On the Asymptotic Nature of First Order Mean Field Games

Markus Fischer,Francisco J Silva

doi:10.1007/s00245-020-09711-1

Abstract

For a class of finite horizon first order mean field games and associated N-player games, we give a simple proof of convergence of symmetric N-player Nash equilibria in distributed open-loop strategies to solutions of the mean field game in Lagrangian form. Lagrangian solutions are then connected with those determined by the usual mean field game system of two coupled first order PDEs, and convergence of Nash equilibria in distributed Markov strategies is established.

Highlights

The purpose of this article is to illustrate a simple way of establishing convergence of open-loop Nash equilibria in the case of first-order non-stationary Mean Field Games (MFGs)
Applied Mathematics & Optimization (2021) 84:2327–2357 rigorous is to show that a solution of the mean field game yields approximate Nash equilibria for the N -player games, with approximation error vanishing as N → ∞
Following [16], we link the notion of Lagrangian MFG equilibrium to the well known PDE characterization of mean field games in terms of two coupled first order partial differential equations, namely a backward first order Hamilton-Jacobi-Bellman equation and a forward continuity equation; see equation (M F G) in Sect. 4.1 below

Summary

Introduction

The purpose of this article is to illustrate a simple way of establishing convergence of open-loop Nash equilibria in the case of first-order non-stationary Mean Field Games (MFGs). In the second part of this article, we consider additional second order assumptions on the data and we assume that the initial distribution is absolutely continuous with respect to the Lebesgue measure In this framework, and following [16], we link the notion of Lagrangian MFG equilibrium to the well known PDE characterization of mean field games in terms of two coupled first order partial differential equations, namely a backward first order Hamilton-Jacobi-Bellman equation and a forward continuity equation; see equation (M F G) in Sect. The corresponding convergence results are given in Theorem 4.1 and Corollary 4.1, respectively

The N-Player Game

Convergence to a Mean Field Game Equilibrium

The First Order Mean Field Game system

Lagrangian MFG Equilibria and the MFG PDE System

The N-Player Equilibria