(1)a;∈r(a-) 2)a-i(∏ An action ai E Af is said to be k-correlated rationalizable; an action ai E nk>1 Af is said to be correlated rationalizable Part(2)in the above definition captures the idea that, at each step k we are willing to impose the assumption that players are able to perform k-l steps of eductive reasoning about their opponents. Although this is not explicitly required in the definition, this corresponds to progressively more stringent assumptions about rationality; if action sets are finite, this implies that correlated rationalizable strategies exist(why?) Proposition 0. 2 Fix a game G=(N, (Ai, Ti, uiieN). Then, for every i E N and k>1 AFCA Proof: The claim is true for k= 1. Assume it is true for some k, fix i E N and pick ∈A by definition, there exists △(A-a,B(T ch that a;∈r(a a-i(Aki)=1. Since by the induction hypothesis Ak C A-, this implies a_i(Ak- )=1 hence,a;∈A, as claimed.■ Independent rationalizability, or(in keeping with standard terminology)"rationalizabil ity" tout-court, incorporates the additional restriction that players beliefs are actually in- dependent product measures on their opponents'action profiles; that players believe that their opponents' beliefs are independent; that they believe that their opponents believe that their respective opponents' beliefs are independent; and so on and so forth Definition 5 Fix a game G=(N, (Ai, Ti, uiieN). For every player i E N, let r=A;. For k≥1 and for every i∈l,leta;∈ Ri iff there exists a-a∈△(A-,B(T-) such that (2)a-(I≠:B-1)=1 (3)a-i is a product measure on A-i, BT-i) An action a,E Rk is said to be k-rationalizable; an action a; E nk>I Ri is said to be rationalizable I conclude this section with two notes pertaining to the interpretation of (correlated) ationalizability First, I emphasize that, although we have motivated(correlated )rationalizability with informal assumptions about rationality and beliefs, we have been quite vague in definin what we mean by statements such as, "Player i believes that Player -i is rational. We have claimed that this "implies"that Player i's beliefs should assign positive probability only to those strategies of Player -i which can themselves be justified by some belief.(1) ai ∈ ri(α−i); (2) α−i( Q j6=i A k−1 j ) = 1. An action ai ∈ Ak i is said to be k-correlated rationalizable; an action ai ∈ T k≥1 Ak i is said to be correlated rationalizable. Part (2) in the above definition captures the idea that, at each step k, we are willing to impose the assumption that players are able to perform k−1 steps of eductive reasoning about their opponents. Although this is not explicitly required in the definition, this corresponds to progressively more stringent assumptions about rationality; if action sets are finite, this implies that correlated rationalizable strategies exist (why?). Proposition 0.2 Fix a game G = (N,(Ai , Ti , ui)i∈N ). Then, for every i ∈ N and k ≥ 1, Ak i ⊂ A k−1 i . Proof: The claim is true for k = 1. Assume it is true for some k, fix i ∈ N and pick ai ∈ A k+1 i . By definition, there exists α−i ∈ ∆(A−i , B(T−i)) such that ai ∈ ri(α−i) and α−i(Ak −i ) = 1. Since by the induction hypothesis Ak −i ⊂ A k−1 −i , this implies α−i(A k−1 −i ) = 1; hence, ai ∈ Ak i , as claimed. Independent rationalizability, or (in keeping with standard terminology) “rationalizabil￾ity” tout-court, incorporates the additional restriction that players’ beliefs are actually in￾dependent product measures on their opponents’ action profiles; that players believe that their opponents’ beliefs are independent; that they believe that their opponents believe that their respective opponents’ beliefs are independent; and so on and so forth. Definition 5 Fix a game G = (N,(Ai , Ti , ui)i∈N ). For every player i ∈ N, let R0 i = Ai . For k ≥ 1 and for every i ∈ I, let ai ∈ Rk i iff there exists α−i ∈ ∆(A−i , B(T−i)) such that (1) ai ∈ ri(α−i); (2) α−i( Q j6=i R k−1 j ) = 1; (3) α−i is a product measure on A−i , B(T−i). An action ai ∈ Rk i is said to be k-rationalizable; an action ai ∈ T k≥1 Rk i is said to be rationalizable. I conclude this section with two notes pertaining to the interpretation of (correlated) rationalizability. First, I emphasize that, although we have motivated (correlated) rationalizability with informal assumptions about rationality and beliefs, we have been quite vague in defining what we mean by statements such as, “Player i believes that Player −i is rational.” We have claimed that this “implies” that Player i’s beliefs should assign positive probability only to those strategies of Player −i which can themselves be justified by some belief. 6