Supplement to Common Knowledge
Proof of Proposition 3.11
Proposition 3.11 (Aumann 1987)If each agent i ∈ N is ω-Bayes rational at each possible world ω ∈ Ω, then the agents are following an Aumann correlated equilibrium. If the CPA is satisfied, then the correlated equilibrium is objective.
Proof.
We must show that s : Ω → S as defined by the
Hi-measurable
si's of the Bayesian rational agents
is an objective Aumann
correlated equilibrium. Let i ∈ n and
ω ∈ Ω be
given, and let gi
: Ω → Si be any
function that is a function of si. Since
si is
constant over each cell of
Hi,
gi must be as well,
that is, gi is
Hi-measurable.
By Bayesian
rationality,
E(uis | Hi)(ω) ≥ E (ui(gi,s−i) | Hi)(ω)
Since ω was chosen arbitrarily, we can take iterated expectations to get
E(E(uis | Hi)(ω)) ≥ E(E(ui(gi,s−i) | Hi)(ω))
which implies that
E(uis) ≥ E(ui(gi,s−i))
so s is an Aumann correlated equilibrium.
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>