readily, and it is easy to add stochastic independence as an explicit restriction on beliefs Indeed, any restriction on first-order beliefs can be easily added to the characterization Let me remind you of the definition first. Let Ag= A; for i= 1, 2. Next, for k 1, say that a∈ A iff there exists a-;∈△(A-) such that a-(41)=1anda∈r(a-) We now define a sequence of event B=R;Wk≥1,B=B(B-1) Thus, b="Every body is certain that every body is rational;B="Everybody is certain that everybody is certain that everybody is rational; and so on Proposition 0.5 Fix a two-player game G=(N, (Ai, uiieN), with N=(1, 2, and a profile of actions a=(a1, a2) 1)If there exists a model M=(S, (Ti, ai, piie) for G and a state w in the model with (w)=ai for i=1, 2 such that ∩B then a E A+l. Hence, if there is a model M and a state w in that model with a;(w)=ai for i=1, 2 such that wE neso b, then a is rationalizable 2)If a E A(k> 1), then there exists a model M=(S2, ( Ti, ai, pi)ieN) for G and a state w in the model such that ai(w)=ai, i=1, 2 and w Ene-o b. If a is rationalizable, there exists a model M and a state w such that a(u)=a1,i=1,2andu∈∩≌0B Proof: Note first that. for k>1 k RnAB(B-D=R B(0B) E=1 (1,k=0)Fixu∈B=R. Then trivially a;=a(u)∈rl(a-l(a),soa1∈ (1,k>0) By induction, suppose the claim is true for (=0,., k-1. Fixw E B the above decomposition, wER, and wE B: (ne=o Be). By Lemma 0.3, suppp: ( lt (w))C ne=o be; but for all w'E neo Be, by the induction hypothesis a-i(w )E A=-. Hence, ax-i(w)(Aki)=l, and we are done. The claim concerning rationalizable profiles follows from the fact that, since the game is finite, there exists K such that k> K implies A'=AK for all i∈N. (2): left as an exercise for the interested reader. B ts The characterization result is quite straightforward. I do point out that the assumption at w E U-o b has behavioral implications: what players actually do at w is consistent with k+1 steps of the iterative procedure defining rationalizability. As I have remarked many times, this is not the case with Nash equilibriumreadily, and it is easy to add stochastic independence as an explicit restriction on beliefs. Indeed, any restriction on first-order beliefs can be easily added to the characterization. Let me remind you of the definition first. Let A0 i = Ai for i = 1, 2. Next, for k ≥ 1, say that ai ∈ Ak i iff there exists α−i ∈ ∆(A−i) such that α−i(A k−1 −i ) = 1 and ai ∈ ri(α−i). We now define a sequence of events: B 0 = R; ∀k ≥ 1, Bk = B(B k−1 ) Thus, B1 = “Everybody is certain that everybody is rational”; B2 = “Everybody is certain that everybody is certain that everybody is rational”; and so on. Proposition 0.5 Fix a two-player game G = (N,(Ai , ui)i∈N ), with N = {1, 2}, and a profile of actions a = (a1, a2). (1) If there exists a model M = (Ω,(Ti , ai , pi)i∈N ) for G and a state ω in the model with ai(ω) = ai for i = 1, 2 such that ω ∈ \ k `=0 B ` then a ∈ Ak+1. Hence, if there is a model M and a state ω in that model with ai(ω) = ai for i = 1, 2 such that ω ∈ T∞ `=0 B` , then a is rationalizable. (2) If a ∈ Ak (k ≥ 1), then there exists a model M = (Ω,(Ti , ai , pi)i∈N ) for G and a state ω in the model such that ai(ω) = ai , i = 1, 2 and ω ∈ Tk−1 `=0 B` . If a is rationalizable, there exists a model M and a state ω such that ai(ω) = ai , i = 1, 2 and ω ∈ T∞ `=0 B` . Proof: Note first that, for k ≥ 1, \ k `=0 B ` = R ∩ \ k `=1 B(B `−1 ) = R ∩ B k \−1 `=0 B ` ! (1, k = 0) Fix ω ∈ B0 = R. Then trivially ai = ai(ω) ∈ ri(α−i(ω)), so ai ∈ A1 i . (1, k > 0) By induction, suppose the claim is true for ` = 0, . . . , k−1. Fix ω ∈ Tk `=0 Bk ; by the above decomposition, ω ∈ Ri and ω ∈ Bi Tk−1 `=0 B`  . By Lemma 0.3, supp pi(·|ti(ω)) ⊂ Tk−1 `=0 B` ; but for all ω 0 ∈ Tk−1 `=0 B` , by the induction hypothesis a−i(ω 0 ) ∈ A k−1 −i . Hence, α−i(ω)(A k−1 −i ) = 1, and we are done. The claim concerning rationalizable profiles follows from the fact that, since the game is finite, there exists K such that k ≥ K implies Ak i = AK i for all i ∈ N. (2) : left as an exercise for the interested reader. The characterization result is quite straightforward. I do point out that the assumption that ω ∈ Sk `=0 B` has behavioral implications: what players actually do at ω is consistent with k + 1 steps of the iterative procedure defining rationalizability. As I have remarked many times, this is not the case with Nash equilibrium. 6