And, of course, if Player 2 expects this, he will play his part and choose L. Finally, if Player 1 anticipates this, she will choose OTT The conclusion is that Player 1 will not need to burn anything, but the mere possibility of doing so makes her strong and allows her to force the( T, l) equilibrium I should add that some people find this conclusion puzzling; i personally dont but you may differ Forward Induction and iterated weak dominance This is a subsection I wish I could avoid writing. But, since you will likely encounter statements like "Iterated Weak Dominance captures forward induction, I guess I really have to suffer through it So, here goes. First of all, the definition Definition 1 Fix a finite game G=(N, (Ai, uiieN) and a player i E N. An action i is weakly dominated for Player i iff there exists a; E A(Ai) such that 2 ui(al,a-i)ai(ai)2ui(ai, a-i and there exists a-i E A-i such that ui(an, a-iai(ai>ui(ai, a-i) ∈A It turns out that an action is weakly dominated iff it is not a best response to any strictly positive probability distribution over opponents' action profiles. You need not worry about this now, anyway Definition 2 Fix a finite game G=(N, (Ai, ui)ieN. For every player i E N, let WD=A;next,fork≥1, and for every i∈N, say that a∈ Wd iff a; is not weakly dominated in the game G-l=(N, (WDi-, ui- ieN)(where ui-denotes the appropriate restriction of ui) That is: at each round, we eliminate all weakly dominated actions for all players; then we look at the residual game. and continue until no further eliminations are possible I Having disposed of the formalities, let us write down the normal form of the burn- ng money game; actually, let's write the reduced normal form, deleting redundant I The emphasis on eliminating all weakly dominated actions is warranted: the order and extent of elimination does matter. This is but one of troubling aspects of iterated weak domAnd, of course, if Player 2 expects this, he will play his part and choose L. Finally, if Player 1 anticipates this, she will choose 0TT. The conclusion is that Player 1 will not need to burn anything, but the mere possibility of doing so makes her strong and allows her to force the (T,L) equilibrium. I should add that some people find this conclusion puzzling; I personally don’t, but you may differ. Forward Induction and Iterated Weak Dominance This is a subsection I wish I could avoid writing. But, since you will likely encounter statements like “Iterated Weak Dominance captures forward induction,” I guess I really have to suffer through it. So, here goes. First of all, the definition. Definition 1 Fix a finite game G = (N,(Ai , ui)i∈N ) and a player i ∈ N. An action ai is weakly dominated for Player i iff there exists αi ∈ ∆(Ai) such that ∀a−i ∈ A−i , X a 0 i∈Ai ui(a 0 i , a−i)αi(ai) ≥ ui(ai , a−i) and there exists a−i ∈ A−i such that X a 0 i∈Ai ui(a 0 i , a−i)αi(ai) > ui(ai , a−i). It turns out that an action is weakly dominated iff it is not a best response to any strictly positive probability distribution over opponents’ action profiles. You need not worry about this now, anyway. Definition 2 Fix a finite game G = (N,(Ai , ui)i∈N ). For every player i ∈ N, let WD0 i = Ai ; next, for k ≥ 1, and for every i ∈ N, say that ai ∈ WDk i iff ai is not weakly dominated in the game Gk−1 = (N,(WDk−1 i , uk−1 i )i∈N ) (where u k−1 i denotes the appropriate restriction of ui). That is: at each round, we eliminate all weakly dominated actions for all players; then we look at the residual game, and continue until no further eliminations are possible.1 Having disposed of the formalities, let us write down the normal form of the burn￾ing money game; actually, let’s write the reduced normal form, deleting redundant 1The emphasis on eliminating all weakly dominated actions is warranted: the order and extent of elimination does matter. This is but one of troubling aspects of iterated weak dominance. 4