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

Yu Si, Jingtao Shi

TL;DR

The paper addresses a linear-quadratic mean-field Stackelberg stochastic differential game with partial information and common noise, involving a leader and a large follower population. It develops a limiting mean-field formulation as $N\to\infty$ with a frozen limit $z(t)$ and derives open-loop and state-feedback decentralized strategies for both followers and the leader using the stochastic maximum principle and optimal filtering, under partial information. Central to the approach are CMF-FBSDEs and Riccati equations that yield explicit follower and leader strategies, with a dimension-expansion technique delivering a feedback form for the leader. The main result proves that these decentralized strategies constitute an $\\varepsilon$-Stackelberg-Nash equilibrium for the original finite-population game, with rigorous error bounds showing mean-field convergence, and numerical experiments corroborating the theoretical findings.

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.

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

TL;DR

The paper addresses a linear-quadratic mean-field Stackelberg stochastic differential game with partial information and common noise, involving a leader and a large follower population. It develops a limiting mean-field formulation as with a frozen limit and derives open-loop and state-feedback decentralized strategies for both followers and the leader using the stochastic maximum principle and optimal filtering, under partial information. Central to the approach are CMF-FBSDEs and Riccati equations that yield explicit follower and leader strategies, with a dimension-expansion technique delivering a feedback form for the leader. The main result proves that these decentralized strategies constitute an -Stackelberg-Nash equilibrium for the original finite-population game, with rigorous error bounds showing mean-field convergence, and numerical experiments corroborating the theoretical findings.

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 -Stackelberg-Nash equilibrium.
Paper Structure (10 sections, 12 theorems, 133 equations, 4 figures)

This paper contains 10 sections, 12 theorems, 133 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Limiting Stackelberg-Nash equilibria
  4. Open-loop decentralized strategies of the followers
  5. State feedback decentralized strategies of the followers
  6. Open loop decentralized strategy of the leader
  7. Feedback decentralized strategy of the leader
  8. $\varepsilon$-Stackelberg-Nash equilibria analysis
  9. Numerical examples
  10. Conclusion

Key Result

Theorem 3.1

Let Assumptions A2.1 and A2.2 hold. For $i=1,2, \cdots$ and any given $u_0(\cdot) \in \mathcal{U}_0^{l,d}$, suppose $\bar{x}_0(\cdot)$ is the solution to (leader limiting state), then the open-loop decentralized optimal strategy of $i$-th follower $\mathcal{A}_i$ is given by where we denote $\hat{p}_i:=\mathbb{E}\left[\left.p_i\right\rvert \mathcal{F}_t^{W_i}\right]$, $\hat{q}_i:=\mathbb{E}\left[

Figures (4)

  • Figure 1: The solution curve of $P_1,P_2$
  • Figure 2:
  • Figure 3: The solution curve of $x^{(N)}$, $z$
  • Figure 4: The solution curve of $\epsilon$

Theorems & Definitions (27)

  • Theorem 3.1
  • proof
  • Remark 3.1
  • Proposition 3.1
  • Remark 3.2
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Remark 3.3
  • Theorem 3.4
  • ...and 17 more