Ti-measurable by the definition of a frame, they represent message-contingent action plans finally, I and T model an abstract, general randomizing(or "correlating ")device. The idea is that, upon observing w E Q, the outside observer sends the message ti(w) to every Player i∈N It is clear that every correlated equilibrium a according to Definition 5 can as an extended correlated equilibrium as per Definition 6: let Q= supp a, i. e. the set of action profiles that get played in equilibrium; then define type partitions indirectly, via the possibility correspondence, assuming that, at each state w=(ai, a-i)E S, Player i is told what her action must be: t(a,a-)={a}×{a-a:(a,a-)∈ supp c} Since ti(ai, a-i) actually depends on a; only, I denote this type by ti. Finally, let a(ai, a-i) ai and T= a. With these definitions, note that, for every ai E Ai, and for every w E ti, ai(w)=ai and wlti)=a(a_ilai). This implies that()in Definition 6 must hold mes yy the revelation principle, the converse is also true. Intuitively, instead of sending the essage ti () to Player i whenever w is realized, the outside observer could simply instruct Player i to play a(w). If it was unprofitable to deviate from ai ()in the original messaging Formally, given an extended correlated equilibrium(F, T), definewell setting, then it must be unprofitable to do so in the simplified game as (a)=丌({u:Wi∈NM,a(u)=a} now observe that, for any ai E Ai, a( x A-i)>0 iff there exists a(maximal) collection of types ti, ..., tK E T such that, at all states w E U_, th, ai ( w)=ai. Now(3)in Definition 6 implies that ∑1(a,(a1(u)1)r(叫lUt4)≥∑a(a1(a1(u)≠)r(叫∪t) u∈ k=1 for all a E Ai, because all types have positive probability. We can clearly rewrite the above summations as follows ∑u(an,a-)π∩a=∪≥∑a(a2-)(∩a,=】∪ and since(n+ila,=aillUk-1ti)=a(a-ilai)by construction, a is a correlated equilibrium according to Proposition 0.2Ti-measurable by the definition of a frame, they represent message-contingent action plans; finally, Ω and π model an abstract, general randomizing (or “correlating”) device. The idea is that, upon observing ω ∈ Ω, the outside observer sends the message ti(ω) to every Player i ∈ N. It is clear that every correlated equilibrium α according to Definition 5 can be interpreted as an extended correlated equilibrium as per Definition 6: let Ω = supp α, i.e. the set of action profiles that get played in equilibrium; then define type partitions indirectly, via the possibility correspondence, assuming that, at each state ω = (ai , a−i) ∈ Ω, Player i is told what her action must be: ti(ai , a−i) = {ai} × {a 0 −i : (ai , a0 −i ) ∈ supp α}. Since ti(ai , a−i) actually depends on ai only, I denote this type by t ai i . Finally, let ai(ai , a−i) = ai and π = α. With these definitions, note that, for every ai ∈ Ai , and for every ω = (ai , a−i) ∈ t ai i , ai(ω) = ai and π(ω|t ai i ) = α(a−i |ai). This implies that (3) in Definition 6 must hold. By the Revelation principle, the converse is also true. Intuitively, instead of sending the message ti(ω) to Player i whenever ω is realized, the outside observer could simply instruct Player i to play ai(ω). If it was unprofitable to deviate from ai(ω) in the original messaging setting, then it must be unprofitable to do so in the simplified game as well. Formally, given an extended correlated equilibrium (F, π), define α(a) = π({ω : ∀i ∈ N, ai(ω) = ai}); now observe that, for any ai ∈ Ai , α({ai}×A−i) > 0 iff there exists a (maximal) collection of types t 1 i , . . . , tK i ∈ Ti such that, at all states ω ∈ SK k=1 t k i , ai(ω) = ai . Now (3) in Definition 6 implies that X ω∈Ω ui(ai ,(aj (ω))j6=i)π(ω| [ K k=1 t k i ) ≥ X ω∈Ω ui(a 0 i ,(aj (ω))j6=i)π(ω| [ K k=1 t k i ) for all a 0 i ∈ Ai , because all types have positive probability. We can clearly rewrite the above summations as follows: X a−i∈A−i ui(ai , a−i)π( \ j6=i [aj = aj ]| [ K k=1 t k i ) ≥ X a−i∈A−i ui(a 0 i , a−i)π( \ j6=i [aj = aj ]| [ K k=1 t k i ) and since π( T j6=i [aj = aj ]| SK k=1 t k i ) = α(a−i |ai) by construction, α is a correlated equilibrium according to Proposition 0.2. 7