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Repeated Games and the Central Limit Theorem

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  • DE MEYER , Bernard

    (CORE, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium)

Abstract

The value vn ( P ) of the n times repeated zero sum game with incomplete information on one side is a concave function on the simplex p(K) that decreases to cav(u)( p ) as n grows. The rate of convergence 1/[ square root n ] that was given in Aumann's demonstration (See [A-M 68]) using a rough bound on martingale variation was proved to be the true one by Mertens and Zamir (See [M-Z 76] and [M-Z 77]) who analyzed a particular game with two states of nature, for which '[ psi_n ]( P ) = [ squareroot_n ][ vn[vn( P ) - cav(u)( p )] was showed to converge to a limit [ psi]( P ) related to the normal density. In our previous paper [DM-89]' we generalized the Mertens and Zamir's reasoning to a class of games [ delta_sigma_0 ] : there we show how the recurrence formula for vn rewritten as one for [ psi_n] becomes a partial differential equation (the heuristic equation) for [ psi] and proved that any solution of this differential problem with some boundary conditions was necessarily the limit of the [ psi_n]. We next proved that for a subclass R[ sigma] of [ delta_sigma_0 ] the heuristic equation had, as in the Mertens and Zamir's game, a solution related to the normal density. In this paper we explain the occurrence of the normal density as a consequence of the Central Limit Theorem.

Suggested Citation

  • DE MEYER , Bernard, 1993. "Repeated Games and the Central Limit Theorem," LIDAM Discussion Papers CORE 1993003, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:1993003
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    Citations

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    Cited by:

    1. Laraki, Rida & Sorin, Sylvain, 2015. "Advances in Zero-Sum Dynamic Games," Handbook of Game Theory with Economic Applications,, Elsevier.
    2. Fedor Sandomirskiy, 2014. "Repeated games of incomplete information with large sets of states," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(4), pages 767-789, November.
    3. Fabien Gensbittel, 2015. "Extensions of the Cav( u ) Theorem for Repeated Games with Incomplete Information on One Side," Mathematics of Operations Research, INFORMS, vol. 40(1), pages 80-104, February.
    4. VIEILLE, Nicolas & ROSENBERG, Dinah & SOLAN, Eilon, 2002. "Stochastic games with a single controller and incomplete information," HEC Research Papers Series 754, HEC Paris.
    5. Bernard de Meyer & Alexandre Marino, 2005. "Duality and optimal strategies in the finitely repeated zero-sum games with incomplete information on both sides," Post-Print halshs-00193996, HAL.
    6. Bernard De Meyer & Alexandre Marino, 2005. "Duality and optimal strategies in the finitely repeated zero-sum games with incomplete information on both sides," Cahiers de la Maison des Sciences Economiques b05027, Université Panthéon-Sorbonne (Paris 1).
    7. R. Buckdahn & P. Cardaliaguet & M. Quincampoix, 2011. "Some Recent Aspects of Differential Game Theory," Dynamic Games and Applications, Springer, vol. 1(1), pages 74-114, March.
    8. Sylvain Sorin, 2011. "Zero-Sum Repeated Games: Recent Advances and New Links with Differential Games," Dynamic Games and Applications, Springer, vol. 1(1), pages 172-207, March.
    9. Fedor Sandomirskiy, 2018. "On Repeated Zero-Sum Games with Incomplete Information and Asymptotically Bounded Values," Dynamic Games and Applications, Springer, vol. 8(1), pages 180-198, March.
    10. Fabien Gensbittel & Miquel Oliu-Barton, 2020. "Optimal Strategies in Zero-Sum Repeated Games with Incomplete Information: The Dependent Case," Dynamic Games and Applications, Springer, vol. 10(4), pages 819-835, December.
    11. Alexandre Marino & Bernard De Meyer, 2005. "Continuous versus Discrete Market Games," Cowles Foundation Discussion Papers 1535, Cowles Foundation for Research in Economics, Yale University.
    12. P. Cardaliaguet, 2008. "Representations Formulas for Some Differential Games with Asymmetric Information," Journal of Optimization Theory and Applications, Springer, vol. 138(1), pages 1-16, July.
    13. Bernard De Meyer & Ehud Lehrer & Dinah Rosenberg, 2010. "Evaluating Information in Zero-Sum Games with Incomplete Information on Both Sides," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 851-863, November.

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