(
) By Lemma 2.12, 
( 
)
is common knowledge at
, so E is common knowledge at by Proposition 2.4.
| This is a file in the archives of the Stanford Encyclopedia of Philosophy. |
be the meet of the agents' partitions
i
for each i
N.
A proposition E
is common knowledge for the agents of N at
iff
()
E.
In Aumann (1976), E is defined to be common knowledge at
iff
()
E.
Proof.
() By Lemma 2.12, ( )
is common knowledge at
, so E is common knowledge at by Proposition 2.4.
(
) We must show that K *N ( E ) implies that ( ) E.
Suppose that there exists 
( )
such that 
E.
Since ( ),
is reachable from ,
so there exists a sequence 0, 1, . . . , m - 1 with associated states 1, 2, . . . , m and information sets i k( k ) such that 0 = , m = , and k i k( k + 1). But at information set i k( m ), agent i k does not know event E. Working backwards on k, we see that event E cannot be common knowledge, that is, agent i 1 cannot rule out the possibility that agent i 2 thinks that . . . that agent i m - 1 thinks that agent i m does not know E. 
| Peter Vanderschraaf peterv@cyrus.andrew.cmu.edu |