Supplement to Common Knowledge

Proof of Proposition 3.7

Proposition 3.7
Assume that the probabilities
μ = (μ1,…,μn) ∈ Δ1(S−1) × … × Δn(Sn)

are common knowledge. Then common knowledge of Bayesian rationality is satisfied if, and only if, μ 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 μi(skj) > 0 for each agent ik, then skj must be optimal for k given some distribution σk ∈ Δk(Sk). Since the agents' distributions are common knowledge, this distribution is precisely μk , so (3.iii) is satisfied for k. (3.iii) is similarly established for each other agent ik, so μ is an endogenous correlated equilibrium.

Now suppose that μ 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.

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Copyright © 2013 by
Peter Vanderschraaf <pvanderschraaf@ucmerced.edu>
Giacomo Sillari <gsillari@sas.upenn.edu>

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