Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Leader-Follower Mean Field LQG Games with Multiplicative Noise

Bing-Chang Wang, Huanshui Zhang, Ji-Feng Zhang

TL;DR

The paper studies leader-follower mean-field LQG games with multiplicative noise, employing a direct approach to derive open-loop and decentralized feedback strategies. Followers’ social optimization is solved via MF forward-backward SDEs, with mean-field approximations enabling tractable follower controls; the leader’s problem is then solved through decoupling and matrix maximum principle, yielding explicit strategies and Riccati-based cost characterizations. The results establish conditions for existence and provide $(oldsymbol{ ext{ε}}_1,oldsymbol{ ext{ε}}_2)$-Stackelberg equilibria with explicit convergence rates in the large-population limit, along with simulations illustrating MF effects. A key novelty is the appearance of cross terms due to MF coupling and the handling of indefinite costs, offering practical design insights for large-scale hierarchical stochastic systems with multiplicative disturbances.

Abstract

This paper studies open-loop and feedback solutions to leader-follower mean field linear-quadratic-Gaussian games with multiplicative noise by the direct approach. The leader-follower game involves a leader and many followers, where the state and control weight matrices in their costs are not limited to be positive definite. From variational analysis with mean field approximations, we obtain a set of open-loop controls in terms of solutions to mean field forward-backward stochastic differential equations. By applying the matrix maximum principle, a set of decentralized feedback strategies is constructed. Distinct from traditional works, a cross term has appeared in derivation due to the presence of mean field terms. For open-loop and feedback solutions, the corresponding optimal costs of all players are explicitly given in terms of the solutions to two Riccati equations, respectively.

Leader-Follower Mean Field LQG Games with Multiplicative Noise

TL;DR

The paper studies leader-follower mean-field LQG games with multiplicative noise, employing a direct approach to derive open-loop and decentralized feedback strategies. Followers’ social optimization is solved via MF forward-backward SDEs, with mean-field approximations enabling tractable follower controls; the leader’s problem is then solved through decoupling and matrix maximum principle, yielding explicit strategies and Riccati-based cost characterizations. The results establish conditions for existence and provide -Stackelberg equilibria with explicit convergence rates in the large-population limit, along with simulations illustrating MF effects. A key novelty is the appearance of cross terms due to MF coupling and the handling of indefinite costs, offering practical design insights for large-scale hierarchical stochastic systems with multiplicative disturbances.

Abstract

This paper studies open-loop and feedback solutions to leader-follower mean field linear-quadratic-Gaussian games with multiplicative noise by the direct approach. The leader-follower game involves a leader and many followers, where the state and control weight matrices in their costs are not limited to be positive definite. From variational analysis with mean field approximations, we obtain a set of open-loop controls in terms of solutions to mean field forward-backward stochastic differential equations. By applying the matrix maximum principle, a set of decentralized feedback strategies is constructed. Distinct from traditional works, a cross term has appeared in derivation due to the presence of mean field terms. For open-loop and feedback solutions, the corresponding optimal costs of all players are explicitly given in terms of the solutions to two Riccati equations, respectively.

Paper Structure

This paper contains 16 sections, 13 theorems, 146 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Contribution and Novelty
  4. Organization and Notation
  5. Problem Formulation
  6. Open-loop Solutions to Leader-Follower MF Games
  7. The MF Social Control Problem for $N$ Followers
  8. Optimization for the Leader
  9. Feedback Solutions to MF Leader-Follower Games
  10. The MF Social Control Problem for $N$ Followers
  11. Optimization for the Leader
  12. Simulation
  13. Concluding Remarks
  14. Proof of Theorems \ref{['thm3.1']} and \ref{['thm4.2']}
  15. Proof of Theorem \ref{['thm3.5']}
  16. ...and 1 more sections

Key Result

Theorem 3.1

\newlabelthm3.10 Problem (P1) admits an optimal control if and only if ${J}_{\rm soc}$ is convex in $u$ and the following system of FBSDEs admits a set of adapted solutions $\{x_i,p_i,q_i^j,i,j=1,\cdots,N\}$: where $p^{(N)}=\frac{1}{N}\sum_{j=1}^Np_j$, $q^{(N)}=\frac{1}{N}\sum_{j=1}^Nq_j^j$, and the optimal control laws of followers $\check{u}_i$ satisfy

Figures (4)

  • Figure 1: The solution to the Riccati equation \ref{['eq18']}, and $P_{i,j}$ is the entry in $i$th row $j$th column of $\mathcal{P}$.
  • Figure 2: The solutions to \ref{['eq73']} and \ref{['eq832']}.
  • Figure 3: Followers' state averages and MF effects under open-loop and feedback controls.
  • Figure 4: States of the leader under open-loop and feedback controls.

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • Remark 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Proposition 3.6
  • Remark 3.7
  • Theorem 3.8
  • ...and 10 more