instead that ai is not a best reply to any belief, so the max program is infeasible; again, this implies that the min program must be unbounded, i. e. w can be made arbitrarily negative (positive values cannot be optimal, because 0 is a feasible value). But this means that ai is strictly dominated: for any K<0 there exists a feasible pair (y, w) such that w< K: but for this y,∑u(an,a-)y2-u(a1,a-1)>-K>0 for any a-∈A-■ Let me emphasize that it is crucial to look at domination by mixtures of strategies for the equivalence result to hold. If one is not willing to allow for actual randomization, then iterated strict dominance as defined above may make little sense, and one may wish to consider domination by pure strategies: Tilman Borgers, among others, has worked on this problem I will ask you for an alternative proof of the preceding proposition, based on the Sepa- rating Hyperplane theorem. I will also ask you to prove the following(easy) result, which is one reason why correlated rationalizability is worth mentioning (even if you believe in independent beliefs) First. consider the following definition Definition 7 Fix a finite game G=(N, (A;, ui)ieN) For every player E N, let SD,=Ai next, for k 1, and for every i E N, say that ai E SD iff a; is not strictly dominated in the game Gk-1=(N, (SD-I,uk -ien)(where u-I denotes the appropriate restriction of u;) The idea is to discard strictly dominated strategies, then look at the game resulting from g by removing them, and iterate the definition Proposition 0.4 Fix a finite game G=(N, (Ai, uiiEN). Then, for every player iE N, and for every k>1, A=SDA I will have more to say about weak dominance and iterated weak dominance when we discuss extensive gamesinstead that ai is not a best reply to any belief, so the max program is infeasible; again, this implies that the min program must be unbounded, i.e. w can be made arbitrarily negative (positive values cannot be optimal, because 0 is a feasible value). But this means that ai is strictly dominated: for any K < 0 there exists a feasible pair (y, w) such that w ≤ K: but for this y, Pui(a n i , a−i)y n − ui(ai , a−i) > −K > 0 for any a−i ∈ A−i . Let me emphasize that it is crucial to look at domination by mixtures of strategies for the equivalence result to hold. If one is not willing to allow for actual randomization, then iterated strict dominance as defined above may make little sense, and one may wish to consider domination by pure strategies: Tilman Borgers, among others, has worked on this problem. I will ask you for an alternative proof of the preceding proposition, based on the Sepa￾rating Hyperplane theorem. I will also ask you to prove the following (easy) result, which is one reason why correlated rationalizability is worth mentioning (even if you believe in independent beliefs). First, consider the following definition: Definition 7 Fix a finite game G = (N,(Ai , ui)i∈N ). For every player i ∈ N, let SD0 i = Ai ; next, for k ≥ 1, and for every i ∈ N, say that ai ∈ SDk i iff ai is not strictly dominated in the game Gk−1 = (N,(SDk−1 i , uk−1 i )i∈N ) (where u k−1 i denotes the appropriate restriction of ui). The idea is to discard strictly dominated strategies, then look at the game resulting from G by removing them, and iterate the definition. Proposition 0.4 Fix a finite game G = (N,(Ai , ui)i∈N ). Then, for every player i ∈ N, and for every k ≥ 1, Ak i = SDk i . I will have more to say about weak dominance and iterated weak dominance when we discuss extensive games. 9