# Radially Symmetric Mean-Field Games with Congestion

Research paper by David Evangelista, Diogo A. Gomes, Levon Nurbekyan

Indexed on: 22 Mar '17Published on: 22 Mar '17Published in: arXiv - Mathematics - Analysis of PDEs

#### 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.