Common learning

Martin W. Cripps, Jeffrey C. Ely, George J. Mailath, Larry Samuelson

Research output: Contribution to journalArticlepeer-review

24 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)909-933
Number of pages25
Issue number4
StatePublished - Jul 2008


  • Common belief
  • Common learning
  • Private beliefs
  • Private signals

ASJC Scopus subject areas

  • Economics and Econometrics


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