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Nonzero-Sum Stochastic Games and Mean-Field Games with Impulse Controls

Author

Listed:
  • Matteo Basei

    (Department of Industrial Engineering and Operations Research, University of California, Berkeley, Berkeley, California 94720-1777; EDF R&D, 75008 Paris, France)

  • Haoyang Cao

    (Department of Industrial Engineering and Operations Research, University of California, Berkeley, Berkeley, California 94720-1777; The Alan Turing Institute, British Library, NW1 2DB London, United Kingdom)

  • Xin Guo

    (Department of Industrial Engineering and Operations Research, University of California, Berkeley, Berkeley, California 94720-1777)

Abstract

We consider a general class of nonzero-sum N -player stochastic games with impulse controls, where players control the underlying dynamics with discrete interventions. We adopt a veri?cation approach and provide su?cient conditions for the Nash equilibria (NEs) of the game. We then consider the limiting situation when N goes to in?nity, that is, a suitable mean-?eld game (MFG) with impulse controls. We show that under appropriate technical conditions, there exists a unique NE solution to the MFG, which is an ϵ-NE approximation to the N -player game, with ϵ = O 1 N . As an example, we analyze in detail a class of two-player stochastic games which extends the classical cash management problem to the game setting. In particular, we present numerical analysis for the cases of the single player, the two-player game, and the MFG, showing the impact of competition on the player’s optimal strategy, with sensitivity analysis of the model parameters.

Suggested Citation

  • Matteo Basei & Haoyang Cao & Xin Guo, 2022. "Nonzero-Sum Stochastic Games and Mean-Field Games with Impulse Controls," Mathematics of Operations Research, INFORMS, vol. 47(1), pages 341-366, February.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:1:p:341-366
    DOI: 10.1287/moor.2021.1131
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    References listed on IDEAS

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    1. Haoyang Cao & Jodi Dianetti & Giorgio Ferrari, 2023. "Stationary Discounted and Ergodic Mean Field Games with Singular Controls," Mathematics of Operations Research, INFORMS, vol. 48(4), pages 1871-1898, November.
    2. Cannerozzi, Federico & Ferrari, Giorgio, 2024. "Cooperation, Correlation and Competition in Ergodic $N$-Player Games and Mean-Field Games of Singular Controls: A Case Study," Center for Mathematical Economics Working Papers 691, Center for Mathematical Economics, Bielefeld University.
    3. Federico Cannerozzi & Giorgio Ferrari, 2024. "Cooperation, Correlation and Competition in Ergodic N-player Games and Mean-field Games of Singular Controls: A Case Study," Papers 2404.15079, arXiv.org, revised Apr 2025.
    4. Nacira Agram & Giulia Pucci & Bernt Øksendal, 2024. "Impulse Control of Conditional McKean–Vlasov Jump Diffusions," Journal of Optimization Theory and Applications, Springer, vol. 200(3), pages 1100-1130, March.
    5. Guohui Guan & Zongxia Liang & Yi Xia, 2024. "Many-insurer robust games of reinsurance and investment under model uncertainty in incomplete markets," Papers 2412.09157, arXiv.org.
    6. Xie, Meiling & Li, Yuhan & Feng, Minyu & Kurths, Jürgen, 2023. "Contact-dependent infection and mobility in the metapopulation SIR model from a birth–death process perspective," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    7. Guohui Guan & Zongxia Liang & Yi Xia, 2025. "Many-insurer robust games of reinsurance and investment under model uncertainty in incomplete markets," Mathematics and Financial Economics, Springer, volume 19, number 4, December.
    8. Sadana, Utsav & Zaccour, Georges, 2026. "Subsidizing a new technology: An impulse Stackelberg game approach," European Journal of Operational Research, Elsevier, vol. 329(3), pages 1015-1029.

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