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Common Learning

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  • 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.

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File URL: http://hdl.handle.net/10.1111/j.1468-0262.2008.00862.x
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Bibliographic Info

Article provided by Econometric Society in its journal Econometrica.

Volume (Year): 76 (2008)
Issue (Month): 4 (07)
Pages: 909-933

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Handle: RePEc:ecm:emetrp:v:76:y:2008:i:4:p:909-933

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References

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  1. Stephen Morris, . "Approximate Common Knowledge Revisited," Penn CARESS Working Papers 6be11f49fbded40b2a623aebf, Penn Economics Department.
  2. Carlsson, H. & Damme, E.E.C. van, 1990. "Global games and equilibrium selection," Discussion Paper 1990-52, Tilburg University, Center for Economic Research.
  3. Martin W. Cripps & Larry Samuelson, 2004. "Disappearing Private Reputations in Long-Run Relationships," PIER Working Paper Archive 04-031, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 28 Jul 2004.
  4. Daron Acemoglu & Victor Chernozhukov & Muhamet Yildiz, 2006. "Learning and Disagreement in an Uncertain World," NBER Working Papers 12648, National Bureau of Economic Research, Inc.
  5. Samet, Dov, 1998. "Iterated Expectations and Common Priors," Games and Economic Behavior, Elsevier, vol. 24(1-2), pages 131-141, July.
  6. Rubinstein, Ariel, 1989. "The Electronic Mail Game: Strategic Behavior under "Almost Common Knowledge."," American Economic Review, American Economic Association, vol. 79(3), pages 385-91, June.
  7. Thomas Wiseman, 2005. "A Partial Folk Theorem for Games with Unknown Payoff Distributions," Econometrica, Econometric Society, vol. 73(2), pages 629-645, 03.
  8. Monderer, Dov & Samet, Dov, 1989. "Approximating common knowledge with common beliefs," Games and Economic Behavior, Elsevier, vol. 1(2), pages 170-190, June.
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Cited by:
  1. Wiseman, Thomas, 2009. "Reputation and exogenous private learning," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1352-1357, May.
  2. Jakub Steiner & Colin Stewart, 2010. "Communication, Timing, and Common Learning," Working Papers tecipa-389, University of Toronto, Department of Economics.
  3. Amil Dasgupta & Jakub Steiner & Colin Stewart, 2007. "Efficient dynamic coordination with individual learning," LSE Research Online Documents on Economics 24498, London School of Economics and Political Science, LSE Library.
  4. Chong Huang, 2011. "Coordination and Social Learning," PIER Working Paper Archive 11-021, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
  5. Dasgupta, Amil & Steiner, Jakub & Stewart, Colin, 2012. "Dynamic coordination with individual learning," Games and Economic Behavior, Elsevier, vol. 74(1), pages 83-101.
  6. Jakub Steiner & Colin Stewart, 2008. "Communication Can Destroy Common Learning," ESE Discussion Papers 184, Edinburgh School of Economics, University of Edinburgh.
  7. Daron Acemoglu & Asuman Ozdaglar, 2011. "Opinion Dynamics and Learning in Social Networks," Dynamic Games and Applications, Springer, vol. 1(1), pages 3-49, March.
  8. 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.
  9. Martin W. Cripps & Jeffrey C. Ely & George J. Mailath & Larry Samuelson, 2011. "Common Learning with Intertemporal Dependence," PIER Working Paper Archive 11-012, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
  10. Daron Acemoglu & Victor Chernozhukov & Muhamet Yildiz, 2006. "Learning and Disagreement in an Uncertain World," NBER Working Papers 12648, National Bureau of Economic Research, Inc.

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