On the existence of solutions for stationary mean-field games with congestion

TL;DR

This paper proves the existence of stationary solutions for mean-field games with congestion, handling singularities near zero density and extending results to general subquadratic Hamiltonians.

Contribution

It develops robust estimates to establish existence of solutions for stationary MFGs with congestion beyond quadratic Hamiltonians.

Findings

Existence of stationary solutions for congestion MFGs proven.

Robust estimates applicable to general subquadratic Hamiltonians.

Extension of previous quadratic Hamiltonian results to broader cases.

Abstract

Mean-field games (MFGs) are models of large populations of rational agents who seek to optimize an objective function that takes into account their location and the distribution of the remaining agents. Here, we consider stationary MFGs with congestion and prove the existence of stationary solutions. Because moving in congested areas is difficult, agents prefer to move in non-congested areas. As a consequence, the model becomes singular near the zero density. The existence of stationary solutions was previously obtained for MFGs with quadratic Hamiltonians thanks to a very particular identity. Here, we develop robust estimates that give the existence of a solution for general subquadratic Hamiltonians.