#### Supplement to Common Knowledge

## 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.