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

In this paper, we consider discrete-time <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$N$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>N</mi> </math> </EquationSource> </InlineEquation>-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, <CitationRef CitationID="CR5">2006</CitationRef>) 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, <CitationRef CitationID="CR5">2006</CitationRef>), 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, <CitationRef CitationID="CR5">2006</CitationRef>), 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