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Proof of Proposition 3.7

Proposition 3.7
Assume that the probabilities
boldface-mu = ( mu 1, . . . , mu n ) in Delta 1 ( S - 1 ) x . . . x Delta n ( S - n )

are common knowledge. Then common knowledge of Bayesian rationality is satisfied if, and only if, boldface-mu 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 in N, if mu i( s k j ) > 0 for each agent i not equal to k, then s k j must be optimal for k given some distribution sigmak in Delta k ( S - k ). Since the agents' distributions are common knowledge, this distribution is precisely mu k , so ( 3.iii ) is satisfied for k. ( 3.iii ) is similarly established for each other agent i not equal to k, so boldface-mu is an endogenous correlated equilibrium.

Now suppose that boldface-mu 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.

Copyright © 2001 by
Peter Vanderschraaf
peterv@cyrus.andrew.cmu.edu

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First published: August 27, 2001
Content last modified: August 27, 2001