Convergence of one-dimensional stationary mean field games with vanishing potential

TL;DR

This paper studies the convergence behavior of one-dimensional stationary mean-field games as the potential diminishes, establishing conditions for convergence and quantifying the rate at which solutions approach the integrable case.

Contribution

It demonstrates the convergence of regular solutions of stationary MFG systems to integrable systems as potential vanishes, including convergence rates, for both increasing and decreasing coupling cases.

Findings

Solutions converge to integrable MFG solutions as potential approaches zero.

Convergence rates are explicitly derived.

Results hold for both increasing and decreasing coupling scenarios.

Abstract

We consider the one-dimensional stationary first-order mean-field game (MFG) system with the coupling between the Hamilton-Jacobi equation and the transport equation. In both cases that the coupling is strictly increasing and decreasing with respect to the density of the population, we show that when the potential vanishes the regular solution of MFG system converges to the one of the corresponding integrable MFG system. Furthermore, we obtain the convergence rate of such limit.