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On Nonterminating Stochastic Games


  • A. J. Hoffman

    (IBM Watson Research Center)

  • R. M. Karp

    (IBM Watson Research Center)


A stochastic game is played in a sequence of steps; at each step the play is said to be in some state i, chosen from a finite collection of states. If the play is in state i, the first player chooses move k and the second player chooses move l, then the first player receives a reward a kl i , and, with probability p kl ij , the next state is j. The concept of stochastic games was introduced by Shapley with the proviso that, with probability 1, play terminates. The authors consider the case when play never terminates, and show properties of such games and offer a convergent algorithm for their solution. In the special case when one of the players is a dummy, the nonterminating stochastic game reduces to a Markovian decision process, and the present work can be regarded as the extension to a game theoretic context of known results on Markovian decision processes.

Suggested Citation

  • A. J. Hoffman & R. M. Karp, 1966. "On Nonterminating Stochastic Games," Management Science, INFORMS, vol. 12(5), pages 359-370, January.
  • Handle: RePEc:inm:ormnsc:v:12:y:1966:i:5:p:359-370

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    Cited by:

    1. Hörner, Johannes & Takahashi, Satoru & Vieille, Nicolas, 2014. "On the limit perfect public equilibrium payoff set in repeated and stochastic games," Games and Economic Behavior, Elsevier, vol. 85(C), pages 70-83.
    2. Endre Boros & Khaled Elbassioni & Vladimir Gurvich & Kazuhisa Makino, 2013. "On Canonical Forms for Zero-Sum Stochastic Mean Payoff Games," Dynamic Games and Applications, Springer, vol. 3(2), pages 128-161, June.
    3. Johannes H�rner & Satoru Takahashi & Nicolas Vieille, 2012. "On the Limit Equilibrium Payoff Set in Repeated and Stochastic Games," Working Papers 1397, Princeton University, Department of Economics, Econometric Research Program..
    4. Johannes Horner & Takuo Sugaya & Satoru Takahashi & Nicolas Vieille, 2009. "Recursive Methods in Discounted Stochastic Games: An Algorithm for delta Approaching 1 and a Folk Theorem," Cowles Foundation Discussion Papers 1742, Cowles Foundation for Research in Economics, Yale University, revised Aug 2010.
    5. Frederic H. Murphy, 1972. "Row Dropping Procedures for Cutting Plane Algorithms," Discussion Papers 16, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    6. Krishnendu Chatterjee & Rupak Majumdar & Thomas Henzinger, 2008. "Stochastic limit-average games are in EXPTIME," International Journal of Game Theory, Springer;Game Theory Society, vol. 37(2), pages 219-234, June.

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