The probability of nontrivial common knowledge

• Published in: Games and economic behavior 2012-11, Vol.76 (2), p.556-570
• Main Authors: Collevecchio, Andrea, LiCalzi, Marco
• Format: Article
• Language: English
• Subjects: Case studies; Cognition; Cognition & reasoning; Economic models; Epistemic game theory; Game theory; Interest rates; Knowledge; Meet of partitions; Phase transitions; Probability; Random partitions; Studies
• ISSN: 0899-8256; 1090-2473
• DOI: 10.1016/j.geb.2012.07.014

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. ► 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.