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Common Learning

Author

Listed:
  • Martin W. Cripps
  • Jeffrey C. Ely
  • George J. Mailath
  • Larry Samuelson

Abstract

Consider two agents who learn the value of an unknown parameter by observing a sequence of private signals. The signals are independent and identically distributed across time but not necessarily across agents. We show that when each agent's signal space is finite, the agents will commonly learn the value of the parameter, that is, that the true value of the parameter will become approximate common knowledge. The essential step in this argument is to express the expectation of one agent's signals, conditional on those of the other agent, in terms of a Markov chain. This allows us to invoke a contraction mapping principle ensuring that if one agent's signals are close to those expected under a particular value of the parameter, then that agent expects the other agent's signals to be even closer to those expected under the parameter value. In contrast, if the agents' observations come from a countably infinite signal space, then this contraction mapping property fails. We show by example that common learning can fail in this case. Copyright Copyright 2008 by The Econometric Society.

Suggested Citation

  • Martin W. Cripps & Jeffrey C. Ely & George J. Mailath & Larry Samuelson, 2008. "Common Learning," Econometrica, Econometric Society, vol. 76(4), pages 909-933, July.
    • Handle: RePEc:ecm:emetrp:v:76:y:2008:i:4:p:909-933
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    References listed on IDEAS

    as
    1. Carlsson, Hans & van Damme, Eric, 1993. "Global Games and Equilibrium Selection," Econometrica, Econometric Society, vol. 61(5), pages 989-1018, September.
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    4. Cripps, Martin W. & Mailath, George J. & Samuelson, Larry, 2007. "Disappearing private reputations in long-run relationships," Journal of Economic Theory, Elsevier, vol. 134(1), pages 287-316, May.
    5. Daron Acemoglu & Victor Chernozhukov & Muhamet Yildiz, 2006. "Learning and Disagreement in an Uncertain World," NBER Working Papers 12648, National Bureau of Economic Research, Inc.
    6. Monderer, Dov & Samet, Dov, 1989. "Approximating common knowledge with common beliefs," Games and Economic Behavior, Elsevier, vol. 1(2), pages 170-190, June.
    7. Samet, Dov, 1998. "Iterated Expectations and Common Priors," Games and Economic Behavior, Elsevier, vol. 24(1-2), pages 131-141, July.
    8. Thomas Wiseman, 2005. "A Partial Folk Theorem for Games with Unknown Payoff Distributions," Econometrica, Econometric Society, vol. 73(2), pages 629-645, March.
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    Citations

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    Cited by:

    1. Antonio Jiménez-Martínez, 2015. "A model of belief influence in large social networks," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 59(1), pages 21-59, May.
    2. Martin Cripps & Jeffrey Ely & George Mailath & Larry Samuelson, 2013. "Common learning with intertemporal dependence," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(1), pages 55-98, February.
    3. Fudenberg, Drew & Takahashi, Satoru, 2011. "Heterogeneous beliefs and local information in stochastic fictitious play," Games and Economic Behavior, Elsevier, vol. 71(1), pages 100-120, January.
    4. Amil Dasgupta & Jakub Steiner & Colin Stewart, 2007. "Efficient Dynamic Coordination with Individual Learning," Working Papers tecipa-301, University of Toronto, Department of Economics.
    5. Chong Huang, 2011. "Coordination and Social Learning," PIER Working Paper Archive 11-021, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    6. Arieli, Itai & Levy, Yehuda John, 2015. "Determinacy of games with Stochastic Eventual Perfect Monitoring," Games and Economic Behavior, Elsevier, vol. 91(C), pages 166-185.
    7. Morris, Stephen, 2014. "Coordination, timing and common knowledge," Research in Economics, Elsevier, vol. 68(4), pages 306-314.
    8. Jakub Steiner & Colin Stewart, 2008. "Communication Can Destroy Common Learning," Edinburgh School of Economics Discussion Paper Series 184, Edinburgh School of Economics, University of Edinburgh.
    9. Daron Acemoglu & Asuman Ozdaglar, 2011. "Opinion Dynamics and Learning in Social Networks," Dynamic Games and Applications, Springer, vol. 1(1), pages 3-49, March.
    10. Eeckhout, Jan & Weng, Xi, 2015. "Common value experimentation," Journal of Economic Theory, Elsevier, vol. 160(C), pages 317-339.
    11. Sharma, Priyanka, 2017. "Is more information always better? A case in credit markets," Journal of Economic Behavior & Organization, Elsevier, vol. 134(C), pages 269-283.
    12. Daron Acemoglu & Victor Chernozhukov & Muhamet Yildiz, 2006. "Learning and Disagreement in an Uncertain World," NBER Working Papers 12648, National Bureau of Economic Research, Inc.
    13. Chong Huang, 2018. "Coordination and social learning," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 65(1), pages 155-177, January.
    14. Steiner, Jakub & Stewart, Colin, 2011. "Communication, timing, and common learning," Journal of Economic Theory, Elsevier, vol. 146(1), pages 230-247, January.
    15. Wiseman, Thomas, 2009. "Reputation and exogenous private learning," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1352-1357, May.
    16. Dasgupta, Amil & Steiner, Jakub & Stewart, Colin, 2012. "Dynamic coordination with individual learning," Games and Economic Behavior, Elsevier, vol. 74(1), pages 83-101.

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    More about this item

    JEL classification:

    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

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