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Proof of Proposition 3.11

Proof.
We must show that s : Ω → S as defined by the _i-measurable s_i's of the Bayesian rational agents is an objective Aumann correlated equilibrium. Let i ∈ n and ω ∈ Ω be given, and let g_i : Ω → S_i be any function that is a function of s_i. Since s_i is constant over each cell of _i, g_i must be as well, that is, g_i is _i-measurable. By Bayesian rationality,

E(uis | i)(ω) ≥ E (ui(gi,s−i) | i)(ω)

Since ω was chosen arbitrarily, we can take iterated expectations to get

E(E(uis | i)(ω)) ≥ E(E(ui(gi,s−i) | i)(ω))

which implies that

E(uis) ≥ E(ui(gi,s−i))

so s is an Aumann correlated equilibrium.

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