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Dynamic Set Values for Nonzero-Sum Games with Multiple Equilibriums

Author

Listed:
  • Zachary Feinstein

    (Stevens Institute of Technology, School of Business, Hoboken, New Jersey 07030)

  • Birgit Rudloff

    (Vienna University of Economics and Business, Institute for Statistics and Mathematics, 1020 Vienna, Austria)

  • Jianfeng Zhang

    (Department of Mathematics, University of Southern California, Los Angeles, California 90089)

Abstract

Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero-sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value ∅ ). Similar to the standard value function in control literature, it enjoys many nice properties, such as regularity, stability, and more importantly, the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls); and (ii) we must allow for path dependent controls, even if the problem is in a state-dependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter, we will also provide a duality approach through certain standard PDE (or path-dependent PDE), which is quite efficient for numerically computing the set value of the game.

Suggested Citation

  • Zachary Feinstein & Birgit Rudloff & Jianfeng Zhang, 2022. "Dynamic Set Values for Nonzero-Sum Games with Multiple Equilibriums," Mathematics of Operations Research, INFORMS, vol. 47(1), pages 616-642, February.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:1:p:616-642
    DOI: 10.1287/moor.2021.1143
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    References listed on IDEAS

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    Cited by:

    1. H. Dharma Kwon & Jan Palczewski, 2025. "Exit Game with Private Information," Mathematics of Operations Research, INFORMS, vol. 50(4), pages 2433-2469, November.
    2. Atiqah Almuzaini & c{C}au{g}{i}n Ararat & Jin Ma, 2025. "Superhedging under Proportional Transaction Costs in Continuous Time," Papers 2511.18169, arXiv.org.
    3. Zachary Feinstein, 2022. "Continuity and sensitivity analysis of parameterized Nash games," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 10(2), pages 233-249, October.
    4. Bixing Qiao & Jianfeng Zhang, 2025. "A New Approach for the Continuous Time Kyle-Back Strategic Insider Equilibrium Problem," Papers 2506.12281, arXiv.org.
    5. Gabriela Kováčová & Birgit Rudloff, 2025. "Approximations of unbounded convex projections and unbounded convex sets," Journal of Global Optimization, Springer, vol. 91(4), pages 787-805, April.
    6. H. Dharma Kwon & Jan Palczewski, 2022. "Exit game with private information," Papers 2210.01610, arXiv.org, revised Oct 2023.

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