Existence of Weak Solutions to Stationary Mean-Field Games through Variational Inequalities

Abstract

Here, we consider stationary monotone mean-field games (MFGs) and study the existence of weak solutions. First, we introduce a regularized problem that preserves the monotonicity. Next, using variational inequality techniques, we prove the existence of solutions to the regularized problem. Then, using Minty's method, we establish the existence of solutions for the original MFG. Finally, we examine the properties of these weak solutions in several examples. Our methods provide a general framework to construct weak solutions to stationary MFGs with local, nonlocal, or congestion terms.

A remark on Uzawa’s algorithm and an application to mean field games systems

“…Let us remark that although the Hamilton-Jacobi-Bellmann equation in (10) is a quasi-variational inequality, the equation we have to solve at each iteration in (12) to update the lagrange multiplier u n is a variational inequality, which is in principle easier to solve than a quasi-variational inequality.…”

“…This results can be found in [5]. The iterations (12) are then the ones from the use of the classical Uzawa's algorithm on L.…”

“…The main novelties of our work is to consider the non potential case and that we consider the cases of optimal stopping and impulse control. Furthermore, we mention the papers [12,2] of R. Ferreira, D. Gomes and al. in which the first order MFG system of continuous control is interpreted as a system of variational inequalities and solve numerically.…”

Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions

“…The theory for first-order MFGs is less developed. The existence of solutions for first or second-order stationary MFGs was examined in [16] (also see [2]) using monotone operators and, using a variational approach, certain first-order MFGs with congestion were examined in [12].…”

“…In [16], the method of continuity was used to prove the existence of a weak solution to stationary monotone MFGs with periodic boundary conditions. Here, we use a different approach: we apply Schaeffer's fixed-point theorem and extend the results in [16] to Dirichlet boundary conditions.…”