LINEAR-QUADRATIC MEAN FIELD STACKELBERG GAMES WITH STATE AND CONTROL DELAYS

In this article, we consider a linear-quadratic mean field game between a leader (dominating player) and a group of followers (agents) under the Stackelberg game setting as proposed in [A. Bensoussan, M. Chau, and S. Yam, Appl. Math. Optim., 74 (2016), pp. 91{128], so that the evolution of each individual follower is now also subjected to delay effects from both his/her state and control variables, as well as those of the leader. The overall Stackelberg game is solved by tackling three subproblems hierarchically. Their resolution corresponds to the establishment of the existence and uniqueness of the solutions of three different forward-backward stochastic functional differential equations, which we manage by applying the unified continuation method as first developed in, for example, [Y. Hu and S. Peng, Probab. Theory Related Fields, 103 (1995), pp. 273{283] and [X. Xu, Fully Coupled Forward-Backward Stochastic Functional Differential Equations and Applications to Quadratic Optimal Control, preprint, arXiv: 1310.6846, 2013]. In particular, by first regarding the mean field term and the delay influence of the leader as exogenous, we use the adjoint equation approach to solve the optimal control of each follower. Next, we utilize the fixed point property to get the desired mean field equilibrium, with which we propose a time independent sufficient condition that warrants its existence and uniqueness. Finally, we solve the optimal control of the leader and conclude that its presence would not interfere with the original existence of the equilibrium of the community. Our present setting introduces a much different challenge than that in the former work [A. Bensoussan, M. Chau, and S. Yam, Appl. Math. Optim., 74 (2016), pp. 91{128], which stems from the delay effects from each follower's state and control, which result in a more complicated system of in finite dimensional forward-backward stochastic functional differential equations. To find the optimal control of the leader, we need to develop a thorough understanding of a special linear operator resulting from the delay effects of the leader's control on the dynamics of each follower, which hinges on the invertibility of another certain functional operator. This was not considered in [A. Bensoussan, M. Chau, and S. Yam, Appl. Math. Optim., 74 (2016), pp. 91{128].