IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v204y2025i3d10.1007_s10957-024-02603-2.html
   My bibliography  Save this article

Zero-Sum Semi-Markov Games with the Risk-Sensitive Average Reward Criterion

Author

Listed:
  • Fang Chen

    (Peking University)

  • Xin Guo

    (Sun Yat-Sen University)

Abstract

This paper studies the risk-sensitive average reward criterion for the semi-Markov game with compact state and action spaces. Under some suitable conditions (slightly weaker than the existing ones), we introduce a parametric operator, verify that the corresponding spectral radius is an eigenvalue of it by the nonlinear Krein-Rutman theorem, and further show the continuity of the spectral radius in the parameters. By the continuity and the intermediate value property, we prove that the Shapley equation admits a non-trivial solution, and then establish the existence of the value and a stationary saddle point. Furthermore, we present an iteration algorithm for computing (at least approximating) the value of the game. Finally, we give two examples to illustrate our conditions and algorithm.

Suggested Citation

  • Fang Chen & Xin Guo, 2025. "Zero-Sum Semi-Markov Games with the Risk-Sensitive Average Reward Criterion," Journal of Optimization Theory and Applications, Springer, vol. 204(3), pages 1-30, March.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:3:d:10.1007_s10957-024-02603-2
    DOI: 10.1007/s10957-024-02603-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-024-02603-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-024-02603-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. A. Jaśkiewicz, 2009. "Zero-Sum Ergodic Semi-Markov Games with Weakly Continuous Transition Probabilities," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 321-347, May.
    2. Ghosh, Mrinal K. & Golui, Subrata & Pal, Chandan & Pradhan, Somnath, 2023. "Discrete-time zero-sum games for Markov chains with risk-sensitive average cost criterion," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 40-74.
    3. Andrzej S. Nowak & Anna Jaśkiewicz, 2005. "Nonzero-sum semi-Markov games with the expected average payoffs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 62(1), pages 23-40, September.
    4. Selene Chávez-Rodríguez & Rolando Cavazos-Cadena & Hugo Cruz-Suárez, 2016. "Controlled Semi-Markov Chains with Risk-Sensitive Average Cost Criterion," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 670-686, August.
    5. Bäuerle, Nicole & Rieder, Ulrich, 2017. "Zero-sum risk-sensitive stochastic games," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 622-642.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Bhabak, Arnab & Saha, Subhamay, 2022. "Risk-sensitive semi-Markov decision problems with discounted cost and general utilities," Statistics & Probability Letters, Elsevier, vol. 184(C).
    2. Tomasz R. Bielecki & Igor Cialenco & Andrzej Ruszczy'nski, 2022. "Risk Filtering and Risk-Averse Control of Markovian Systems Subject to Model Uncertainty," Papers 2206.09235, arXiv.org.
    3. Yonghui Huang & Xianping Guo & Xinyuan Song, 2011. "Performance Analysis for Controlled Semi-Markov Systems with Application to Maintenance," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 395-415, August.
    4. Chen, Fang & Guo, Xianping, 2023. "Two-person zero-sum risk-sensitive stochastic games with incomplete reward information on one side," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 218-245.
    5. Hubert Asienkiewicz & Łukasz Balbus, 2019. "Existence of Nash equilibria in stochastic games of resource extraction with risk-sensitive players," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(3), pages 502-518, October.
    6. Carlos Camilo-Garay & Rolando Cavazos-Cadena & Hugo Cruz-Suárez, 2022. "Contractive Approximations in Risk-Sensitive Average Semi-Markov Decision Chains on a Finite State Space," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 271-291, January.
    7. A. Jaśkiewicz & A. S. Nowak, 2006. "Approximation of Noncooperative Semi-Markov Games," Journal of Optimization Theory and Applications, Springer, vol. 131(1), pages 115-134, October.
    8. Qingda Wei & Xian Chen, 2021. "Nonzero-sum Risk-Sensitive Average Stochastic Games: The Case of Unbounded Costs," Dynamic Games and Applications, Springer, vol. 11(4), pages 835-862, December.
    9. Yonghui Huang & Zhaotong Lian & Xianping Guo, 2023. "Zero-sum infinite-horizon discounted piecewise deterministic Markov games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 97(2), pages 179-205, April.
    10. Andrzej Nowak, 2006. "Remarks on sensitive equilibria in stochastic games with additive reward and transition structure," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(3), pages 481-494, December.
    11. Łukasz Balbus & Kevin Reffett & Łukasz Woźny, 2013. "Markov Stationary Equilibria in Stochastic Supermodular Games with Imperfect Private and Public Information," Dynamic Games and Applications, Springer, vol. 3(2), pages 187-206, June.
    12. Julio Saucedo-Zul & Rolando Cavazos-Cadena & Hugo Cruz-Suárez, 2020. "A Discounted Approach in Communicating Average Markov Decision Chains Under Risk-Aversion," Journal of Optimization Theory and Applications, Springer, vol. 187(2), pages 585-606, November.
    13. Subrata Golui & Chandan Pal & Subhamay Saha, 2022. "Continuous-Time Zero-Sum Games for Markov Decision Processes with Discounted Risk-Sensitive Cost Criterion," Dynamic Games and Applications, Springer, vol. 12(2), pages 485-512, June.
    14. Anna Jaśkiewicz & Andrzej S. Nowak, 2016. "Stationary Almost Markov Perfect Equilibria in Discounted Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 430-441, May.
    15. Anna Jaśkiewicz & Andrzej Nowak, 2015. "On pure stationary almost Markov Nash equilibria in nonzero-sum ARAT stochastic games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 81(2), pages 169-179, April.
    16. Wenzhao Zhang & Congying Liu, 2024. "Discrete-time stopping games with risk-sensitive discounted cost criterion," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(2), pages 437-466, October.
    17. Andrzej Nowak, 2007. "On stochastic games in economics," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(3), pages 513-530, December.
    18. Qingda Wei & Xian Chen, 2019. "Risk-Sensitive Average Equilibria for Discrete-Time Stochastic Games," Dynamic Games and Applications, Springer, vol. 9(2), pages 521-549, June.
    19. Elena Parilina & Stepan Akimochkin, 2021. "Cooperative Stochastic Games with Mean-Variance Preferences," Mathematics, MDPI, vol. 9(3), pages 1-15, January.
    20. Ghosh, Mrinal K. & Golui, Subrata & Pal, Chandan & Pradhan, Somnath, 2023. "Discrete-time zero-sum games for Markov chains with risk-sensitive average cost criterion," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 40-74.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:204:y:2025:i:3:d:10.1007_s10957-024-02603-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.