One-dimensional, non-local, first-order, stationary mean-field games with congestion: a Fourier approach

TL;DR

This paper introduces a Fourier-based approach to analyze one-dimensional non-local mean-field games with congestion, simplifying the problem to finite dimensions when using trigonometric polynomial kernels and approximating more general kernels.

Contribution

It develops a Fourier expansion method for solving mean-field games with congestion, including a reduction to finite dimensions and kernel approximation techniques.

Findings

Reduction to finite-dimensional systems for polynomial kernels

Effective approximation of general kernels using Fourier methods

Framework applicable to a broad class of non-local mean-field games

Abstract

Here, we study a one-dimensional, non-local mean-field game model with congestion. When the kernel in the non-local coupling is a trigonometric polynomial we reduce the problem to a finite dimensional system. Furthermore, we treat the general case by approximating the kernel with trigonometric polynomials. Our technique is based on Fourier expansion methods.