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.

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