From infinity to one: The reduction of some mean field games to a global control problem

TL;DR

This paper explores how certain mean field games, especially with common noise, can be transformed into a global control problem, simplifying analysis and solution derivation.

Contribution

It introduces a novel approach to reduce mean field games with common noise to a single global control problem using the Master equation framework.

Findings

Reduction of mean field games to a global control problem

Introduction of a PDE called the Master equation

Application to games on graphs

Abstract

This paper presents recent results from Mean Field Game theory underlying the introduction of common noise that imposes to incorporate the distribution of the agents as a state variable. Starting from the usual mean field games equations introduced by J.M. Lasry and P.L. Lions and adapting them to games on graphs, we introduce a partial differential equation, often referred to as the Master equation, from which the MFG equations can be deduced. Then, this Master equation can be reinterpreted using a global control problem inducing the same behaviors as in the non-cooperative initial mean field game.