Quadratic Mean Field Games

TL;DR

This paper introduces Quadratic Mean Field Games, highlighting their connection to the non-linear Schrödinger equation, and provides a foundation for physicists to understand this emerging interdisciplinary field.

Contribution

It offers an accessible introduction to Quadratic Mean Field Games and explores their deep link with the non-linear Schrödinger equation, facilitating cross-disciplinary understanding.

Findings

Connection established between quadratic mean field games and non-linear Schrödinger equation

Provides approximation schemes for analyzing mean field games

Serves as an entry point for physicists into the field

Abstract

Mean field games were introduced independently by J-M. Lasry and P-L. Lions, and by M. Huang, R.P. Malham\'e and P. E. Caines, in order to bring a new approach to optimization problems with a large number of interacting agents. The description of such models split in two parts, one describing the evolution of the density of players in some parameter space, the other the value of a cost functional each player tries to minimize for himself, anticipating on the rational behavior of the others. Quadratic Mean Field Games form a particular class among these systems, in which the dynamics of each player is governed by a controlled Langevin equation with an associated cost functional quadratic in the control parameter. In such cases, there exists a deep relationship with the non-linear Schr\"odinger equation in imaginary time, connexion which lead to effective approximation schemes as well as a better understanding of the behavior of Mean Field Games. The aim of this paper is to serve as an introduction to Quadratic Mean Field Games and their connexion with the non-linear Schr\"odinger equation, providing to physicists a good entry point into this new and exciting field.