Minimax Approach to First-Order Mean Field Games

TL;DR

This paper introduces a minimax-based framework for solving first-order mean field games with atomic distributions, establishing existence of generalized solutions and their relation to approximate Nash equilibria.

Contribution

It proposes a new minimax solution concept for mean field games with atoms and proves its existence and connection to finite-player Nash equilibria.

Findings

Existence of generalized minimax solutions for the mean field game system.

Minimax solutions yield ε-Nash equilibria in finite-player differential games.

Framework accommodates distributions with atoms in mean field games.

Abstract

The paper is devoted to the first-order mean field game system in the case when the distribution of players can contain atoms. The proposed definition of a generalized solution is based on the minimax approach to the Hamilton-Jacobi equation. We prove the existence of the generalized (minimax) solution of the mean filed game system using the Nash equilibrium in the auxiliary differential game with infinitely many identical players. We show that the minimax solution of the original system provide the $\varepsilon$-Nash equilibrium in the differential game with finite number of players.