Remarks on potential mean field games

TL;DR

This paper provides an overview of potential mean field games, introduces a new proof for equilibrium characterization, offers criteria for potential existence, and discusses the selection problem among multiple equilibria.

Contribution

It presents a novel Lagrangian proof for potential minimizers being equilibria and offers new criteria to identify potential mean field games.

Findings

Minimizers of the potential are equilibria

Criteria to determine potential existence

Discussion on selecting among multiple Nash equilibria

Abstract

In this expository article, we give an overview of the concept of potential mean field games of first order. We give a new proof that minimizers of the potential are equilibria by using a Lagrangian formulation. We also provide criteria to determine whether or not a game has a potential. Finally, we discuss in some depth the selection problem in mean field games, which consists in choosing one out of multiple Nash equilibria.