Proof of Proposition 3.7

Proposition 3.7
Assume that the probabilities

\[ \boldsymbol{\mu} = (\mu_1 ,\ldots ,\mu_n) \in \Delta_1(S_{-1}) \times \ldots \times \Delta_n(S_{-n}) \]

are common knowledge. Then common knowledge of Bayesian rationality is satisfied if, and only if, \(\boldsymbol{\mu}\) is an endogenous correlated equilibrium.

Proof.
Suppose first that common knowledge of Bayesian rationality is satisfied. Then, by Proposition 3.4, for a given agent \(k \in N,\) if \(\mu_i(s_{kj}) \gt 0\) for each agent \(i \ne k\), then \(s_{kj}\) must be optimal for \(k\) given some distribution \(\sigma_{k} \in \Delta_k(S_{-k}).\) Since the agents’ distributions are common knowledge, this distribution is precisely \(\mu_k\), so (3.iii) is satisfied for \(k\). (3.iii) is similarly established for each other agent \(i \ne k\), so \(\boldsymbol{\mu}\) is an endogenous correlated equilibrium.

Now suppose that \(\boldsymbol{\mu}\) is an endogenous correlated equilibrium. Then, since the distributions are common knowledge, (3.i) is common knowledge, so common knowledge of Bayesian rationality is satisfied by Proposition 3.4.

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