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Strong 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

    (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique, Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres)

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 strong uniform value. This solves two open problems. First, this shows that for any > 0, the decision-maker has a pure strategy σ which is-optimal in any n-stage problem, provided that n is big enough (this result was only known for behavior strategies, that is, strategies which use randomization). Second, for any > 0, the decision-maker can guarantee the limit of the n-stage value minus in the infinite problem where the payoff is the expectation of the inferior limit of the time average payoff.

Suggested Citation

  • Xavier Venel & Bruno Ziliotto, 2016. "Strong Uniform Value in Gambling Houses and Partially Observable Markov Decision Processes," Post-Print hal-01395429, HAL.
    • Handle: RePEc:hal:journl:hal-01395429
    DOI: 10.1137/15M1043340
    Note: View the original document on HAL open archive server: https://hal.science/hal-01395429
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    References listed on IDEAS

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    1. 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.
    2. Eitan Altman, 1994. "Denumerable Constrained Markov Decision Processes and Finite Approximations," Mathematics of Operations Research, INFORMS, vol. 19(1), pages 169-191, February.
    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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