Linear quadratic mean field Stackelberg differential games

We consider linear–quadratic mean field Stackelberg differential games with the adapted open-loop information structure of the leader. There are one leader and N followers, where N is arbitrarily large. The leader holds a dominating position in the game in the sense that the leader first chooses and then announces the optimal strategy to which the N followers respond by playing a Nash game, i.e., choosing their optimal strategies noncooperatively and simultaneously based on the leader's observed strategy. In our setting, the followers are coupled with each other through the mean field term included in their cost functions, and are strongly influenced by the leader's open-loop strategy included in their cost functions and dynamics. From the leader's perspective, he is coupled with the N followers through the mean field term included in his cost function. To circumvent the complexity brought about by the coupling nature among the leader and the followers with large N, which makes the use of the direct approach almost impossible, our approach in this paper is to characterize an approximated stochastic mean field process by solving a local optimal control problem of the followers with leader's control taken as an exogenous stochastic process. We show that for each fixed strategy of the leader, the followers’ local optimal decentralized strategies lead to an ϵ-Nash equilibrium. The paper then solves the leader's local optimal control problem, as a nonstandard constrained optimization problem, with constraints being induced by the approximated mean field process determined by Nash followers (which also depend on the leader's control). We show that the local optimal decentralized controllers for the leader and the followers constitute an (ϵ₁,ϵ₂)-Stackelberg–Nash equilibrium for the original game, where ϵ₁ and ϵ₂ both converge to zero as N→∞. Numerical examples are provided to illustrate the theoretical results.