common knowledge

Definition (common knowledge)

intuition

Fact $F$ is common knowledge among players of a game if all players know $F$, akll the players know that all players know $F$, all players know that all players know that all players know $F$, etc. (for finite number of levels).

formal definition

Let $(N, Y, (\mathcal{F}_i)_{i \in N},\mathfrak{s})$ be an Aumann model of incomplete information, let $A \subseteq Y$ be an event, and let $\omega \in Y$ be a state of the world. The event $A$ is common knowledge in $\omega$ if for every finite sequence of players $i_1, i_2, ..., i_l$,

i.e. event $A$ is common knowledge at state of the world $\omega$ if in $\omega$ every player knows event $A$, every player knows that every player knows event $A$, etc.

Theorem (common knowledge for connected component of graph)

#incomplete

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, p. 331.
2. https://en.wikipedia.org/wiki/Common_knowledge_(logic)