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Purification, Saturation and the Exact Law of Large Numbers

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  • Wang, Jianwei
  • Zhang, Yongchao

Abstract

Purification results are important in game theory and statistical decision theory. The purpose of this paper is to prove a general purification theorem that generalizes many authors' results. The key idea of our proof is to make use of the exact law of large numbers. As an application, we show that every mixed strategy in games with finite players, general action spaces, and diffused, conditionally independent incomplete information has many strong purifications.

Suggested Citation

  • Wang, Jianwei & Zhang, Yongchao, 2010. "Purification, Saturation and the Exact Law of Large Numbers," MPRA Paper 22119, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:22119
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    References listed on IDEAS

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    Cited by:

    1. , & , P. & , & ,, 2015. "Strategic uncertainty and the ex-post Nash property in large games," Theoretical Economics, Econometric Society, vol. 10(1), January.
    2. Grant, Simon & Meneghel, Idione & Tourky, Rabee, 2013. "Savage Games: A Theory of Strategic Interaction with Purely Subjective Uncertainty," Risk and Sustainable Management Group Working Papers 151501, University of Queensland, School of Economics.
    3. Khan, M. Ali & Rath, Kali P. & Sun, Yeneng & Yu, Haomiao, 2013. "Large games with a bio-social typology," Journal of Economic Theory, Elsevier, vol. 148(3), pages 1122-1149.
    4. He, Wei & Sun, Xiang, 2014. "On the diffuseness of incomplete information game," Journal of Mathematical Economics, Elsevier, vol. 54(C), pages 131-137.
    5. Lei Qiao & Yeneng Sun & Zhixiang Zhang, 2016. "Conditional exact law of large numbers and asymmetric information economies with aggregate uncertainty," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 62(1), pages 43-64, June.
    6. M. Ali Khan & Yongchao Zhang, 2017. "Existence of pure-strategy equilibria in Bayesian games: a sharpened necessity result," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(1), pages 167-183, March.
    7. Grant, Simon & Meneghel, Idione & Tourky, Rabee, 2016. "Savage games," Theoretical Economics, Econometric Society, vol. 11(2), May.
    8. Michael Greinecker & Konrad Podczeck, 2013. "Purification and Independence," Working Papers 2013-18, Faculty of Economics and Statistics, Universität Innsbruck.
    9. Wei He & Xiang Sun & Yeneng Sun & Yishu Zeng, 2021. "Characterization of equilibrium existence and purification in general Bayesian games," Papers 2106.08563, arXiv.org.
    10. He, Wei & Yannelis, Nicholas C., 2015. "Discontinuous games with asymmetric information: An extension of Reny's existence theorem," Games and Economic Behavior, Elsevier, vol. 91(C), pages 26-35.
    11. Khan, M. Ali & Zhang, Yongchao, 2014. "On the existence of pure-strategy equilibria in games with private information: A complete characterization," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 197-202.
    12. Michael Greinecker & Christopher Kah, 2021. "Pairwise Stable Matching in Large Economies," Econometrica, Econometric Society, vol. 89(6), pages 2929-2974, November.
    13. Barelli, Paulo & Duggan, John, 2015. "Purification of Bayes Nash equilibrium with correlated types and interdependent payoffs," Games and Economic Behavior, Elsevier, vol. 94(C), pages 1-14.
    14. Yeneng Sun & Lei Wu & Nicholas C. Yannelis, 2013. "Incentive compatibility of rational expectations equilibrium in large economies: a counterexample," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 1(1), pages 3-10, May.
    15. Michael Greinecker & Konrad Podczeck, 2015. "Purification and roulette wheels," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 58(2), pages 255-272, February.

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    More about this item

    Keywords

    Exact law of large numbers; Fubini extension; Incomplete information; Purification; Saturated probability space;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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