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Nonzero-sum constrained discrete-time Markov games: the case of unbounded costs

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  • Wenzhao Zhang
  • Yonghui Huang
  • Xianping Guo

Abstract

In this paper, we consider discrete-time $$N$$ N -person constrained stochastic games with discounted cost criteria. The state space is denumerable and the action space is a Borel set, while the cost functions are admitted to be unbounded from below and above. Under suitable conditions weaker than those in (Alvarez-Mena and Hernández-Lerma, Math Methods Oper Res 63:261–285, 2006 ) for bounded cost functions, we also show the existence of a Nash equilibrium for the constrained games by introducing two approximations. The first one, which is as in (Alvarez-Mena and Hernández-Lerma, Math Methods Oper Res 63:261–285, 2006 ), is to construct a sequence of finite games to approximate a (constrained) auxiliary game with an initial distribution that is concentrated on a finite set. However, without hypotheses of bounded costs as in (Alvarez-Mena and Hernández-Lerma, Math Methods Oper Res 63:261–285, 2006 ), we also establish the existence of a Nash equilibrium for the auxiliary game with unbounded costs by developing more shaper error bounds of the approximation. The second one, which is new, is to construct a sequence of the auxiliary-type games above and prove that the limit of the sequence of Nash equilibria for the auxiliary-type games is a Nash equilibrium for the original constrained games. Our results are illustrated by a controlled queueing system. Copyright Sociedad de Estadística e Investigación Operativa 2014

Suggested Citation

  • Wenzhao Zhang & Yonghui Huang & Xianping Guo, 2014. "Nonzero-sum constrained discrete-time Markov games: the case of unbounded costs," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 1074-1102, October.
    • Handle: RePEc:spr:topjnl:v:22:y:2014:i:3:p:1074-1102
    DOI: 10.1007/s11750-013-0313-9
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    References listed on IDEAS

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    1. Andrzej S. Nowak, 1999. "Sensitive equilibria for ergodic stochastic games with countable state spaces," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 50(1), pages 65-76, August.
    2. 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.
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    Cited by:

    1. Qiuli Liu & Wai-Ki Ching & Xianping Guo, 2023. "Zero-sum stochastic games with the average-value-at-risk criterion," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(3), pages 618-647, October.
    2. Wenzhao Zhang, 2019. "Discrete-Time Constrained Average Stochastic Games with Independent State Processes," Mathematics, MDPI, vol. 7(11), pages 1-18, November.

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