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Robust Incentive Stackelberg Mean Field Stochastic Linear-Quadratic Differential Game with Model Uncertainty

Na Xiang, Jingtao Shi

Abstract

This paper investigates a robust incentive Stackelberg stochastic differential game problem for a linear-quadratic mean field system, where the model uncertainty appears in the drift term of the leader's state equation. Moreover, both the state average and control averages enter into the leader's dynamics and cost functional. Based on the zero-sum game approach, mean field approximation and duality theory, firstly the representation of the leader's limiting cost functional and the closed-loop representation of decentralized open-loop saddle points are given, via decoupling methods. Then by convex analysis and the variational method, the decentralized strategies of the followers' auxiliary limiting problems and the corresponding consistency condition system are derived. Finally, applying decoupling technique, the leader's approximate incentive strategy set is obtained, under which the asymptotical robust incentive optimality of the decentralized mean field strategy is verified. A numerical example is given to illustrate the theoretical results.

Robust Incentive Stackelberg Mean Field Stochastic Linear-Quadratic Differential Game with Model Uncertainty

Abstract

This paper investigates a robust incentive Stackelberg stochastic differential game problem for a linear-quadratic mean field system, where the model uncertainty appears in the drift term of the leader's state equation. Moreover, both the state average and control averages enter into the leader's dynamics and cost functional. Based on the zero-sum game approach, mean field approximation and duality theory, firstly the representation of the leader's limiting cost functional and the closed-loop representation of decentralized open-loop saddle points are given, via decoupling methods. Then by convex analysis and the variational method, the decentralized strategies of the followers' auxiliary limiting problems and the corresponding consistency condition system are derived. Finally, applying decoupling technique, the leader's approximate incentive strategy set is obtained, under which the asymptotical robust incentive optimality of the decentralized mean field strategy is verified. A numerical example is given to illustrate the theoretical results.

Paper Structure

This paper contains 9 sections, 21 theorems, 205 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The limiting leader's team-optimal problem
  4. The auxiliary limiting problem for followers
  5. Asymptotic optimality
  6. Leader's asymptotic robust team optimality
  7. Followers' asymptotic Nash equilibrium
  8. A numerical example
  9. Conclusions

Key Result

Proposition 3.1

Let (A1)-(A2) hold. There exist two bounded self-adjoint linear operators $M_1:L_{\mathcal{G}^0}^2(0,T;\mathbb{R}^{m_L+m_F})\to L_{\mathcal{G}^0}^2(0,T;\mathbb{R}^{m_L+m_F})$, $M_4:\mathcal{U}_{vd}\to \mathcal{U}_{vd}$, bounded operator $M_2:\mathcal{U}_{vd}\to L_{\mathcal{G}^0}^2(0,T;\mathbb{R}^{m_ where with $y^i(\cdot)$, $z^i(\cdot)$, $p^i(\cdot)$, $i=1,2,3$, satisfy the following backward-for

Figures (9)

  • Figure 1: Relationships
  • Figure 2: The numerical solutions of Riccati equations of $P_1(\cdot)$, $\Pi_1(\cdot)$, $P_2(\cdot)$ and $\Pi_2(\cdot)$
  • Figure 3: The optimal state trajectories $x_0^*(\cdot)$ and $m^*(\cdot)$ of Problem (L2)
  • Figure 4: The decentralized optimal control $(\bar{u}^*_0(\cdot),\bar{u}^*_1(\cdot))$ of Problem (L2)
  • Figure 5: The worst-case disturbance $v^*(\cdot)$ for the leader
  • ...and 4 more figures

Theorems & Definitions (52)

  • Remark 2.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.2
  • Definition 3.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 42 more