#### Supplement to Common Knowledge

## Proof of Proposition 3.7

**Proposition 3.7**

Assume that the probabilities

μ= (μ_{1},…,μ_{n}) ∈ Δ_{1}(S_{−1}) × … × Δ_{n}(S_{−n})

are common knowledge. Then common knowledge of Bayesian rationality
is satisfied if, and only if, **μ** 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*
∈ N, if
μ_{i}(*s*_{kj})
> 0 for each agent *i* ≠ *k*, then
*s*_{kj} must be optimal for
*k* given some distribution σ_{k} ∈
Δ_{k}(S_{−k}). Since the
agents' distributions are common knowledge, this distribution is
precisely μ_{k} , so (3.iii) is satisfied for
*k*. (3.iii) is similarly established for each other agent
*i* ≠ *k*, so **μ** is an endogenous
correlated equilibrium.

Now suppose that **μ** 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.

Peter Vanderschraaf <

*pvanderschraaf@ucmerced.edu*>

Giacomo Sillari <

*gsillari@sas.upenn.edu*>