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## Proof of Proposition 3.7

Proposition 3.7
Assume that the probabilities
 = ( 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.