Stanford Encyclopedia of Philosophy
Supplement to Common Knowledge

Proof of Proposition 3.4

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 _i(s_{k j}) > 0, then s_{k j} 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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