Supplement to Common Knowledge
Proof of Lemma 2.16
Lemma 2.16.(ω) is common knowledge for the agents of N at ω.
Proof.
Since
is a coarsening of
i for
each i ∈ N,
Ki((ω)).
Hence,
K1N((ω) ), and since by definition
Ki((ω)) =
{ ω |
i(ω)
⊆
(ω)}
=
(ω),
K1N((ω)) =
∩
i ∈ NKi((ω)) = (ω)
Applying the recursive definition of mutual knowledge, for any m ≥ 1,
KmN((ω)) = |
∩ i ∈ N |
Ki(Km−1N((ω)) = |
∩ i ∈ N |
Ki((ω)) = (ω) |
so, since ω ∈ (ω), by definition we have ω ∈ K*N((ω)). □
Copyright © 2007 by
Peter Vanderschraaf <pvanderschraaf@gmail.com>
Giacomo Sillari <gsillari@sas.upenn.edu>
Peter Vanderschraaf <pvanderschraaf@gmail.com>
Giacomo Sillari <gsillari@sas.upenn.edu>