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
H_{i} for
each i ∈ N,
K_{i}(M(ω)).
Hence,
K^{1}_{N}(M(ω) ), and since by definition
K_{i}(M(ω)) =
{ ω |
H_{i}(ω)
⊆
M(ω)}
=
M(ω),
K^{1}_{N}(M(ω)) =
∩
i ∈ NK_{i}(M(ω)) = M(ω)
Applying the recursive definition of mutual knowledge, for any m ≥ 1,
K^{m}_{N}(M(ω)) = |
∩ i ∈ N |
K_{i}(K^{m−1}_{N}(M(ω)) = |
∩ i ∈ N |
K_{i}(M(ω)) = M(ω) |
so, since ω ∈ M(ω), by definition we have ω ∈ K*_{N}(M(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>