Existence of weak solutions to stationary mean-field games through variational inequalities

TL;DR

This paper proves the existence of weak solutions for stationary monotone mean-field games using variational inequalities and Minty's method, providing a general framework applicable to various types of MFGs.

Contribution

The paper introduces a regularized problem and employs variational inequalities and Minty's method to establish the existence of weak solutions for stationary MFGs, extending the theoretical understanding.

Findings

Existence of weak solutions for stationary MFGs proven.

Framework applicable to local, nonlocal, and congestion terms.

Properties of solutions analyzed in several examples.

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 inequalities 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.