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Zero-Sum Markov Games with Random State-Actions-Dependent Discount Factors: Existence of Optimal Strategies

Author

Listed:
  • David González-Sánchez

    (CONACYT–Universidad de Sonora)

  • Fernando Luque-Vásquez

    (Universidad de Sonora)

  • J. Adolfo Minjárez-Sosa

    (Universidad de Sonora)

Abstract

This work deals with a class of discrete-time zero-sum Markov games under a discounted optimality criterion with random state-actions-dependent discount factors of the form $$\tilde{\alpha }(x_{n},a_{n},b_{n},\xi _{n+1})$$ α ~ ( x n , a n , b n , ξ n + 1 ) , where $$x_{n}, a_{n}, b_{n}$$ x n , a n , b n , and $$\xi _{n+1}$$ ξ n + 1 are the state, the actions of players, and a random disturbance at time n, respectively, taking values in Borel spaces. Assuming possibly unbounded payoff, we prove the existence of a value of the game as well as a stationary pair of optimal strategies.

Suggested Citation

  • David González-Sánchez & Fernando Luque-Vásquez & J. Adolfo Minjárez-Sosa, 2019. "Zero-Sum Markov Games with Random State-Actions-Dependent Discount Factors: Existence of Optimal Strategies," Dynamic Games and Applications, Springer, vol. 9(1), pages 103-121, March.
    • Handle: RePEc:spr:dyngam:v:9:y:2019:i:1:d:10.1007_s13235-018-0248-8
    DOI: 10.1007/s13235-018-0248-8
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    References listed on IDEAS

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    1. Yonghui Huang & Qingda Wei & Xianping Guo, 2013. "Constrained Markov decision processes with first passage criteria," Annals of Operations Research, Springer, vol. 206(1), pages 197-219, July.
    2. Nahum Shimkin & Adam Shwartz, 1995. "Asymptotically Efficient Adaptive Strategies in Repeated Games Part I: Certainty Equivalence Strategies," Mathematics of Operations Research, INFORMS, vol. 20(3), pages 743-767, August.
    3. Hugo Cruz-Suárez & Rocio Ilhuicatzi-Roldán & Raúl Montes-de-Oca, 2014. "Markov Decision Processes on Borel Spaces with Total Cost and Random Horizon," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 329-346, July.
    4. Nahum Shimkin & Adam Shwartz, 1996. "Asymptotically Efficient Adaptive Strategies in Repeated Games Part II. Asymptotic Optimality," Mathematics of Operations Research, INFORMS, vol. 21(2), pages 487-512, May.
    5. 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.
    6. Martin J. Osborne & Ariel Rubinstein, 1994. "A Course in Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262650401, April.
    7. Nowak, Andrzej S. & Szajowski, Krzysztof, 1998. "Nonzero-sum Stochastic Games," MPRA Paper 19995, University Library of Munich, Germany, revised 1999.
    8. He, Wei & Sun, Yeneng, 2017. "Stationary Markov perfect equilibria in discounted stochastic games," Journal of Economic Theory, Elsevier, vol. 169(C), pages 35-61.
    9. Alexander Krausz & Ulrich Rieder, 1997. "Markov games with incomplete information," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 46(2), pages 263-279, June.
    10. Dutta, Prajit K & Sundaram, Rangarajan, 1992. "Markovian Equilibrium in a Class of Stochastic Games: Existence Theorems for Discounted and Undiscounted Models," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(2), pages 197-214, April.
    11. Wei He & Yeneng Sun, 2013. "Stationary Markov Perfect Equilibria in Discounted Stochastic Games," Papers 1311.1562, arXiv.org, revised Jan 2017.
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    Cited by:

    1. John Stachurski & Junnan Zhang, 2019. "Dynamic Programming with State-Dependent Discounting," Papers 1908.08800, arXiv.org, revised Oct 2020.
    2. Carmen G. Higuera-Chan & J. Adolfo Minjárez-Sosa, 2021. "A Mean Field Approach for Discounted Zero-Sum Games in a Class of Systems of Interacting Objects," Dynamic Games and Applications, Springer, vol. 11(3), pages 512-537, September.

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