Proof of Proposition 3.7

Proposition 3.7
Assume that the probabilities

μ = (μ1,…,μn) ∈ Δ1(S−1) × … × Δn(S−n)

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 μ_i(s_kj) > 0 for each agent i ≠ k, then s_kj must be optimal for k given some distribution σ_k ∈ Δ_k(S_−k). Since the agents’ distributions are common knowledge, this distribution is precisely μ_k , so (3.iii) is satisfied for k. (3.iii) is similarly established for each other agent i ≠ k, 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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