Optimal control and potential games in the mean field

We study a mean-field optimal control problem with general non-Markovian dynamics that allow both common noise and jumps. We show that its minimizers are Nash equilibria of an associated mean field game of controls. These types of games are potential, and the resulting Nash equilibria are closely linked to McKean-Vlasov equations of Langevin type. Furthermore, as a byproduct of our main result, we obtain the existence of strong common noise equilibria for potential games. To illustrate the general theory, we present several examples, including a mean field game of controls with interactions through a price variable, as well as mean-field Cucker-Smale Flocking and Kuramoto models. We also establish the invariance property of the value function, which is a key ingredient used in our proofs.