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Purification and roulette wheels

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  • Michael Greinecker
  • Konrad Podczeck

Abstract

We use concepts introduced by Aumann more than 30 years ago to throw new light on purification in games with extremely dispersed private information. We show that one can embed payoff-irrelevant randomization devices in the private information of players and use these randomization devices to implement mixed strategies as deterministic functions of the private information. This approach gives rise to very short and intuitive proofs for a number of purification results that previously required sophisticated methods from functional analysis or nonstandard analysis. We use our methods to prove the first general purification theorem for games with private information in which a player’s payoffs can depend in arbitrary ways on events in the private information of other players and in which we allow for shared information in a general way. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joecth:v:58:y:2015:i:2:p:255-272
    DOI: 10.1007/s00199-014-0815-1
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    References listed on IDEAS

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    1. Aumann, Robert J., 1974. "Subjectivity and correlation in randomized strategies," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 67-96, March.
    2. Govindan, Srihari & Reny, Philip J. & Robson, Arthur J., 2003. "A short proof of Harsanyi's purification theorem," Games and Economic Behavior, Elsevier, vol. 45(2), pages 369-374, November.
    3. SCHMEIDLER, David, 1973. "Equilibrium points of nonatomic games," LIDAM Reprints CORE 146, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Rustichini, Aldo, 1993. "Mixing on Function Spaces," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 3(1), pages 183-191, January.
    5. Konrad Podczeck, 2010. "On existence of rich Fubini extensions," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 45(1), pages 1-22, October.
    6. Sun, Yeneng, 2006. "The exact law of large numbers via Fubini extension and characterization of insurable risks," Journal of Economic Theory, Elsevier, vol. 126(1), pages 31-69, January.
    7. Paul R. Milgrom & Robert J. Weber, 1985. "Distributional Strategies for Games with Incomplete Information," Mathematics of Operations Research, INFORMS, vol. 10(4), pages 619-632, November.
    8. Carmona, Guilherme & Podczeck, Konrad, 2014. "Existence of Nash equilibrium in games with a measure space of players and discontinuous payoff functions," Journal of Economic Theory, Elsevier, vol. 152(C), pages 130-178.
    9. Jianwei Wang & Yongchao Zhang, 2012. "Purification, saturation and the exact law of large numbers," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 50(3), pages 527-545, August.
    10. Carmona, Guilherme & Podczeck, Konrad, 2009. "On the existence of pure-strategy equilibria in large games," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1300-1319, May.
    11. Nicholas Yannelis, 2009. "Debreu’s social equilibrium theorem with asymmetric information and a continuum of agents," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 38(2), pages 419-432, February.
    12. M. Khan & Kali Rath & Yeneng Sun, 2006. "The Dvoretzky-Wald-Wolfowitz theorem and purification in atomless finite-action games," International Journal of Game Theory, Springer;Game Theory Society, vol. 34(1), pages 91-104, April.
    13. Khan, M. Ali & Rath, Kali P., 2009. "On games with incomplete information and the Dvoretsky-Wald-Wolfowitz theorem with countable partitions," Journal of Mathematical Economics, Elsevier, vol. 45(12), pages 830-837, December.
    14. Konrad Podczeck, 2009. "On purification of measure-valued maps," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 38(2), pages 399-418, February.
    15. Haifeng Fu, 2008. "Mixed-strategy equilibria and strong purification for games with private and public information," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 37(3), pages 521-532, December.
    16. He, Wei & Sun, Xiang, 2014. "On the diffuseness of incomplete information game," Journal of Mathematical Economics, Elsevier, vol. 54(C), pages 131-137.
    17. Roy Radner & Robert W. Rosenthal, 1982. "Private Information and Pure-Strategy Equilibria," Mathematics of Operations Research, INFORMS, vol. 7(3), pages 401-409, August.
    18. Erik J. Balder, 1988. "Generalized Equilibrium Results for Games with Incomplete Information," Mathematics of Operations Research, INFORMS, vol. 13(2), pages 265-276, May.
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    Cited by:

    1. Khan, M. Ali & Zhang, Yongchao, 2018. "On pure-strategy equilibria in games with correlated information," Games and Economic Behavior, Elsevier, vol. 111(C), pages 289-304.

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    More about this item

    Keywords

    Purification; Games with incomplete information; Roulette wheels; C72;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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