Here is the main definition Definition5LetS"=S; for all i∈N.Next, for every i∈ n and s,∈S, say that s1∈Snis∈Sn1 and there exists a CPS u on(S-,) such that si∈r(p)and, for all l∈T Sm71∩S-1(1)≠0→(S|S-1(I)=1 (I am implicitly using our conventions for product sets The key condition is the restriction on the CPS justifying a strategy at step n. It basically says that, whenever possible, beliefs should be concentrated on the previously This business of"whenever possible" can be understood by reference to the game Figure 2, Burning Money. 2 You should convince yourselves that the sequence of elimination I have used in the analysis is the one prescribed by EFR. Then, after step 5, no strategy of Player 1 reaches the LhS subgame. But there is no reason to stop at Step 6: as in the definition of normal-form rationalizability, what should happen is that S=s But then, what should Player 2 believe in the LHS subgame? This is the reason for 2.(b): Sin SI(LHS subgame)=0, so no restrictions apply Then is Player 2 free to choose any best reply to any belief? Well, not quite, because candidate strategies at step 6 must have survived step 5, by condition 1. In turn, this implies that they must be best replies to CPSs that assign probability one to Si(not Si) whenever possible; and this is possible in the LHS subgame In fact, here' s an alternative definition of EFR (that is, this definition and the e one above identify the same sets of strategies. Let Si=S; for all i E N. Next, say that SiE Si iff there exists a CPS u such that 1.s:∈r;(P 2. For all l∈五;andm=0,…,n-1:Sm∩S-:(1)≠0→μ(Sml|S-(D)=1 Thus, at any point in the game, players attribute the highest degree of strategic sophistication to their opponents, where "strategic sophistication"is interpreted as levels of efr.” i conclude by noting that, in generic perfect infor er games,EFR yields the BI outcome. And, last but not least, there exists an epistemic characterization of EFR which is just as satisfactory (in the sense we have discussed in class)as the To be really precise, the representation of the game Burning Money suggests a game with observable actions but possibly simultaneous moves. To apply the definitions in the text, represent the battle-of-the-sexes subgames assuming that Player 1 moves first, and Player 2 moves ne without observing I' s choice(other than r or 0, of courseHere is the main definition. Definition 5 Let S 0 i = Si for all i ∈ N. Next, for every i ∈ N and si ∈ Si , say that si ∈ S n i iff si ∈ S n−1 −i and there exists a CPS µ on (S−i , Ii) such that si ∈ ri(µ) and, for all I ∈ Ii , S n−1 −i ∩ S−i(I) 6= ∅ ⇒ µ(S n−1 −i |S−i(I)) = 1. (I am implicitly using our conventions for product sets). The key condition is the restriction on the CPS justifying a strategy at step n. It basically says that, whenever possible, beliefs should be concentrated on the previously defined set of level-(n − 1) EFR strategies. This business of “whenever possible” can be understood by reference to the game in Figure 2, Burning Money.2 You should convince yourselves that the sequence of elimination I have used in the analysis is the one prescribed by EFR. Then, after step 5, no strategy of Player 1 reaches the LHS subgame. But there is no reason to stop at Step 6: as in the definition of normal-form rationalizability, what should happen is that S 6 = S 5 . But then, what should Player 2 believe in the LHS subgame? This is the reason for 2.(b): S 5 1 ∩ S1(LHS subgame) = ∅, so no restrictions apply. Then is Player 2 free to choose any best reply to any belief? Well, not quite, because candidate strategies at step 6 must have survived step 5, by condition 1. In turn, this implies that they must be best replies to CPSs that assign probability one to S 4 1 (not S 5 1 ) whenever possible; and this is possible in the LHS subgame. In fact, here’s an alternative definition of EFR (that is, this definition and the one above identify the same sets of strategies.) Let S 0 i = Si for all i ∈ N. Next, say that si ∈ S n i iff there exists a CPS µ such that: 1. si ∈ ri(µ) 2. For all I ∈ Ii and m = 0, . . . , n − 1: S m −i ∩ S−i(I) 6= ∅ ⇒ µ(S m −i |S−i(I)) = 1. Thus, at any point in the game, players attribute the highest degree of strategic sophistication to their opponents, where “strategic sophistication” is interpreted as “levels of EFR.” I conclude by noting that, in generic perfect information games, EFR yields the BI outcome. And, last but not least, there exists an epistemic characterization of EFR which is just as satisfactory (in the sense we have discussed in class) as the 2To be really precise, the representation of the game Burning Money suggests a game with observable actions but possibly simultaneous moves. To apply the definitions in the text, represent the battle-of-the-sexes subgames assuming that Player 1 moves first, and Player 2 moves next, without observing 1’s choice (other than x or 0, of course!) 7