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## Stanford Encyclopedia of Philosophy

Supplement to Common Knowledge

In a game , 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
_{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.

*First published: August 27, 2001*

*Content last modified: August 27, 2001*