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

Wenzhao Zhang (), Yonghui Huang () and Xianping Guo ()

TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, 2014, vol. 22, issue 3, 1074-1102

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

Keywords: Markov games; Occupation measure; Unbounded costs; Constrained Nash equilibria; 90C40; 91A25 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s11750-013-0313-9

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