An introduction to monotonicity methods in mean-field games

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.