Supplement to Common Knowledge
Proof of Proposition 3.11Proposition 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.
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.