Mean field game and team with stochastic leader-follower interaction

Abstract

The thesis is concerned with the application of mean field game (MFG) and mean field team (MFT) in the leader-follower (LF) interaction. We first introduce the LF game and MFT, separately, which can be treated as two preliminary chapters, then the LF game combine with MFT and MFG are investigated. More details about the four topics in this thesis are introduced as follows. The first topic studies a mixed linear quadratic (LQ) stochastic LF game with input constraint, where the model involves two agents with the same hierarchy in decision making and each agent has two controls which act as a leader and a follower, respectively. By solving a follower problem, we obtain a Nash equilibrium. Then a leader problem with constrained controls is tackled and the optimal controls are presented by projection mappings. Moreover, we consider the case that the control weights are singular. In this case, a sufficient condition for the uniform convexity of the cost functional is given and a minimizing sequence of solutions with non-degenerate control weights is constructed to investigate the weak convergence of the corresponding personal cost functionals. The second topic investigates the robust LQ MFT control under a direct approach, where a global uncertainty drift is involved for a large number of weakly-coupled interactive agents. All agents treat the uncertainty as an adversarial agent to obtain a "worst case" disturbance. Using variational analysis, we first obtain the centralized controls by a set of forward-backward stochastic differential equations. Then the decentralized controls are designed by mean field heuristics. Finally, the proof of asymptotically social optimality is given. The third topic combines the LF problem and the MFT problem, which involves one leader and a large number of weakly-coupled interactive followers. All agents cooperate to optimize the social cost functional. Unlike the second topic, we apply the fixed point approach in this topic to solve the problem and obtain a set of decentralized social optimality strategies (the asymptotical Stackelberg equilibrium) through a consistency condition (CC) system. The fourth topic is a new game by combing three factors: hierarchical structure for iterative decision, model uncertainty with asymmetric information, and weak-coupling in a large population system. In particular, two classes of agents involved are denoted as leaders and followers, who sequentially make decisions with a hierarchical structure. As a consequence, the information structures between different hierarchies become asymmetric due to their iterative positions. Model uncertainty then arises in their decisions since the lacking of communication among non-cooperative leaders/followers. Moreover, all agents are framed within a weakly-coupled large population system with complex interrelations. Thus, leaders or followers play a Nash game with each other in their own hierarchy, while leaders and followers play a Stackelberg game between the two hierarchies. Applying the MFG theory, we obtain an asymptotic Stackelberg-Nash-Cournot equilibrium based on a CC system. The well-posedness of such consistency is derived by the fixed point analysis under mild conditions.