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Operator approach to values of stochastic games with varying stage duration

Author

Listed:
  • Sylvain Sorin

    (Sorbonne Universités, UPMC Univ Paris 06, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Univ Paris Diderot)

  • Guillaume Vigeral

    (Université Paris-Dauphine, CEREMADE)

Abstract

We study the links between the values of stochastic games with varying stage duration h, the corresponding Shapley operators $$\mathbf{T}$$ T and $$\mathbf{T}_h= h\mathbf{T}+ (1-h ) Id$$ T h = h T + ( 1 - h ) I d and the solution of the evolution equation $$\dot{f}_t = (\mathbf{T}- Id )f_t$$ f ˙ t = ( T - I d ) f t . Considering general non expansive maps we establish two kinds of results, under both the discounted or the finite length framework, that apply to the class of “exact” stochastic games. First, for a fixed length or discount factor, the value converges as the stage duration go to 0. Second, the asymptotic behavior of the value as the length goes to infinity, or as the discount factor goes to 0, does not depend on the stage duration. In addition, these properties imply the existence of the value of the finite length or discounted continuous time game (associated to a continuous time jointly controlled Markov process), as the limit of the value of any time discretization with vanishing mesh.

Suggested Citation

  • Sylvain Sorin & Guillaume Vigeral, 2016. "Operator approach to values of stochastic games with varying stage duration," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(1), pages 389-410, March.
    • Handle: RePEc:spr:jogath:v:45:y:2016:i:1:d:10.1007_s00182-015-0512-8
    DOI: 10.1007/s00182-015-0512-8
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    References listed on IDEAS

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    1. Abraham Neyman, 2013. "Stochastic Games with Short-Stage Duration," Dynamic Games and Applications, Springer, vol. 3(2), pages 236-278, June.
    2. Neyman, Abraham, 2017. "Continuous-time stochastic games," Games and Economic Behavior, Elsevier, vol. 104(C), pages 92-130.
    3. Mertens,Jean-François & Sorin,Sylvain & Zamir,Shmuel, 2015. "Repeated Games," Cambridge Books, Cambridge University Press, number 9781107030206.
      • Mertens,Jean-François & Sorin,Sylvain & Zamir,Shmuel, 2015. "Repeated Games," Cambridge Books, Cambridge University Press, number 9781107662636.
    4. Nowak, Andrzej S. & Szajowski, Krzysztof, 1998. "Nonzero-sum Stochastic Games," MPRA Paper 19995, University Library of Munich, Germany, revised 1999.
    5. Guillaume Vigeral, 2013. "A Zero-Sum Stochastic Game with Compact Action Sets and no Asymptotic Value," Dynamic Games and Applications, Springer, vol. 3(2), pages 172-186, June.
    6. repec:dau:papers:123456789/10880 is not listed on IDEAS
    7. Abraham Neyman & Sylvain Sorin, 2010. "Repeated games with public uncertain duration process," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 29-52, March.
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    Cited by:

    1. Jan-Henrik Steg, 2018. "On Preemption in Discrete and Continuous Time," Dynamic Games and Applications, Springer, vol. 8(4), pages 918-938, December.
    2. Sylvain Sorin, 2018. "Limit Value of Dynamic Zero-Sum Games with Vanishing Stage Duration," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 51-63, February.

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