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dc.contributor.authorVigeral, Guillaume
dc.subjectdynamic programmingen
dc.subjectShapley operatoren
dc.subjectnonexpansive mappingsen
dc.subjectasymptotic propertiesen
dc.subjectzero-sum stochastic gamesen
dc.titleIterated monotonic nonexpansive operators and asymptotic properties of zero-sum stochastic gamesen
dc.typeDocument de travail / Working paper
dc.description.abstractenWe consider an operator $\Ps$ defined on a set of real valued functions and satisfying two properties of monotonicity and additive homogeneity. This is motivated by the case of zero sum stochastic games, for which the Shapley operator is monotone and additively homogeneous. We study the asymptotic of the trajectories defined by $v_n=\frac{\Ps^n(0)}{n}$ ($n\in N , n \rightarrow \infty$) and $v_\lambda=\lambda\Ps\left(\frac{1-\lambda}{\lambda}v_\lambda\right)$ ($\lambda \in (0,1], \lambda \rightarrow 0$). Examining the iterates of $\Ps$, we exhibit analytical conditions on the operator that imply that $v_n$ and $v_\lambda$ have at most one accumulation point for the uniform norm. In particular this establishes the uniform convergence of $v_n$ and $v_\lambda$ to the same limit for a large subclass of the class of games where only one player control the transitions. We also study the general case of two players controlling the transitions, giving a sufficient condition for convergence.en
dc.publisher.nameUniversité Paris-Dauphineen
dc.subject.ddclabelProbabilités et mathématiques appliquéesen

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