Social Choice_Problem Set 8_Solution

Social Choice_Problem Set 8_Solution - 1 1 1 1-1 ABCE 1 1...

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Social Choice: Manipulation and Analysis (Econ 497/ Econ 597) Problem Set 8: Power in voting 1. Suppose there are 5 men and 7 women. (a) How many di/erent ways can we arrange these 12 people in a line? 12! (b) How many di/erent committees of 4 people can be formed? ° 12 4 ± (c) How many di/erent committees of 4 committees can be formed with exactly two men and two women? ° 5 2 ±° 7 2 ± (d) How many di/erent committees of 6 committees can be formed with exactly 3 men and 3 women? ° 5 3 ±° 7 3 ± 2. Suppose we have a weighted voting system with 5 voters °named A, B, C, D, and E. The voters have the following number of votes: A 8 Votes B 6 Votes C 4 Votes D 2 Votes E 2 Votes Passage of a bill requires 18 votes. (a) Find all of the winning coalitions. Answer: All of the coalitions whose sum adds up to 18 or more. ABC = 18 ABDE = 18 ABCD = 20 ABCE = 20 ABCDE = 22 (b) Calculate the Banzhaf Index of Power for each of the 5 voters. Answer A B C D E ABC 1 1 1 -1 -1 ABDE 1 1 -1 1
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Unformatted text preview: 1 1 1 1-1 ABCE 1 1 1-1 1 ABCDE 1 1 1 1 1 TBP 5 5 3 1 1 BI 1 3 1 3 1 5 1 15 1 15 (c) (5 Points) Calculate the Shapley-Shubik Index of Power for voter C. Hint: 5!=120. ..Don±t try and do this the long way! 1 Answer: Voter C is pivotal if preceded by a coalition with 14, 15, 16, or 17 votes. Since all individual votes are even, it is impossible to have 15 or 17. There fore we only need to add up the orderings where C is preceded by 14 or 16 votes. There is one coalition to that produces 14 votes: coalition AB. The number of orderings where C is preceded by AB is 2!1!2! = 4 : There are 2 coalitions that yield 16 votes: ABD, ABE. Thus, there are 3!1!1! = 6 orderings where C is preceded by ABD and 3!1!1! = 6 orderings where C is preceded by ABE . The total number of pivotal orderings for C is 16 : Thus, C &s SSI is 16 120 = 2 15 : 2...
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