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## 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

 ( ) = k H ik,

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

 ( E ( ) )
=
 ( E k H ik )
=
 ( k E H ik )
=
 k ( E H ik )
=
 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 ).