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Pathwise uniform value in gambling houses and Partially Observable Markov Decision Processes

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  • Xavier Venel

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Bruno Ziliotto

    (GREMAQ - Groupe de recherche en économie mathématique et quantitative - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - INRA - Institut National de la Recherche Agronomique - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique)

Abstract

In several standard models of dynamic programming (gambling houses, MDPs, POMDPs), we prove the existence of a robust notion of value for the infinitely repeated problem, namely the pathwise uniform value. This solves two open problems. First, this shows that for any ǫ > 0, the decision-maker has a pure strategy σ which is ǫ-optimal in any n-stage game, provided that n is big enough (this result was only known for behavior strategies, that is, strategies which use randomization). Second, the strategy σ can be chosen such that under the long-run average payoff criterion, the decision-maker has more than the limit of the n-stage values.

Suggested Citation

  • Xavier Venel & Bruno Ziliotto, 2016. "Pathwise uniform value in gambling houses and Partially Observable Markov Decision Processes," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01302567, HAL.
    • Handle: RePEc:hal:cesptp:hal-01302567
    Note: View the original document on HAL open archive server: https://hal.science/hal-01302567
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    References listed on IDEAS

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    1. Eitan Altman, 1994. "Denumerable Constrained Markov Decision Processes and Finite Approximations," Mathematics of Operations Research, INFORMS, vol. 19(1), pages 169-191, February.
    2. Dinah Rosenberg & Eilon Solan & Nicolas Vieille, 2000. "Blackwell Optimality in Markov Decision Processes with Partial Observation," Discussion Papers 1292, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    3. Renault, Jérôme & Venel, Xavier, 2017. "A distance for probability spaces, and long-term values in Markov Decision Processes and Repeated Games," TSE Working Papers 17-748, Toulouse School of Economics (TSE).
    4. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, January.
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    Cited by:

    1. Venel, Xavier, 2021. "Regularity of dynamic opinion games," Games and Economic Behavior, Elsevier, vol. 126(C), pages 305-334.

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