20 01 2022 E TS

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GAME THEORY - January 20, 2022 Last name: First name: ID #: SOL VE THE EXERCISES AND ANSWER THE QUESTIONS ON THESE SHEETS Exercise 1 5 points Given the zero-sum game described by the following matrix: A =   3 5 4 2 3 1 5 2 3   find a pair of optimal strategies for the players and the value of the game. Answer of exercise 1 First of all, notice that vI ̸= vII , therefore there are no optimal pure strategies. Then we observe tha t the second row is strictly dominated by the first one. Thus the game can be redu ced to A = ( 3 5 4 5 2 3 ) If Pl1 plays (p, 0, 1 − p), then the utilities of the three columns are respectively (5 − 2p, 3p + 2, p + 3). Then Pl1 plays p = 2/3 and Pl2 plays (q, 0, 1 − q). By the indifference principle, q = 1/3 and then the value is 11/3. Exercise 2 5 points Let W = {Alessia, Barbara } and M = {Charlie, David, Ethan }. Suppose that Alessia prefers David to Charlie and Charlie to Ethan, while Barbara prefers Charlie to Ethan and Ethan to David. 1. If the ”women visiting men” algorithm is followed, what is t he stable matching? Do we need to know the preferences of the men? 2. Choose a preference profile for the men such that there is anothe r stable set (not necessarily as a result of a visiting algorithm). 3. Assume that all the men prefer to be paired than to be alone. Gi ven the preferences of the women, is there a man that will be alone in all the stable sets, no matter what th e men wish? Explain your answer. Answer of exercise 2 1. The stable set is (David, Alessia), (Charlie, Barbara), Etha n alone. The preferences of the men are irrelevant, since women choose, and they both get their first choice. 2. For instance if the preferences are David : Barbara ≻ ∅ ≻ Alessia Charlie : Alessia ≻ Barbara ≻ ∅ Ethan : Alessia ≻ Barbara ≻ ∅ (where ∅ stands…