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_{} = ( _{ 1}, . . . , _{ n} ) _{ 1} ( S_{ - 1} ) x . . . x _{ n} ( S_{ - n} )
are common knowledge. Then common knowledge of Bayesian rationality is satisfied if, and only if, _{} 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
_{ i}( s_{ k j} ) > 0
for each agent i
k,
then s_{ k j} must be optimal for k given some distribution
_{k}
_{ k} ( S_{ - k} ).
Since the agents' distributions are common knowledge, this
distribution is precisely
_{ k} ,
so ( 3.iii ) is satisfied for k. ( 3.iii ) is similarly
established for each other agent i
k, so
_{} is an
endogenous correlated equilibrium.
Now suppose that
_{}
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.
First published: August 27, 2001
Content last modified: August 27, 2001