Mean field multilayer stackelberg differential games in multi-agent systems

Abstract

In this paper, we study linear-quadratic hierarchical mean field Stackelberg differential games with decentralized adapted open-loop information structure. In this game model, there are three levels of decision making, with a leader at the top level, sub-leaders at the intermediate level, and a large population of followers at lowest level. Accordingly, the leader cannot influence the followers' actions directly, but instead sub-leaders link up followers to the global leader as an intermediate layer. The leader plays a Stackelberg game with the sub-leaders, and the sub-leaders play a Stackelberg game of the mean field type with the followers. The followers are (weakly) coupled through a mean field term, which only affects the followers' individual costs. One of the contributions of this work is to consider the infinite population limit of the finite-follower multi-layer game model. We establish the existence of Stackelberg equilibrium in the limiting case, which is expected to be an approximate Stackelberg equilibrium by the law of large numbers when the population of followers is finite, but sufficiently large. We show the effectiveness of the proposed method through a numerical example.