Elsevier

Games and Economic Behavior

Volume 76, Issue 2, November 2012, Pages 556-570
Games and Economic Behavior

The probability of nontrivial common knowledge

https://doi.org/10.1016/j.geb.2012.07.014Get rights and content

Abstract

We study the probability that two or more agents can attain common knowledge of nontrivial events when the size of the state space grows large. We adopt the standard epistemic model where the knowledge of an agent is represented by a partition of the state space. Each agent is endowed with a partition generated by a random scheme consistent with his cognitive capacity. Assuming that agentsʼ partitions are independently distributed, we prove that the asymptotic probability of nontrivial common knowledge undergoes a phase transition. Regardless of the number of agents, when their cognitive capacity is sufficiently large, the probability goes to one; and when it is small, it goes to zero. Our proofs rely on a graph-theoretic characterization of common knowledge that has independent interest.

Highlights

► We study the probability that two or more agents attain nontrivial common knowledge. ► Each agent has a random knowledge partition consistent with his cognitive capacity. ► When the size of the state space grows, nontrivial common knowledge undergoes a phase transition. ► Our proofs rely on a novel graph-theoretic characterization of common knowledge.

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We acknowledge insightful comments from the referees, and useful feedback from audiences at Paris-Sorbonne, UECE Lisbon 2011, and Amases 2011. We thank Lucia Milone for graciously sharing one of her amazingly fast computer codes.

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