Potential Mean-Field Games and Gradient Flows

We consider a mean-field optimal control problem with general dynamics including common noise and jumps and show that its minimizers are Nash equilibria of an associated mean-field game of controls. These types of games are necessarily potential, and the Nash equilibria obtained as the minimizers of the control problems are naturally related to the McKean-Vlasov equations of Langevin type. We provide several examples to motivate the general theory including the Cucker-Smale Flocking and Kuramoto mean-field games. The invariance property of the value function of these control problems, utilized in our proof, is also proved.