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Optimal Linear Closed-Loop Stackelberg Strategy with Asymmetric Information

Author

Listed:
  • Yue Sun

    (Shandong University)

  • Hongdan Li

    (Shandong University of Science and Technology)

  • Juanjuan Xu

    (Shandong University)

  • Huanshui Zhang

    (Shandong University of Science and Technology)

Abstract

In this paper, we consider the optimal linear closed-loop Stackelberg strategy for stochastic system where the state is unknown but observed though linear stochastic measurement systems. Particularly, the information structure is asymmetric where the leader can obtain the follower’s measurement information and its historical control inputs, while the follower can only obtain its own measurement information and historical inputs, which have led to major challenge since the 1970s. It is well known that the closed-loop Stackelberg optimal control in general is a hard problem and no explicit solution can be derived. Instead, this paper aims to present a linear closed-loop solution to the problem. However, how to give a rational definition for the linear closed-loop Stackelberg controller is a difficulty. By using innovation, this paper defines the linear closed-loop Stackelberg controller with specified given gain matrices. Then the analytical solutions for the leader and the follower are given, respectively, which are the feedback form of the Kalman filtering and the optimal gain matrices for the leader and follower are obtained by decoupled solving two Riccati equations. Finally, by using the forward iteration algorithm, the coupled forward and backward Riccati equations are solved in infinite horizon, which overcomes the obstacles that the separation principle becomes invalid.

Suggested Citation

  • Yue Sun & Hongdan Li & Juanjuan Xu & Huanshui Zhang, 2023. "Optimal Linear Closed-Loop Stackelberg Strategy with Asymmetric Information," Journal of Optimization Theory and Applications, Springer, vol. 199(3), pages 1158-1187, December.
    • Handle: RePEc:spr:joptap:v:199:y:2023:i:3:d:10.1007_s10957-023-02316-y
    DOI: 10.1007/s10957-023-02316-y
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    References listed on IDEAS

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    1. Øksendal, Bernt & Sandal, Leif & Ubøe, Jan, 2013. "Stochastic Stackelberg equilibria with applications to time-dependent newsvendor models," Journal of Economic Dynamics and Control, Elsevier, vol. 37(7), pages 1284-1299.
    2. Jørgensen, Steffen & Zaccour, Georges, 2014. "A survey of game-theoretic models of cooperative advertising," European Journal of Operational Research, Elsevier, vol. 237(1), pages 1-14.
    3. Wu, Chengfeng & Liu, Xin & Li, Annan, 2021. "A loss-averse retailer–supplier supply chain model under trade credit in a supplier-Stackelberg game," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 353-365.
    4. Pietro Giovanni & Georges Zaccour, 2019. "A selective survey of game-theoretic models of closed-loop supply chains," 4OR, Springer, vol. 17(1), pages 1-44, March.
    5. A.R. Kian & J.B. Cruz & M. A. Simaan, 2002. "Stochastic Discrete-Time Nash Games with Constrained State Estimators," Journal of Optimization Theory and Applications, Springer, vol. 114(1), pages 171-188, July.
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