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Marginal Values of a Stochastic Game

Author

Listed:
  • Luc Attia

    (Université Paris-Dauphine, Paris Sciences and Lettres Research University, Centre National de la Recherche Scientifique, Centre de Recherche en Mathématiques de la Décision, 75775 Paris, France)

  • Miquel Oliu-Barton

    (Université Paris-Dauphine, Paris Sciences and Lettres Research University, Centre National de la Recherche Scientifique, Centre de Recherche en Mathématiques de la Décision, 75775 Paris, France)

  • Raimundo Saona

    (Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria)

Abstract

Zero-sum stochastic games are parameterized by payoffs, transitions, and possibly a discount rate. In this article, we study how the main solution concepts, the discounted and undiscounted values, vary when these parameters are perturbed. We focus on the marginal values, introduced by Mills in 1956 in the context of matrix games—that is, the directional derivatives of the value along any fixed perturbation. We provide a formula for the marginal values of a discounted stochastic game. Further, under mild assumptions on the perturbation, we provide a formula for their limit as the discount rate vanishes and for the marginal values of an undiscounted stochastic game. We also show, via an example, that the two latter differ in general.

Suggested Citation

  • Luc Attia & Miquel Oliu-Barton & Raimundo Saona, 2025. "Marginal Values of a Stochastic Game," Mathematics of Operations Research, INFORMS, vol. 50(1), pages 482-505, February.
    • Handle: RePEc:inm:ormoor:v:50:y:2025:i:1:p:482-505
    DOI: 10.1287/moor.2023.0297
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    References listed on IDEAS

    as
    1. Truman Bewley & Elon Kohlberg, 1978. "On Stochastic Games with Stationary Optimal Strategies," Mathematics of Operations Research, INFORMS, vol. 3(2), pages 104-125, May.
    2. Bruno Ziliotto, 2016. "A Tauberian Theorem for Nonexpansive Operators and Applications to Zero-Sum Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 41(4), pages 1522-1534, November.
    3. Miquel Oliu-Barton, 2021. "New Algorithms for Solving Zero-Sum Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 255-267, February.
    4. Miquel Oliu-Barton, 2018. "The Splitting Game: Value and Optimal Strategies," Dynamic Games and Applications, Springer, vol. 8(1), pages 157-179, March.
    5. Miquel Oliu-Barton, 2022. "Weighted-average stochastic games with constant payoff," Operational Research, Springer, vol. 22(3), pages 1675-1696, July.
    6. Guillaume Vigeral, 2013. "A Zero-Sum Stochastic Game with Compact Action Sets and no Asymptotic Value," Dynamic Games and Applications, Springer, vol. 3(2), pages 172-186, June.
    7. Truman Bewley & Elon Kohlberg, 1976. "The Asymptotic Theory of Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 1(3), pages 197-208, August.
    8. Luc Attia & Miquel Oliu-Barton, 2021. "Shapley–Snow Kernels, Multiparameter Eigenvalue Problems, and Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 46(3), pages 1181-1202, August.
    9. Luc Attia & Miquel Oliu-Barton, 2019. "A formula for the value of a stochastic game," Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, vol. 116(52), pages 26435-26443, December.
    10. Eilon Solan, 2003. "Continuity of the Value of Competitive Markov Decision Processes," Journal of Theoretical Probability, Springer, vol. 16(4), pages 831-845, October.
    11. Miquel Oliu-Barton, 2014. "The Asymptotic Value in Finite Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 39(3), pages 712-721, August.
    12. repec:dau:papers:123456789/10880 is not listed on IDEAS
    13. Abraham Neyman & Sylvain Sorin, 2010. "Repeated games with public uncertain duration process," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 29-52, March.
    Full references (including those not matched with items on IDEAS)

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