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Common Knowledge, Regained

Yannai A. Gonczarowski, Yoram Moses

TL;DR

This paper introduces a new definition of common knowledge that relaxes the simultaneity requirement in dynamic settings with timing frictions, resolving the Halpern--Moses paradox while preserving key implications of CK. It defines CK through time-invariant events tied to locally observed events and derives an Induction Rule, co-occurrence, and reachability tools to prove CK can emerge in finite time even when traditional CK cannot. The authors prove an agreement theorem in this dynamic setting and show that CK under the new definition characterizes equilibrium behavior in coordinated-attack games, unifying seemingly disparate equilibria. The framework extends to applications in dynamic coordination and equilibria analysis, with potential implications for distributed computing, online platforms, and mechanism design, while acknowledging limitations such as probabilistic delivery and suggesting directions for further work on probabilistic CK notions. CK under this approach is thus powerful yet robust to asynchrony, providing a principled foundation for reasoning about shared knowledge and coordinated action in realistic, time-frictioned environments.

Abstract

For common knowledge to arise in dynamic settings, all players must simultaneously come to know it has arisen. Consequently, common knowledge cannot arise in many realistic settings with timing frictions. This counterintuitive observation of Halpern and Moses (1990) was discussed by Arrow et al. (1987) and Aumann (1989), was called a paradox by Morris (2014), and has evaded satisfactory resolution for four decades. We resolve this paradox by proposing a new definition for common knowledge, which coincides with the traditional one in static settings but is more permissive in dynamic settings. Under our definition, common knowledge can arise without simultaneity, particularly in canonical examples of the Haplern-Moses paradox. We demonstrate its usefulness by deriving for it an agreement theorem à la Aumann (1976), showing it arises in the setting of Geanakoplos and Polemarchakis (1982) with timing frictions added, and applying it to characterize equilibrium behavior in a dynamic coordination game.

Common Knowledge, Regained

TL;DR

This paper introduces a new definition of common knowledge that relaxes the simultaneity requirement in dynamic settings with timing frictions, resolving the Halpern--Moses paradox while preserving key implications of CK. It defines CK through time-invariant events tied to locally observed events and derives an Induction Rule, co-occurrence, and reachability tools to prove CK can emerge in finite time even when traditional CK cannot. The authors prove an agreement theorem in this dynamic setting and show that CK under the new definition characterizes equilibrium behavior in coordinated-attack games, unifying seemingly disparate equilibria. The framework extends to applications in dynamic coordination and equilibria analysis, with potential implications for distributed computing, online platforms, and mechanism design, while acknowledging limitations such as probabilistic delivery and suggesting directions for further work on probabilistic CK notions. CK under this approach is thus powerful yet robust to asynchrony, providing a principled foundation for reasoning about shared knowledge and coordinated action in realistic, time-frictioned environments.

Abstract

For common knowledge to arise in dynamic settings, all players must simultaneously come to know it has arisen. Consequently, common knowledge cannot arise in many realistic settings with timing frictions. This counterintuitive observation of Halpern and Moses (1990) was discussed by Arrow et al. (1987) and Aumann (1989), was called a paradox by Morris (2014), and has evaded satisfactory resolution for four decades. We resolve this paradox by proposing a new definition for common knowledge, which coincides with the traditional one in static settings but is more permissive in dynamic settings. Under our definition, common knowledge can arise without simultaneity, particularly in canonical examples of the Haplern-Moses paradox. We demonstrate its usefulness by deriving for it an agreement theorem à la Aumann (1976), showing it arises in the setting of Geanakoplos and Polemarchakis (1982) with timing frictions added, and applying it to characterize equilibrium behavior in a dynamic coordination game.
Paper Structure (23 sections, 17 theorems, 7 equations)

This paper contains 23 sections, 17 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Further Related Literature
  3. Preliminaries
  4. Model
  5. Model Primitives
  6. Events, Knowledge and Local Events
  7. Traditionally Defined, "Instantaneous" Common Knowledge
  8. The Halpern--Moses Paradox
  9. Overcoming the Tyranny of the Clock
  10. Common Knowledge, Redefined
  11. Induction Rule
  12. Co-occurrence
  13. Reachability
  14. Getting to Common Knowledge despite Timing Frictions
  15. Leveraging Common Knowledge: Agreement as a Case Study
  16. ...and 8 more sections

Key Result

Theorem 1

Consider any BDTF. Let $\omega$ be a history such that $d^\alpha_\omega+d^\beta_\omega>2$. If a fact is not common knowledge (as traditionally defined) between $\alpha$ and $\beta$ at time $0$ in $\omega$, then it is never common knowledge between them at any later time in $\omega$.

Theorems & Definitions (41)

  • Definition 1: Bilateral DCMAK with Timing Frictions (BDTF)
  • Theorem 1: No New Common Knowledge with Timing Frictions HalpernM90SteinerS11
  • proof : Proof of \ref{['no-ck']}
  • Definition 2: Common Knowledge
  • Lemma 1
  • proof
  • Lemma 2: Locality of Common Knowledge
  • proof
  • Lemma 3: Common Knowledge as a Fixed Point
  • proof
  • ...and 31 more