The bottom line is that we can actually take Definition 6 as our basic notion of correlated equilibrium(Or does this, for example): there is no added generality But then we get an epistemic characterization of correlated equilibrium almost for free. Observe that Condition (3)in Definition 6 implies that, at each state w E ti, Player is action ai(w)is a best response to her first-order beliefs, given by a(w(a-i=w: Vi+ (w)=aili(w)). Hence, if we reinterpret an extended correlated equilibrium(F, T)as a model M=(F, (pilin) in which pi =T for all iE N, we get Proposition 0. 3 Fix a gameG=(N, (Ai, lilieN) and a model M=(, (Ti, ai, piie forG. If there exists T∈△(92) such that p;=丌 for all i∈N,andR=!.,then ( O, (Ti, aieN), T) is an extended correlated equilibrium of G Conversely, if a is a correlated equilibrium of G, there exists a model M=(Q, (Ti, ai, pilieN) for g in which n=R The model alluded to in the second part of the Proposition is of course the one constructed above. Observe that in that model a is indeed a common prior You may feel a bit cheated by this result. After all, it seems all we have done is change our interpretation of the relevant objects. This is of course entirely correct! It is certainly the case that Proposition 0.3 characterizes correlated equilibrium beliefs common prior in a model with the feature that every player is rational at every stale t is a More precisely, a distribution over action profiles is a correlated equilibrium belief if As I have mentioned several times, this may or may not have any behavioral implications but at least Proposition 0. 3 provides an arguably more palatable rationale for correlated equilibrium beliefs than the "outside observer"storThe bottom line is that we can actually take Definition 6 as our basic notion of correlated equilibrium (OR does this, for example): there is no added generality. But then we get an epistemic characterization of correlated equilibrium almost for free. Observe that Condition (3) in Definition 6 implies that, at each state ω ∈ ti , Player i’s action ai(ω) is a best response to her first-order beliefs, given by α π −i (ω)(a−i) = π({ω 0 : ∀j 6= i, aj (ω 0 ) = aj}|ti(ω)). Hence, if we reinterpret an extended correlated equilibrium (F, π) as a model M = (F,(pi)i∈N ) in which pi = π for all i ∈ N, we get: Proposition 0.3 Fix a game G = (N,(Ai , ui)i∈N ) and a model M = (Ω,(Ti , ai , pi)i∈N ) for G. If there exists π ∈ ∆(Ω) such that pi = π for all i ∈ N, and R = Ω, then ((Ω,(Ti , ai)i∈N ), π) is an extended correlated equilibrium of G. Conversely, if α is a correlated equilibrium of G, there exists a model M = (Ω,(Ti , ai , pi)i∈N ) for G in which Ω = R. The model alluded to in the second part of the Proposition is of course the one constructed above. Observe that in that model α is indeed a common prior. You may feel a bit cheated by this result. After all, it seems all we have done is change our interpretation of the relevant objects. This is of course entirely correct! It is certainly the case that Proposition 0.3 characterizes correlated equilibrium beliefs. More precisely, a distribution over action profiles is a correlated equilibrium belief if it is a common prior in a model with the feature that every player is rational at every state. As I have mentioned several times, this may or may not have any behavioral implications, but at least Proposition 0.3 provides an arguably more palatable rationale for correlated equilibrium beliefs than the “outside observer” story. 8