Linear Quadratic Mean Field Stackelberg Games: Open-Loop and Feedback Solutions

This article investigates open-loop and feedback solutions to linear quadratic mean field (MF) games with a leader and many followers. The leader first gives its strategy and then all the followers cooperate to optimize the social cost as the sum of their costs. By variational analysis with MF approximations, we obtain a set of open-loop controls of players in terms of solutions to MF forward-backward stochastic differential equations (FBSDEs), which is further shown be to an asymptotic Stackelberg-team equilibrium. By applying the matrix maximum principle, a set of decentralized feedback strategies is constructed for all the players. For open-loop and feedback solutions, the corresponding costs of all players are explicitly given by virtue of the solutions to two Riccati equations, respectively. The performances of two solutions are compared by the numerical simulation.