Proof of Proposition 3.1
Proposition 3.1.
Let 
be a finite set of states
of the world. Suppose that
( i ) Agents i and j have a common prior probability distribution
(
)
over the events of
such that
(
) > 0
for each
,
and
( ii ) It is common knowledge at
that i's posterior probability of event E is q i( E ) and
that j's posterior probability of E is q j( E ).
Then q i( E ) = q j( E ).
Proof.
Let
be the meet of all the agents' partitions, and let
(
)
be the element of
containing . Since
(
)
consists of cells common to every agents information partition, we can write
where each
H ik
i. Since
i's posterior probability of event E is common knowledge, it is
constant on
(
),
and so
|
q i( E ) =
(
E | H ik ) for all k
|
Hence,
(
E 
H ik ) = q i( E )
(
H ik )
|
and so
|
|
= |
|
= |
|
|
|
=
|
|
=
|
k
|
q i( E )
(
H ik )
|
|
|
|
=
|
|
q i
( E )
|
k
|
(
H ik )
|
|
=
|
|
q i( E )
(
|
k
|
H ik )
|
|
|
|
=
|
q i( E )
(
(
) )
|
Applying the same argument to j, we have
|
( E
(
) ) = q j( E )
(
(
) )
|
so we must have
q i( E ) = q j( E ).
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