IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v97y2023i2d10.1007_s00186-023-00809-0.html
   My bibliography  Save this article

Zero-sum infinite-horizon discounted piecewise deterministic Markov games

Author

Listed:
  • Yonghui Huang

    (Sun Yat-Sen University)

  • Zhaotong Lian

    (University of Macau)

  • Xianping Guo

    (Sun Yat-Sen University)

Abstract

This paper is devoted to zero-sum piecewise deterministic Markov games with Borel state and action spaces, where the expected infinite-horizon discounted payoff criterion is considered. Both the transition rate and payoff rate are allowed to be unbounded. The policies of the two players are history-dependent, and the controls continuously act on the transition rate and the payoff rate. Under suitable conditions, Dynkin’s formula and the comparison theorem are developed in our setup, via which the game is shown to have the value function as the unique solution to the associated Shapley equation. By the Shapley equation in the form of a differential equation, we establish the existence of a saddle point with a very simple form, which only depends on the current state and can be applied at any time. A potential algorithm for computing saddle points is proposed.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:mathme:v:97:y:2023:i:2:d:10.1007_s00186-023-00809-0
    DOI: 10.1007/s00186-023-00809-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00186-023-00809-0
    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/s00186-023-00809-0?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. Tomás Prieto-Rumeau & José Lorenzo, 2015. "Approximation of zero-sum continuous-time Markov games under the discounted payoff criterion," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(3), pages 799-836, October.
    3. Alain Haurie & Jacek B Krawczyk & Georges Zaccour, 2012. "Games and Dynamic Games," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8442, February.
    4. Fabien Gensbittel & Jérôme Renault, 2015. "The Value of Markov Chain Games with Incomplete Information on Both Sides," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 820-841, October.
    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. Javier Frutos & Guiomar Martín-Herrán, 2018. "Selection of a Markov Perfect Nash Equilibrium in a Class of Differential Games," Dynamic Games and Applications, Springer, vol. 8(3), pages 620-636, September.
    2. Bolei Di & Andrew Lamperski, 2022. "Newton’s Method, Bellman Recursion and Differential Dynamic Programming for Unconstrained Nonlinear Dynamic Games," Dynamic Games and Applications, Springer, vol. 12(2), pages 394-442, June.
    3. Massol, Olivier & Rifaat, Omer, 2018. "Phasing out the U.S. Federal Helium Reserve: Policy insights from a world helium model," Resource and Energy Economics, Elsevier, vol. 54(C), pages 186-211.
    4. 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.
    5. Colombo, Luca & Labrecciosa, Paola, 2022. "Product quality differentiation in a renewable resource oligopoly," Journal of Environmental Economics and Management, Elsevier, vol. 111(C).
    6. Mathew P. Abraham & Ankur A. Kulkarni, 2018. "An Approach Based on Generalized Nash Games and Shared Constraints for Discrete Time Dynamic Games," Dynamic Games and Applications, Springer, vol. 8(4), pages 641-670, December.
    7. Fabien Gensbittel & Christine Grün, 2019. "Zero-Sum Stopping Games with Asymmetric Information," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 277-302, February.
    8. Ilko Vrankić & Tomislav Herceg & Mirjana Pejić Bach, 2021. "Dynamics and stability of evolutionary optimal strategies in duopoly," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 29(3), pages 1001-1019, September.
    9. Elena M. Parilina & Puduru Viswanadha Reddy & Georges Zaccour, 2022. "Endogenous Duration of Long-term Agreements in Cooperative Dynamic Games with Nontransferable Utility," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 808-836, December.
    10. Dhruva Kartik & Ashutosh Nayyar, 2021. "Upper and Lower Values in Zero-Sum Stochastic Games with Asymmetric Information," Dynamic Games and Applications, Springer, vol. 11(2), pages 363-388, June.
    11. Arguedas, Carmen & Cabo, Francisco & Martín-Herrán, Guiomar, 2020. "Enforcing regulatory standards in stock pollution problems," Journal of Environmental Economics and Management, Elsevier, vol. 100(C).
    12. Francesco Caruso & Maria Carmela Ceparano & Jacqueline Morgan, 2020. "Best response algorithms in ratio-bounded games: convergence of affine relaxations to Nash equilibria," CSEF Working Papers 593, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy.
    13. Katarzyna Kańska & Agnieszka Wiszniewska-Matyszkiel, 2022. "Dynamic Stackelberg duopoly with sticky prices and a myopic follower," Operational Research, Springer, vol. 22(4), pages 4221-4252, September.
    14. Genc, Talat S. & De Giovanni, Pietro, 2018. "Optimal return and rebate mechanism in a closed-loop supply chain game," European Journal of Operational Research, Elsevier, vol. 269(2), pages 661-681.
    15. Crettez, Bertrand & Hayek, Naila & Zaccour, Georges, 2018. "Brand imitation: A dynamic-game approach," International Journal of Production Economics, Elsevier, vol. 205(C), pages 139-155.
    16. Vladimir Mazalov & Elena Parilina, 2020. "The Euler-Equation Approach in Average-Oriented Opinion Dynamics," Mathematics, MDPI, vol. 8(3), pages 1-16, March.
    17. Carmen Arguedas & Francisco Cabo & Guiomar Martín-Herrán, 2017. "Optimal Pollution Standards and Non-compliance in a Dynamic Framework," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 68(3), pages 537-567, November.
    18. Lu, Lijue & Marín-Solano, Jesús & Navas, Jorge, 2019. "An analysis of efficiency of time-consistent coordination mechanisms in a model of supply chain management," European Journal of Operational Research, Elsevier, vol. 279(1), pages 211-224.
    19. Escobar, Juan F. & Llanes, Gastón, 2018. "Cooperation dynamics in repeated games of adverse selection," Journal of Economic Theory, Elsevier, vol. 176(C), pages 408-443.
    20. Bernhard, Pierre & Deschamps, Marc & Zaccour, Georges, 2023. "Large satellite constellations and space debris: Exploratory analysis of strategic management of the space commons," European Journal of Operational Research, Elsevier, vol. 304(3), pages 1140-1157.

    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:mathme:v:97:y:2023:i:2:d:10.1007_s00186-023-00809-0. 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.