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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. 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.
    2. 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.
    3. Laraki, Rida & Sorin, Sylvain, 2015. "Advances in Zero-Sum Dynamic Games," Handbook of Game Theory with Economic Applications,, Elsevier.
    4. 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.
    5. 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.
    6. 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.
    7. Alexandre Marino & Bernard De Meyer, 2005. "Continuous versus Discrete Market Games," Cowles Foundation Discussion Papers 1535, Cowles Foundation for Research in Economics, Yale University.
    8. VIEILLE, Nicolas & ROSENBERG, Dinah & SOLAN, Eilon, 2002. "Stochastic games with a single controller and incomplete information," HEC Research Papers Series 754, HEC Paris.
    9. Bernard de Meyer & Alexandre Marino, 2005. "Duality and optimal strategies in the finitely repeated zero-sum games with incomplete information on both sides," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00193996, HAL.
    10. 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.
    11. 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.
    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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