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Closed-Loop Nash Equilibrium in the Class of Piecewise Constant Strategies in a Linear State Feedback Form for Stochastic LQ Games

Author

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  • Vasile Drăgan

    (“Simion Stoilow” Institute of Mathematics, Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
    The Academy of the Romanian Scientists, Str. Ilfov, 3, 050044 Bucharest, Romania
    These authors contributed equally to this work.)

  • Ivan Ganchev Ivanov

    (Faculty of Economics and Business Administration, Sofia University St. Kliment Ohridski, 1113 Sofia, Bulgaria
    These authors contributed equally to this work.)

  • Ioan-Lucian Popa

    (Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania
    These authors contributed equally to this work.)

  • Ovidiu Bagdasar

    (School of Computing and Engineering, University of Derby, Derby DE22 1GB, UK
    These authors contributed equally to this work.)

Abstract

In this paper, we examine a sampled-data Nash equilibrium strategy for a stochastic linear quadratic (LQ) differential game, in which admissible strategies are assumed to be constant on the interval between consecutive measurements. Our solution first involves transforming the problem into a linear stochastic system with finite jumps. This allows us to obtain necessary and sufficient conditions assuring the existence of a sampled-data Nash equilibrium strategy, extending earlier results to a general context with more than two players. Furthermore, we provide a numerical algorithm for calculating the feedback matrices of the Nash equilibrium strategies. Finally, we illustrate the effectiveness of the proposed algorithm by two numerical examples. As both situations highlight a stabilization effect, this confirms the efficiency of our approach.

Suggested Citation

  • Vasile Drăgan & Ivan Ganchev Ivanov & Ioan-Lucian Popa & Ovidiu Bagdasar, 2021. "Closed-Loop Nash Equilibrium in the Class of Piecewise Constant Strategies in a Linear State Feedback Form for Stochastic LQ Games," Mathematics, MDPI, vol. 9(21), pages 1-15, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2713-:d:664734
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    References listed on IDEAS

    as
    1. Engwerda, J. C., 1998. "Computational aspects of the open-loop Nash equilibrium in linear quadratic games," Journal of Economic Dynamics and Control, Elsevier, vol. 22(8-9), pages 1487-1506, August.
    2. Drăgan, Vasile & Ivanov, Ivan G. & Popa, Ioan-Lucian, 2020. "On the closed loop Nash equilibrium strategy for a class of sampled data stochastic linear quadratic differential games," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    3. Engwerda, Jacob C., 1998. "On the open-loop Nash equilibrium in LQ-games," Journal of Economic Dynamics and Control, Elsevier, vol. 22(5), pages 729-762, May.
    4. Sun, Jingrui & Yong, Jiongmin, 2019. "Linear–quadratic stochastic two-person nonzero-sum differential games: Open-loop and closed-loop Nash equilibria," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 381-418.
    5. Dockner,Engelbert J. & Jorgensen,Steffen & Long,Ngo Van & Sorger,Gerhard, 2000. "Differential Games in Economics and Management Science," Cambridge Books, Cambridge University Press, number 9780521637329.
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