Proposition 0. 2 draws a connection between the notions of Nash and Correlated equi- librium: recall that, in the former, an action receives positive probability iff it is a best response to the equilibrium belief. Note also that, if a E A(A) is an independent probability distribution, a(a-ilai)=a-i(a-i), the marginal of a on A-i, for all ai E Ai. Thus, every Nash equilibrium is a correlated equilibrium Moreover, Proposition 0. 2 reinforces our interpretation of correlated equilibrium as an attempt to depart from independence of beliefs, while remaining firmly within an equilibrium setting The first step in the epistemic characterization of correlated equilibrium is actually prompted by a more basic question. The story that is most often heard to justify corre- lated equilibrium runs along the following lines: the players bring in an outside observer who randomizes according to the distribution a and prescribes an action to each player Definition 5 then essentially requires that the players find it profitable ex-ante to follow the prescription rather than adopt any alternative prescription- contingent plan (i.e. "if the observer tells me to do X, I shall do Y instead"). Proposition 0.2 shows that this is equivalent to assuming that, upon receiving a prescription, players do not gain by deviating to any other action The basic question that should have occurred to you is whether a richer communication structure"allows for more coordination opportunitiesie. whether there exists expected payoff vectors which may be achieved using a richer structure, but may not be achieved when messages are limited to action prescriptions The answer to this question is actually negative, as follows from a simple application of the Revelation principle. However, the point is that frames may also be used to define correlated equilibria in this extended sense Definition 6 Fix a game G=(, (Ai, ui)iEN). An extended correlated equilibrium is a (1)F=(Q, (Ti, aieN) is a frame for G (2)丌∈△(g) is a probability over S2 such that, for all i∈ N and ti∈T,丌(t)>0; (3) for every player i∈ N and t∈T ∑1(a(,、a1(u)1)r(|t)≥∑(,(()≠)x() u∈ for all c∈A The similarity between 3)and Proposition 0. 2 should be obvious Note that the formal definition of a frame in an extended correlated equilibrium is as in Definition 1. However, the standard interpretation is different: the cells ti E Ti represent possible messages that the observer may send to Player i; since the action functions ai areProposition 0.2 draws a connection between the notions of Nash and Correlated equi￾librium: recall that, in the former, an action receives positive probability iff it is a best response to the equilibrium belief. Note also that, if α ∈ ∆(A) is an independent probability distribution, α(a−i |ai) = α−i(a−i), the marginal of α on A−i , for all ai ∈ Ai . Thus, every Nash equilibrium is a correlated equilibrium. Moreover, Proposition 0.2 reinforces our interpretation of correlated equilibrium as an attempt to depart from independence of beliefs, while remaining firmly within an equilibrium setting. The first step in the epistemic characterization of correlated equilibrium is actually prompted by a more basic question. The story that is most often heard to justify corre￾lated equilibrium runs along the following lines: the players bring in an outside observer who randomizes according to the distribution α and prescribes an action to each player. Definition 5 then essentially requires that the players find it profitable ex-ante to follow the prescription rather than adopt any alternative prescription-contingent plan (i.e. “if the observer tells me to do X, I shall do Y instead”). Proposition 0.2 shows that this is equivalent to assuming that, upon receiving a prescription, players do not gain by deviating to any other action. The basic question that should have occurred to you is whether a richer “communication structure” allows for more coordination opportunities—i.e. whether there exists expected payoff vectors which may be achieved using a richer structure, but may not be achieved when messages are limited to action prescriptions. The answer to this question is actually negative, as follows from a simple application of the Revelation principle. However, the point is that frames may also be used to define correlated equilibria in this extended sense. Definition 6 Fix a game G = (N,(Ai , ui)i∈N ). An extended correlated equilibrium is a tuple (F, π) where: (1) F = (Ω,(Ti , ai)i∈N ) is a frame for G; (2) π ∈ ∆(Ω) is a probability over Ω such that, for all i ∈ N and ti ∈ Ti , π(ti) > 0; (3) for every player i ∈ N and ti ∈ Ti , X ω∈Ω ui(ai(ω),(aj (ω))j6=i)π(ω|ti) ≥ X ω∈Ω ui(a 0 i ,(aj (ω))j6=i)π(ω|ti) for all a 0 i ∈ Ai . The similarity between (3) and Proposition 0.2 should be obvious. Note that the formal definition of a frame in an extended correlated equilibrium is as in Definition 1. However, the standard interpretation is different: the cells ti ∈ Ti represent possible messages that the observer may send to Player i; since the action functions ai(·) are 6