#### Supplement to Common Knowledge

## 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,

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.