Supplement to Common Knowledge
Proof of Lemma 2.16
Lemma 2.16.M(ω) is common knowledge for the agents of N at ω.
Proof.
Since
M
is a coarsening of
Hi for
each i ∈ N,
Ki(M(ω)).
Hence,
K1N(M(ω) ), and since by definition
Ki(M(ω)) =
{ ω |
Hi(ω)
⊆
M(ω)}
=
M(ω),
K1N(M(ω)) =
∩
i ∈ NKi(M(ω)) = M(ω)
Applying the recursive definition of mutual knowledge, for any m ≥ 1,
| KmN(M(ω)) = |
∩ i ∈ N |
Ki(Km−1N(M(ω)) = |
∩ i ∈ N |
Ki(M(ω)) = M(ω) |
so, since ω ∈ M(ω), by definition we have ω ∈ K*N(M(ω)). □
