Large population games with interactions through controls and common noise: convergence results and equivalence between open-loop and closed-loop controls

In the presence of a common noise, we study the convergence problems in mean field game (MFG) and mean field control (MFC) problem where the cost function and the state dynamics depend upon the joint conditional distribution of the controlled state and the control process. In the first part, we consider the MFG setting. We start by recalling the notions of measure-valued MFG equilibria and of approximate closed-loop Nash equilibria associated to the corresponding N-player game. Then, we show that all convergent sequences of approximate closed-loop Nash equilibria, when N -> infinity, converge to measure-valued MFG equilibria. And conversely, any measure-valued MFG equilibrium is the limit of a sequence of approximate closed-loop Nash equilibria. In other words, measure-valued MFG equilibria are the accumulation points of the approximate closed-loop Nash equilibria. Previous work has shown that measure-valued MFG equilibria are the accumulation points of the approximate openloop Nash equilibria. Therefore, we obtain that the limits of approximate closed-loop Nash equilibria and approximate open-loop Nash equilibria are the same. In the second part, we deal with the MFC setting. After recalling the closed-loop and open-loop formulations of the MFC problem, we prove that they are equivalent. We also provide some convergence results related to approximate closed-loop Pareto equilibria.