a large number of competing players. Considering the limit when the number of players is infinite, Nash equilibria are studied by considering the interaction of a typical player with the population's distribution. The situation in which the players cooperate corresponds to Mean Field Control (MFC) problems, which can also be viewed as optimal control problems driven by a McKean-Vlasov dynamics. These two types of problems have found a wide range of potential applications, for which numerical methods play a key role since most models do not have analytical solutions. In these notes, we review several aspects of numerical methods for MFG and MFC. We start by presenting some heuristics in a basic linear-quadratic setting. We then discuss numerical schemes for forward-backward systems of partial differential equations (PDEs), optimization techniques for variational problems driven by a Kolmogorov-Fokker-Planck PDE, an approach based on a monotone operator viewpoint, and stochastic methods relying on machine learning tools.