Some analytically solvable problems of the mean-field games theory

TL;DR

This paper explores specific solvable cases of mean field games equations, reducing them to nonlinear ODE systems, with applications demonstrated in economic models like investor opinion formation.

Contribution

It identifies conditions under which mean field games can be solved analytically by transforming PDEs into nonlinear ODEs, highlighting practical economic applications.

Findings

Reduction of mean field games to nonlinear ODEs under specific data choices

Analytical solutions for certain mean field game problems

Application to economic models like investor opinion dynamics

Abstract

We study the mean field games equations, consisting of the coupled Kolmogorov-Fokker-Planck and Hamilton-Jacobi-Bellman equations. The equations are complemented by initial and terminal conditions. It is shown that with some specific choice of data, this problem can be reduced to solving a quadratically nonlinear system of ODEs. This situation occurs naturally in economic applications. As an example, the problem of forming an investor's opinion on an asset is considered.