#### Supplement to Common Knowledge

## Proof of Proposition 3.4

**Proposition 3.4**.

In a game Γ, common knowledge of Bayesian rationality is satisfied if, and only if, (3.i) is common knowledge.

**Proof.**

Suppose first that common knowledge of Bayesian rationality is
satisfied. Since it is common knowledge that agent i knows that agent k
is Bayesian rational, it is also common knowledge that if
μ_{i}(*s*_{kj})
> 0, then *s*_{kj} must be
optimal for *k* given some belief over S_{-k},
so (3.i) is
common knowledge.

Suppose now that (3.i) is common knowledge. Then, by (3.i), agent
*i* knows that agent *k* is Bayesian rational. Since
(3.i) is common knowledge, all statements of the form ‘For
*i*, *j*, … , *k* ∈ *N*, *i*
knows that *j* knows that … is Bayesian rational’
follow by induction.
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