Common Knowledge, explained.
A fact is common knowledge when everyone knows it, everyone knows that everyone knows it, and this mutual knowledge continues at every level.
Why it happens
Three perfect logicians see other hats but not their own. Each simultaneous silent round eliminates worlds in which someone would already know their hat was blue. A public announcement excludes the all-white world and starts the induction.
A fact is common knowledge when everyone knows it, everyone knows that everyone knows it, and this mutual knowledge continues at every level.
Read the result
Count simultaneous silent rounds under the same public protocol; a public announcement removes a world that private observation alone may not remove.
A worked example
Three blue hats and a public fact
Keep all three hats blue and enable the public announcement that at least one is blue.
Hear two silent rounds as participants eliminate worlds in which someone would have known earlier.
Hear the third-round deduction, then reset without the announcement to see why silence no longer starts the same induction.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Three perfect logicians see other hats but not their own. Each simultaneous silent round eliminates worlds in which someone would already know their hat was blue. A public announcement excludes the all-white world and starts the induction.
Where this idea is useful
A practical use
When coordinating a group, distinguish privately sending the same information from establishing a shared public fact.
A common misconception
“Everyone privately knowing a fact makes it common knowledge.”
Mutual knowledge includes what each person knows about others’ knowledge at every level. A public event can change these higher-order beliefs.
What this explanation leaves out
- Perfect reasoning, simultaneous rounds and common knowledge of the protocol are essential; real communication can be delayed or misunderstood.
Why does nobody act on the first round with three blue hats?
Each blue-hatted person still considers a world with only the other two blue, where those people would also initially be unsure.
Does your coordination message establish that others know everyone received it?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.