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.