Schr\"odinger approach to Mean Field Games with negative coordination

TL;DR

This paper explores the behavior of mean field games with negative coordination by linking the coupled PDEs to nonlinear Schr"odinger equations, especially in long-time limits and varying interaction regimes.

Contribution

It introduces a novel approach by applying the Schr"odinger equation framework to analyze negative coordination in mean field games, revealing insights into different dynamical regimes.

Findings

Identification of regimes where disorder, interactions, and external potential dominate

Analysis of the forward-backward structure in relation to different regimes

Insights into the long-time behavior of the system

Abstract

Mean Field Games provide a powerful framework to analyze the dynamics of a large number of controlled agents in interaction. Here we consider such systems when the interactions between agents result in a negative coordination and analyze the behavior of the associated system of coupled PDEs using the now well established correspondence with the non linear Schr\"odinger equation. We focus on the long optimization time limit and on configurations such that the game we consider goes through different regimes in which the relative importance of disorder, interactions between agents and external potential varies, which makes possible to get insights on the role of the forward-backward structure of the Mean Field Game equations in relation with the way these various regimes are connected.