Radially Symmetric Mean-Field Games with Congestion

TL;DR

This paper analyzes radial solutions for stationary Mean-Field Games with congestion, deriving explicit formulas and variational formulations, and providing numerical approximations for these models.

Contribution

It introduces a tractable approach for solving radial MFGs with congestion, including explicit solutions and a variational framework.

Findings

Explicit formulas for first-order MFG solutions

Variational formulation for elliptic MFGs

Numerical approximations of solutions

Abstract

Here, we study radial solutions for first- and second-order stationary Mean-Field Games (MFG) with congestion on $\mathbb{R}^d$. MFGs with congestion model problems where the agents' motion is hampered in high-density regions. The radial case, which is one of the simplest non one-dimensional MFG, is relatively tractable. As we observe in this paper, the Fokker-Planck equation is integrable with respect to one of the unknowns. Consequently, we obtain a single equation substituting this solution into the Hamilton-Jacobi equation. For the first-order case, we derive explicit formulas; for the elliptic case, we study a variational formulation of the resulting equation. In both cases, we use our approach to compute numerical approximations to the solutions of the corresponding MFG systems.