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Proof of Proposition 3.11

Proof.
We must show that s : S as defined by the _i-measurable s _i's of the Bayesian rational agents is an objective Aumann correlated equilibrium. Let i n and be given, and let g _i : 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 _i-measurable. By Bayesian rationality,

E ( u i s | i )( ) E ( u i ( g i , s - i )| i )( )

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

E ( E ( u i s | i )( ) ) E ( E ( u i ( g i , s - i )| i )( ) )

which implies that

E ( u i s ) E ( u i ( g i , s - i ) )

so s is an Aumann correlated equilibrium.

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