Common knowledge

Common knowledge is information that every member of a group knows, and that every member knows everyone knows, through arbitrarily many levels of recursion.

Mutual knowledge: A fact is mutual knowledge when every member of a group knows it. Common knowledge begins with mutual knowledge but adds infinitely many levels of awareness.
Coordination game: A game in which players benefit from choosing actions that match one another. Players may fail to coordinate when they lack common knowledge of which convention applies.
Propositional logic: A formal system for reasoning about propositions and their logical relations. Common-knowledge definitions apply recursively to propositions that can be true or false.
Public knowledge: Information openly available to members of a group, though its mutual awareness may be unspecified. Public availability alone does not always establish the infinite recursive awareness required here.
Epistemic logic: A formal logic for representing what agents know and believe. Its operators express individual knowledge and the recursive structure of common knowledge.
Battle of the sexes: A two-player coordination game in which players prefer different coordinated outcomes. It illustrates how shared expectations can guide choices when preferences do not fully align.
Possible-world semantics: A method for interpreting modal statements through sets of possible worlds. An agent knows a fact when it holds in every world the agent considers possible.
Common belief: A proposition that every member believes, and that every member believes everyone believes, recursively. It parallels common knowledge but does not require the proposition to be true.
Knowledge operator: A formal operator that states that an agent knows a proposition. Repeated knowledge operators encode who knows that others know the fact.
Aumann's agreement theorem: A result stating that agents with common priors cannot agree to disagree when their posteriors are common knowledge. It shows how public mutual awareness constrains persistent disagreement.

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