This is a file in the archives of the Stanford Encyclopedia of Philosophy. |

## Stanford Encyclopedia of Philosophy

Supplement to Common Knowledge

If each agent i N is -Bayes rational at each possible world , 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 :
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.

*First published: August 27, 2001*

*Content last modified: August 27, 2001*