#### 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.

**Proof.**

We must show that *s* : Ω → 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
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
_{i}-measurable.
By Bayesian
rationality,

E(u_{i}s|_{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_{i}s|_{i})(ω)) ≥E(E(u_{i}(g_{i},s_{−i}) |_{i})(ω))

which implies that

E(u_{i}s) ≥E(u_{i}(g_{i},s_{−i}))

so *s* is an Aumann correlated equilibrium.

Copyright © 2007 by

Peter Vanderschraaf <

Giacomo Sillari <

Peter Vanderschraaf <

*pvanderschraaf@gmail.com*>Giacomo Sillari <

*gsillari@andrew.cmu.edu*>