Proof of Proposition 3.7

Proposition 3.7
Assume that the probabilities

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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