Mean field games of controls: On the convergence of Nash equilibria

Abstract

In this paper, we provide convergence and existence results for mean field games of controls. Mean field games of controls are a class of mean field games where the mean field interactions are achieved through the joint (conditional) distribution of the controlled state and the control process. The framework we are considering allows to control the diffusion coefficient σ, and the controls/strategies are supposed to be of open loop type. Using (controlled) Fokker–Planck equations, we introduce a notion of measure-valued solution of mean field game of controls and prove a relation between these solutions on the one hand, and the approximate Nash equilibria on the other hand. First of all, in the N-player game associated to the mean field game of controls, given a sequence of approximate Nash equilibria, it is shown that, this sequence admits limits as N tends to infinity, and each limit is a measure-valued solution of the corresponding mean field game of controls. Conversely, any measure-valued solution can be obtained as the limit of a sequence of approximate Nash equilibria of the N-player game. In other words, the measure-valued solutions are the accumulation points of the approximate Nash equilibria. Then, by considering an approximate strong solution of mean field game of controls which is the classical strong solution where the optimality is obtained by admitting a small error ε, we prove that the measure-valued solutions are the accumulation points of this type of solutions when ε goes to zero. Finally, the existence of a measure-valued solution of mean field game of controls is proved in the case without common noise.