## Proof of Proposition 3.11

Proposition 3.11 (Aumann 1987)
If each agent $$i \in$$ N is $$\omega$$-Bayes rational at each possible world $$\omega \in \Omega$$, 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 $$s : \Omega \rightarrow S$$ as defined by the $$\mathcal{H}_i$$-measurable $$s_i$$’s of the Bayesian rational agents is an objective Aumann correlated equilibrium. Let $$i \in n$$ and $$\omega \in \Omega$$ be given, and let $$g_i : \Omega \rightarrow S_i$$ be any function that is a function of $$s_i$$. Since $$s_i$$ is constant over each cell of $$\mathcal{H}_i, g_i$$ must be as well, that is, $$g_i$$ is $$\mathcal{H}_i$$-measurable. By Bayesian rationality,

$E(u_i \circ s \mid \mathcal{H}_i)(\omega) \ge E(u_i (g_i,s_{-i}) \mid \mathcal{H}_i)(\omega)$

Since $$\omega$$ was chosen arbitrarily, we can take iterated expectations to get

$E(E(u_i \circ s \mid \mathcal{H}_i)(\omega)) \ge E(E(u_i (g_i,s_{-i}) \mid \mathcal{H}_i)(\omega))$

which implies that

$E(u_i \circ s) \ge E(u_i (g_i,s_{-i}))$

so $$s$$ is an Aumann correlated equilibrium.