Nonlinear Open-Loop Mean field Stackelberg Stochastic Differential Game
Jianhui Huang, Qi Huang
TL;DR
This work analyzes nonlinear open-loop mean field Stackelberg stochastic differential games by framing the leader–follower interaction through fully coupled conditional mean-field forward–backward SDEs and solving via a fixed-point approach. It constructs decentralized optimal control problems for followers and the leader, derives corresponding maximum principles, and proves well-posedness for a novel CMF-FBSDE to obtain existence, uniqueness, and estimates. The decentralized controls are shown to form an $\varepsilon$-Stackelberg equilibrium as the number of followers grows, with two fixed-point schemes and propagation-chaos techniques underpinning the limit analysis. An applied example to a robot control center and a swarm of unicycle-type robots demonstrates the practicality and relevance of the nonlinear mean field Stackelberg framework for complex hierarchical multi-agent systems.
Abstract
This paper studies a nonlinear open-loop mean field Stackelberg stochastic differential game by using the probabilistic method through the FBSDE system and the idea of taking control as the fixed point. We successively construct the decentralized optimal control problems for the followers and the leader, among which the leader's decentralized optimal control problem is a partial information optimal control problem with the fully coupled conditional mean-field forward-backward stochastic differential equation (FBSDE, in short) as the state equation. We successively derive the maximum principles for the corresponding decentralized optimal control problems of the followers and the leader. To obtain the existence, uniqueness and estimations of solutions of the state equation, the variational equation and the adjoint equation for the leader's decentralized optimal control problem, we study the well-posedness of a new form of conditional mean-field FBSDE. And the decentralized optimal controls of the leader and followers are proved to be the approximate Stackelberg equilibrium of the nonlinear mean field Stackelberg game. Finally, we apply the theoretical results developed in this paper to solve a nonlinear mean field Stackelberg game problem between a robot control center and unicycle-type swarm robots.