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Merging with a set of probability measures: a characterization

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    (Department of Economics, Kanto Gakuin University)

Abstract

In this paper, I provide a characterization of a \textit{set} of probability measures with which a prior ``weakly merges.'' In this regard, I introduce the concept of ``conditioning rules'' that represent the \textit{regularities% } of probability measures and define the ``eventual generation'' of probability measures by a family of conditioning rules. I then show that a set of probability measures is learnable (i.e., all probability measures in the set are weakly merged by a prior) if and only if all probability measures in the set are eventually generated by a \textit{countable} family of conditioning rules. I also demonstrate that quite similar results are obtained with ``almost weak merging.'' In addition, I argue that my characterization result can be extended to the case of infinitely repeated games and has some interesting applications with regard to the impossibility result in Nachbar (1997, 2005).

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  • ,, 2015. "Merging with a set of probability measures: a characterization," Theoretical Economics, Econometric Society, vol. 10(2), May.
    • Handle: RePEc:the:publsh:1360
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    References listed on IDEAS

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    1. Fudenberg, Drew & Levine, David, 1998. "Learning in games," European Economic Review, Elsevier, vol. 42(3-5), pages 631-639, May.
    2. Kalai, Ehud & Lehrer, Ehud, 1994. "Weak and strong merging of opinions," Journal of Mathematical Economics, Elsevier, vol. 23(1), pages 73-86, January.
    3. Johannes Hörner & Wojciech Olszewski, 2006. "The Folk Theorem for Games with Private Almost-Perfect Monitoring," Econometrica, Econometric Society, vol. 74(6), pages 1499-1544, November.
    4. Kalai, Ehud & Lehrer, Ehud, 1993. "Rational Learning Leads to Nash Equilibrium," Econometrica, Econometric Society, vol. 61(5), pages 1019-1045, September.
    5. Fudenberg, Drew & Maskin, Eric, 1991. "On the dispensability of public randomization in discounted repeated games," Journal of Economic Theory, Elsevier, vol. 53(2), pages 428-438, April.
    6. Matthew O. Jackson & Ehud Kalai & Rann Smorodinsky, 1999. "Bayesian Representation of Stochastic Processes under Learning: de Finetti Revisited," Econometrica, Econometric Society, vol. 67(4), pages 875-894, July.
    7. Drew Fudenberg & David K. Levine, 1998. "The Theory of Learning in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061945, April.
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    Cited by:

    1. Norman, Thomas W.L., 2022. "The possibility of Bayesian learning in repeated games," Games and Economic Behavior, Elsevier, vol. 136(C), pages 142-152.

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    More about this item

    Keywords

    Bayesian learning; weak merging; conditioning rules; eventual generation; frequency-based prior;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

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