Proof of Proposition 0. 2: (1)Fix a1 E A1 such that al(w(a1)>0; that is, consider an action which, at w, Player 2 expects Player 1 to choose with positive probability. This implies that there exists w such that al(w)= a1 and p2(wt2(w))>0 Our assumptions imply that w E B2(R1)n B2([a2=2 ) B2(Rin a2= o2])(here B2:B2(E∩F)=B2(E)∩B2(F). Hence,by 0.3, supp p2((t2(w))C R1 n(a2=o2]. But then w'E R1n(a2=p2l, which implies that a1=a1(u/)∈r1(a2(u/)=r1(2) Thus, every al such that aw(ai>0 is a best reply to 2. To complete the proof, show that a(w)=1. Note that this is not part of our assumptions; however, our assumptions do imply that w E B2(a1=o1]), and by Lemma 0.4, the required equality follows. The proof of Part(1)is complete (2)Construct a model as follows: let s= lien supp oi and define via the possibility correspondence ti( by lettin v(a1,a2)∈9,t(a1,a2)={a}×supp for i= 1, 2. This is similar to(but simpler than) the construction we used to prove the "revelation principle"in the last lecture notes. States are profiles that have positive proba bility in equilibrium, and each player is informed of her action at any state. This makes it is possible to define ai( Finally, we can use a common prior on S to complete the definition of a model: Pi(a1,a2 P2(a1, a2)=o1(alo(a2). There is nothing special about this; the key point is that, for ev- ery(a1, a2)E Q, Pi(a1, a2 ti (a1,a2))=o-i(a-i) and therefore a-i(a1, a2)=o-i with these definitions. Note that this implies in particular that Ja-i=p-i=@ for i= 1, 2; hence, clearly(a1,a2)∈B(a1=o1]∩[a2=l]) at any state(a1,a2)∈92 Now consider any state w=(a1, a2). Then ai( w)=aiErio-i=ri(a-i(w)), SoWE Ri that is, R1= R2=Q(which is obvious, if you think about it! and therefore, at any state u∈9,u∈B(). We are done■ Observe that, to prove (1), we do not actually need w E Bi(Ri. However, this comes almost for free in part(2). Also, the assumptions imply (via Lemma 0. 4) that players are indeed rational at the state under consideration; this is not used in the proof of Part(1) but again it comes for free in part(2) A key observation is that Aumann and Brandenburger's construction solves another key interpretational problem almost automatically. Mixed actions are viewed as beliefs: thus ne does not even need to invoke randomizations to justify them Rationalizability We now turn to Tan and Werlang's characterization of correlated rationalizability. Actually, I will again present the argument for two-player games, but here everything generalizesProof of Proposition 0.2: (1) Fix a1 ∈ A1 such that α1(ω)(a1) > 0; that is, consider an action which, at ω, Player 2 expects Player 1 to choose with positive probability. This implies that there exists ω 0 such that a1(ω 0 ) = a1 and p2(ω 0 |t2(ω)) > 0. Our assumptions imply that ω ∈ B2(R1) ∩ B2([α2 = φ2]) = B2(R1 ∩ [α2 = φ2]) (here we are using a key property of B2: B2(E ∩ F) = B2(E) ∩ B2(F)). Hence, by Lemma 0.3, supp p2(·|t2(ω)) ⊂ R1 ∩ [α2 = φ2]. But then ω 0 ∈ R1 ∩ [α2 = φ2], which implies that a1 = a1(ω 0 ) ∈ r1(α2(ω 0 )) = r1(φ2). Thus, every a1 such that α1(ω)(a1) > 0 is a best reply to φ2. To complete the proof, we must show that α1(ω) = φ1. Note that this is not part of our assumptions; however, our assumptions do imply that ω ∈ B2([α1 = φ1]), and by Lemma 0.4, the required equality follows. The proof of Part (1) is complete. (2) Construct a model as follows: let Ω = Q i∈N supp φi and define via the possibility correspondence ti(·) by letting ∀(a1, a2) ∈ Ω, ti(a1, a2) = {ai} × supp φ−i for i = 1, 2. This is similar to (but simpler than) the construction we used to prove the “revelation principle” in the last lecture notes. States are profiles that have positive proba￾bility in equilibrium, and each player is informed of her action at any state. This makes it is possible to define ai(a1, a2) = ai . Finally, we can use a common prior on Ω to complete the definition of a model: p1(a1, a2) = p2(a1, a2) = φ1(a1)φ2(a2). There is nothing special about this; the key point is that, for ev￾ery (a1, a2) ∈ Ω, pi(a1, a2|ti(a1, a2)) = φ−i(a−i) and therefore α−i(a1, a2) = φ−i with these definitions. Note that this implies in particular that [α−i = φ−i ] = Ω for i = 1, 2; hence, clearly (a1, a2) ∈ B([α1 = φ1] ∩ [α2 = φ2]) at any state (a1, a2) ∈ Ω. Now consider any state ω = (a1, a2). Then ai(ω) = ai ∈ ri(φ−i) = ri(α−i(ω)), so ω ∈ Ri ; that is, R1 = R2 = Ω (which is obvious, if you think about it!) and therefore, at any state ω ∈ Ω, ω ∈ B(R). We are done. Observe that, to prove (1), we do not actually need ω ∈ Bi(Ri). However, this comes almost for free in part (2). Also, the assumptions imply (via Lemma 0.4) that players are indeed rational at the state under consideration; this is not used in the proof of Part (1), but again it comes for free in part (2). A key observation is that Aumann and Brandenburger’s construction solves another key interpretational problem almost automatically. Mixed actions are viewed as beliefs: thus, one does not even need to invoke randomizations to justify them. Rationalizability We now turn to Tan and Werlang’s characterization of correlated rationalizability. Actually, I will again present the argument for two-player games, but here everything generalizes 5