An introduction to monotonicity methods in mean-field games

TL;DR

This chapter reviews monotonicity techniques in mean-field games, highlighting their use in proving solution uniqueness, constructing numerical methods, and establishing weak solutions for both stationary and time-dependent cases.

Contribution

It introduces the Minty method and regularization strategies to address monotone MFGs, providing new approaches for existence and solution analysis.

Findings

Established existence of weak solutions for stationary and time-dependent MFGs

Applied Minty method and regularization to PDEs in MFGs

Demonstrated monotonicity techniques' effectiveness in MFG analysis

Abstract

This chapter examines monotonicity techniques in the theory of mean-field games(MFGs). Originally, monotonicity ideas were used to establish the uniqueness of solutions for MFGs. Later, monotonicity methods and monotone operators were further exploited to build numerical methods and to construct weak solutions under mild assumptions. Here, after a brief discussion on the mean-field game formulation, we introduce the Minty method and regularization strategies for PDEs. These are then used to address typical stationary and time-dependent monotone MFGs and to establish the existence of weak solutions for such MFGs.