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Subgame-Perfection in Quitting Games with Perfect Information and Differential Equations

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  • Eilon Solan

Abstract

We introduce a new approach to studying subgame-perfect equilibrium payoffs in stochastic games: the differential equations approach. We apply our approach to quitting games with perfect information. Those are sequential games in which at every stage one of n players is chosen; each player is chosen with probability 1/ n . The chosen player i decides whether to quit, in which case the game terminates and the terminal payoff is some vector a i ∈ R n , or whether to continue, in which case the game continues to the next stage. If no player ever quits, the payoff is some vector a * ∈ R n . We define a certain differential inclusion, prove that it has at least one solution, and prove that every vector on a solution of this differential inclusion is a subgame-perfect equilibrium payoff.
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Suggested Citation

  • Eilon Solan, 2002. "Subgame-Perfection in Quitting Games with Perfect Information and Differential Equations," Discussion Papers 1356, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  • Handle: RePEc:nwu:cmsems:1356
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    References listed on IDEAS

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    1. Eran Shmaya & Eilon Solan & Nicolas Vieille, 2001. "An Application of Ramsey Theorem to Stopping Games," Discussion Papers 1323, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    2. Bergemann, Dirk & Valimaki, Juuso, 2003. "Dynamic common agency," Journal of Economic Theory, Elsevier, pages 23-48.
    3. Mertens,Jean-François & Sorin,Sylvain & Zamir,Shmuel, 2015. "Repeated Games," Cambridge Books, Cambridge University Press, number 9781107662636, November.
      • Mertens,Jean-François & Sorin,Sylvain & Zamir,Shmuel, 2015. "Repeated Games," Cambridge Books, Cambridge University Press, number 9781107030206, December.
    4. repec:dau:papers:123456789/6017 is not listed on IDEAS
    5. Vrieze, O J & Thuijsman, F, 1989. "On Equilibria in Repeated Games with Absorbing States," International Journal of Game Theory, Springer;Game Theory Society, pages 293-310.
    6. Frank Thuijsman & Thirukkannamangai E. S. Raghavan, 1997. "Perfect Information Stochastic Games and Related Classes," International Journal of Game Theory, Springer;Game Theory Society, pages 403-408.
    7. Shmaya, Eran & Solan, Eilon & Vieille, Nicolas, 2003. "An application of Ramsey theorem to stopping games," Games and Economic Behavior, Elsevier, vol. 42(2), pages 300-306, February.
    8. Matthew O. Jackson & Ehud Kalai & Rann Smorodinsky, 1999. "Bayesian Representation of Stochastic Processes under Learning: de Finetti Revisited," Econometrica, Econometric Society, pages 875-894.
      • Eilon Solan & Nicolas Vieille, 1998. "Quitting Games," Discussion Papers 1227, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    9. Jeffrey C. Ely & Juuso Välimäki, 2003. "Bad Reputation," The Quarterly Journal of Economics, Oxford University Press, pages 785-814.
    10. Mertens, Jean-Francois, 2002. "Stochastic games," Handbook of Game Theory with Economic Applications,in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 47, pages 1809-1832 Elsevier.
    11. Eilon Solan & Nicolas Vieille, 2001. "Quitting Games," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 265-285, May.
    12. Dinah Rosenberg & Eilon Solan & Nicolas Vieille, 1999. "Stopping Games with Randomized Strategies," Discussion Papers 1258, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    13. repec:dau:papers:123456789/6231 is not listed on IDEAS
    14. VIEILLE, Nicolas & SOLAN, Eilon, 2002. "Perturbed Markov Chains," Les Cahiers de Recherche 757, HEC Paris.
    15. Eilon Solan & Rakesh V. Vohra, 2002. "Correlated equilibrium payoffs and public signalling in absorbing games," International Journal of Game Theory, Springer;Game Theory Society, pages 91-121.
    16. Dinah Rosenberg & Eilon Solan & Nicolas Vieille, 2002. "Stochastic Games with Imperfect Monitoring," Discussion Papers 1341, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
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    Cited by:

    1. Solan, Eilon & Vieille, Nicolas, 2003. "Deterministic multi-player Dynkin games," Journal of Mathematical Economics, Elsevier, vol. 39(8), pages 911-929, November.

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