Linear-Quadratic Mean Field Stackelberg Stochastic Differential Game with Partial Information and Common Noise

TL;DR

This paper develops a framework for a linear-quadratic mean field Stackelberg stochastic differential game with partial information and common noise, deriving decentralized strategies and equilibria in a leader-follower setting.

Contribution

It introduces a novel approach combining stochastic maximum principle and filtering to solve leader-follower games with partial information and common noise.

Findings

Decentralized strategies are derived using stochastic maximum principle.

The strategies form an $ ext{epsilon}$-Stackelberg-Nash equilibrium.

The framework accounts for the influence of followers' states on the leader's dynamics.

Abstract

This paper is concerned with a linear-quadratic mean field Stackelberg stochastic differential game with partial information and common noise, which contains a leader and a large number of followers. To be specific, the followers face a large population Nash game after the leader first announces his strategy, while the leader will then optimize his own cost functional on consideration of the followers' reactions. The state equation of the leader and followers are both general stochastic differential equations, where the diffusion terms contain both the control and state variables. However, the followers' average state terms enter into the drift term of the leader's state equation, reflecting that the leader's state is influenced by the followers' states. By virtue of stochastic maximum principle with partial information and optimal filter technique, we deduce the open-loop adapted decentralized strategies and feedback decentralized strategies of this leader-followers system, and demonstrate that the decentralized strategies are the corresponding $\varepsilon$-Stackelberg-Nash equilibrium.