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The probability of nontrivial common knowledge

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  • Collevecchio, Andrea
  • LiCalzi, Marco

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.

Suggested Citation

  • Collevecchio, Andrea & LiCalzi, Marco, 2012. "The probability of nontrivial common knowledge," Games and Economic Behavior, Elsevier, vol. 76(2), pages 556-570.
    • Handle: RePEc:eee:gamebe:v:76:y:2012:i:2:p:556-570
    DOI: 10.1016/j.geb.2012.07.014
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    1. Rinott, Yosef & Scarsini, Marco, 2000. "On the Number of Pure Strategy Nash Equilibria in Random Games," Games and Economic Behavior, Elsevier, vol. 33(2), pages 274-293, November.
    2. R.J. Aumann & S. Hart (ed.), 2002. "Handbook of Game Theory with Economic Applications," Handbook of Game Theory with Economic Applications, Elsevier, edition 1, volume 3, number 3.
    3. Hellman, Ziv & Samet, Dov, 2012. "How common are common priors?," Games and Economic Behavior, Elsevier, vol. 74(2), pages 517-525.
    4. Dimitri, Nicola, 1993. "Learning partitions," Economics Letters, Elsevier, vol. 42(2-3), pages 195-199.
    5. Geanakoplos, John, 1994. "Common knowledge," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 2, chapter 40, pages 1437-1496, Elsevier.
    6. Marco LiCalzi & Oktay Surucu, 2012. "The Power of Diversity over Large Solution Spaces," Management Science, INFORMS, vol. 58(7), pages 1408-1421, July.
    7. John Geanakoplos, 1992. "Common Knowledge," Journal of Economic Perspectives, American Economic Association, vol. 6(4), pages 53-82, Fall.
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    Cited by:

    1. Marco LiCalzi & Oktay Surucu, 2012. "The Power of Diversity over Large Solution Spaces," Management Science, INFORMS, vol. 58(7), pages 1408-1421, July.
    2. Marco LiCalzi & Lucia Milone, 2012. "Talent management in triadic organizational architectures," Lecture Notes in Economics and Mathematical Systems, in: Andrea Teglio & Simone Alfarano & Eva Camacho-Cuena & Miguel Ginés-Vilar (ed.), Managing Market Complexity, edition 127, chapter 0, pages 169-181, Springer.

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    More about this item

    Keywords

    Epistemic game theory; Random partitions; Meet of partitions;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

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