Supplement to Common Knowledge

Proof of Proposition 3.11

Proposition 3.11 (Aumann 1987)
If each agent i ∈ N is ω-Bayes rational at each possible world ω ∈ Ω, then the agents are following an Aumann correlated equilibrium. If the CPA is satisfied, then the correlated equilibrium is objective.

We must show that s : Ω → S as defined by the Hi-measurable si's of the Bayesian rational agents is an objective Aumann correlated equilibrium. Let in and ω ∈ Ω be given, and let gi : Ω → Si be any function that is a function of si. Since si is constant over each cell of Hi, gi must be as well, that is, gi is Hi-measurable. By Bayesian rationality,

E(uicircles | Hi)(ω) ≥ E (ui(gi,si) | Hi)(ω)

Since ω was chosen arbitrarily, we can take iterated expectations to get

E(E(uicircles | Hi)(ω)) ≥ E(E(ui(gi,si) | Hi)(ω))

which implies that

E(uicircles) ≥ E(ui(gi,si))

so s is an Aumann correlated equilibrium.

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Peter Vanderschraaf
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