## 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.