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 burning 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