## Proof of Proposition 3.1

Proposition 3.1.
Let Ω be a finite set of states of the world. Suppose that
1. Agents i and j have a common prior probability distribution μ(·) over the events of Ω such that μ(ω) > 0 for each ω ∈ Ω, and
2. It is common knowledge at ω that i's posterior probability of event E is qi(E) and that j's posterior probability of E is qj(E).
Then qi(E) = qj(E).

Proof.
Let be the meet of all the agents' partitions, and let ( ω) be the element of containing ω. Since ( ω) consists of cells common to every agents information partition, we can write

 (ω) = ∪ k Hik,

where each Hiki. Since i's posterior probability of event E is common knowledge, it is constant on (ω), and so

 qi(E) = μ(E | Hik) for all k

Hence,

 μ( E ∩ Hik) = qi(E) μ(Hik)

and so

 μ(E ∩ (ω))
=
 μ(E ∩ ∪ k Hik)
=
 μ( ∪ k E ∩ Hik)
=
 Σ k μ(E ∩ Hik)
=
 Σ k qi(E) μ(Hik)
=
 qi(E) Σ k μ(Hik)
=
 qi(E) μ( ∪ k Hik)
= qi(E) μ((ω))

Applying the same argument to j, we have

 μ(E ∩ (ω)) = qj(E) μ((ω))

so we must have qi(E) = qj(E).