It is easier to define EFR if we model beliefs by using conditional probability systems, or CPSs for short. You will remember I defined these objects in our last double class on epistemic foundations The main justification for adopting CPSs as our basic representation of beliefs is that, after all, belief systems and behavioral strategies represent conditional beliefs bout pure strategies chosen by opponents. Thus, we may as well represent these conditional beliefs directly. Recall that Ii denotes the set of information sets of Player i, and that, for I ELi, S(D), Si(I) and S_i(I)denote, respectively, the set of strategy profiles reach 1, the set of strategies of Player i which allow I to be reached, and the set of strategy profiles of i's opponents which allow I to be reached Definition 3 A conditional probability system on(S-i, Ii) is a map u: 25-ix ):I∈}→0,1 such that: (i) For alll∈,p(1S-:()∈△(S-); (ii) For all I EIi, u(S-i(DS-i(D)=1 (i)AcS-(1)cS-() implies u(4|S-1(1)=(4|S-(D)(S-1(D)S-() Let us now define extensive-form rationalizability, or EFR. First of all, CPSs allow us to define(weak) sequential rationality in a very simple way Definition 4 A strategy si E Si is weakly sequentially rational (WSr) given the CPS u on(S-,T), written si∈r(),if, for all I∈五 such that s;∈S(D), and for alls∈S2(), ∑[4(s,s-)-t1(s,s-)l(s-l|S-()≥ I have discussed"weak"sequential rationality during the course. The basic idea is that the usual notion of sequential rationality requires that a strategy be condi- tionally optimal even at information sets that it precludes from being reached. As I have argued a few times during the course, this requirement is not sustainable from a decision-theoretic viewpoint. The only reason it is imposed is that, in equilibrium, a player's actions at information sets that she herself prevents from being reached still serve as her opponents beliefs. And, even granting this, we must invoke back ward induction(or something like it) to force these beliefs to be consistent with rationality-and hence restrict our attention to fully sequentially rational strategies Extensive-form rationalizability takes a much more decision-theoretic view, and therefore needs no more than weak sequential rationality. That is, it uses a notion of rationality applied to plans of action, not strategies(if you remember the difference) In any case, using full sequential rationality in our definition of EFR would not make any difference in terms of outcomesIt is easier to define EFR if we model beliefs by using conditional probability systems, or CPSs for short. You will remember I defined these objects in our last double class on epistemic foundations. The main justification for adopting CPSs as our basic representation of beliefs is that, after all, belief systems and behavioral strategies represent conditional beliefs about pure strategies chosen by opponents. Thus, we may as well represent these conditional beliefs directly. Recall that Ii denotes the set of information sets of Player i, and that, for every I ∈ Ii , S(I), Si(I) and S−i(I) denote, respectively, the set of strategy profiles which reach I, the set of strategies of Player i which allow I to be reached, and the set of strategy profiles of i’s opponents which allow I to be reached. Definition 3 A conditional probability system on (S−i , Ii) is a map µ : 2S−i × {S−i(I) : I ∈ Ii} → [0, 1] such that: (i) For all I ∈ Ii , µ(·|S−i(I)) ∈ ∆(S−i); (ii) For all I ∈ Ii , µ(S−i(I)|S−i(I)) = 1; (iii) A ⊂ S−i(I) ⊂ S−i(I 0 ) implies µ(A|S−i(I 0 )) = µ(A|S−i(I))µ(S−i(I)|S−i(I 0 )). Let us now define extensive-form rationalizability, or EFR. First of all, CPSs allow us to define (weak) sequential rationality in a very simple way. Definition 4 A strategy si ∈ Si is weakly sequentially rational (WSR) given the CPS µ on (S−i , Ii), written si ∈ ri(µ), if, for all I ∈ Ii such that si ∈ Si(I), and for all s 0 i ∈ Si(I), X s−i∈S−i [ui(si , s−i) − ui(s 0 i , s−i)]µ(s−i |S−i(I)) ≥ 0 I have discussed “weak” sequential rationality during the course. The basic idea is that the usual notion of sequential rationality requires that a strategy be condi￾tionally optimal even at information sets that it precludes from being reached. As I have argued a few times during the course, this requirement is not sustainable from a decision-theoretic viewpoint. The only reason it is imposed is that, in equilibrium, a player’s actions at information sets that she herself prevents from being reached still serve as her opponents’ beliefs. And, even granting this, we must invoke back￾ward induction (or something like it) to force these beliefs to be consistent with rationality—and hence restrict our attention to fully sequentially rational strategies. Extensive-form rationalizability takes a much more decision-theoretic view, and therefore needs no more than weak sequential rationality. That is, it uses a notion of rationality applied to plans of action, not strategies (if you remember the difference). [In any case, using full sequential rationality in our definition of EFR would not make any difference in terms of outcomes.] 6