The variational structure and time-periodic solutions for mean-field games systems

TL;DR

This paper introduces a new game-theoretic framework for mean-field game systems, deriving novel variational formulations and proving the existence of time-periodic solutions in complex MFG models.

Contribution

It presents a two-player infinite-dimensional differential game formulation of MFGs, linking to known variational principles and establishing new formulations for congested MFGs.

Findings

New variational formulations for first-order MFGs with congestion

Existence proof for time-periodic solutions in viscous MFGs

Connection between game perspective and variational principles

Abstract

Here, we observe that mean-field game (MFG) systems admit a two-player infinite-dimensional general-sum differential game formulation. We show that particular regimes of this game reduce to previously known variational principles. Furthermore, based on the game-perspective we derive new variational formulations for first-order MFG systems with congestion. Finally, we use these findings to prove the existence of time-periodic solutions for viscous MFG systems with a coupling that is not a non-decreasing function of density.