ON A DIFFERENTIAL GENERALIZED NASH EQUILIBRIUM PROBLEM WITH MEAN FIELD INTERACTION

We consider a class of N-player linear quadratic differential generalized Nash equilibrium problems (GNEPs) with bound constraints on the individual control and state variables. In addition, we assume the individual players' optimal control problems are coupled through their dynamics and objectives via a time-dependent mean field interaction term. This assumption allows us to model the realistic setting that strategic players in large games cannot observe the individual states of their competitors. We observe that the GNEPs require a constraint qualification, which necessitates sufficient robustness of the individuals, in order to prove the existence of an open-loop pure strategy Nash equilibrium and to derive optimality conditions. In order to gain qualitative insight into the N-player game, we assume that players are identical and pass to the limit in N to derive a type of first-order constrained mean field game (MFG). We prove that the mean field interaction terms converge to an absolutely continuous curve of probability measures on the set of possible state trajectories. Using variational convergence methods, we show that the optimal control problems converge to a representative agent problem. Under additional regularity assumptions, we provide an explicit form for the mean field term as the solution of a continuity equation and demonstrate the link back to the N-player GNEP.