Common Knowledge > Proof of Proposition 3.11 (Stanford Encyclopedia of Philosophy/Winter 2008 Edition)

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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.

Proof.
We must show that s : Ω → S as defined by the calligraphic-H _i-measurable s_i's of the Bayesian rational agents is an objective Aumann correlated equilibrium. Let i ∈ n and ω ∈ Ω be given, and let g_i : Ω → S_i be any function that is a function of s_i. Since s_i is constant over each cell of _i, g_i must be as well, that is, g_i is calligraphic-H _i-measurable. By Bayesian rationality,

E(u_is | _i)(ω) ≥ E (u_i(g_i,s_−i) | _i)(ω)

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

E(E(u_is | _i)(ω)) ≥ E(E(u_i(g_i,s_−i) | _i)(ω))

which implies that

E(u_is) ≥ E(u_i(g_i,s_−i))

so s is an Aumann correlated equilibrium.

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