Common Knowledge

An event is common knowledge if everyone knows it, everyone knows that everyone knows it, and so on ad infinitum.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

Formally $E$ is common knowledge if $E\subseteq\bigcap_{n\ge 1}K^n E$ , where $K E=\bigcap_i K_i E$ and $K_i$ denotes player $i$ 's knowledge operator.

In a Kripke model, $E$ is common knowledge at state $\omega$ iff $E$ holds at every state reachable by the reflexive transitive closure of the agents' accessibility relations.

Intuition

It is the infinite tower of mutual knowledge about a fact — not just you know, but you know I know, and I know you know I know, …

Small failures of this tower can dramatically change strategic outcomes.

Worked example

Aumann's "Agreeing to Disagree": common priors plus common knowledge of posteriors implies equal posteriors.

The Muddy Children / Blue-Eyed Islanders puzzle hinges on common knowledge of the puzzle setup.

The math

Rubinstein's e-mail game shows that approximate common knowledge is strictly weaker than full common knowledge.

Monderer-Samet characterize common knowledge via self-evident events.

Where it is used

Foundational for epistemic game theory, speculative trade, and cryptographic coordination.

Critical in the analysis of market microstructure and information cascades.

Go deeper

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Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.