R

USE X=9, Y=20

Problem 4. In a parliamentary system with three parties, a recent election has allotted 40 + X seats to party A, 30 + Y seats to party B, and 25 seats to party 0. In order to form a government, a coalition needs a majority of the seats in total (e. g., 41 out of 80). The parties in the government enjoy a total 10 units of indivisible perks, which they can redistribute among the coalition members in any way they agree on. The parties outside the government get 0 units of perks, and each party tries to maximize the expected value of its own perks. The process of coalition formation is as follows. First, A is given right to form a government. If it fails, then B is given right to form a government, and if B also fails then C is given right to form a government. If C also fails, then the game ends and each gets 0. The party who is given right to form a government, say i, approaches one of the other two parties, say 3', and oers some n E {1, . . . , 10}. If j accepts, then they form the government, 1' gets 10 n and j gets a units of perks. If 3' rejects the offer, then 3' fails to form a government (in which case, as described above, either another party is given right to form a government or the game ends with 0 payoffs for all parties}. a) Calculate the number of different pure strategies of A. Hint: To obtain the number of strate- gies, multiply the number of actions the player has at each of her decision nodes. 3 points b) Applying backward induction, nd a subgameperfect Nash equilibrium outcome (payoffs) of this game. 7 points