Common Learning

  1. Martin W. Cripps1,
  2. Jeffrey C. Ely2,
  3. George J. Mailath3,
  4. Larry Samuelson4,†

Article first published online: 28 JUN 2008

DOI: 10.1111/j.1468-0262.2008.00862.x

Issue

Econometrica

Econometrica

Volume 76, Issue 4, pages 909–933, July 2008

Additional Information

How to Cite

Cripps, M. W., Ely, J. C., Mailath, G. J. and Samuelson, L. (2008), Common Learning. Econometrica, 76: 909–933. doi: 10.1111/j.1468-0262.2008.00862.x

Author Information

  1. 1

    Dept. of Economics, University College London, Gower Street, London WC1E 6BT, United Kingdom; m.cripps@ucl.ac.uk

  2. 2

    Dept. of Economics, Northwestern University, 2003 Sheridan Road, Evanston, IL 60208-2600, U.S.A.; jeffely@northwestern.edu

  3. 3

    Dept. of Economics, University of Pennsylvania, 3718 Locust Walk, Philadelphia, PA 19103-6297, U.S.A.; gmailath@econ.upenn.edu

  4. 4

    Economics Department, Yale University, 30 Hillhouse Avenue, New Haven, CT 06520-8268, U.S.A.; Larry.Samuelson@yale.edu

  1. †

    Cripps thanks the Economic and Social Research Council (U.K.) via ELSE, and Mailath and Samuelson thank the National Science Foundation (Grants SES-0350969 and SES-0549946, respectively) for financial support. We thank Stephen Morris, the editor, and three referees for helpful comments.

Publication History

  1. Issue published online: 28 JUN 2008
  2. Article first published online: 28 JUN 2008
  3. Manuscript received August, 2006; final revision received March, 2008.

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Keywords:

  • Common learning;
  • common belief;
  • private signals;
  • private beliefs

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.

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