Thinking Mathematically (6th Edition)

Published by Pearson

Chapter 13 - Voting and Apportionment - 13.2 Flaws of Voting Methods - Exercise Set 13.2: 15

Answer

(a) Since Candidate B received the most points, Candidate B is declared the winner using the Borda count method. (b) The majority criterion is not satisfied. Although Candidate A has a majority of the first-place votes, Candidate A was not declared the winner. (c) The head-to-head criterion is not satisfied. Candidate A is favored over all other candidates in head-to-head comparison. However, Candidate A was not declared the winner. (d) The irrelevant alternatives criterion is not satisfied. Initially, Candidate B was declared the winner. After Candidate C dropped out, Candidate B was not declared the winner. Instead, Candidate A was declared the winner.

Work Step by Step

(a) With the Borda count method, each candidate receives 1 point for each fourth-place vote, 2 points for each third-place vote, 3 points for each second-place vote, and 4 points for each first-place vote. The candidate with the most points is declared the winner. We can find the total points for each candidate. Candidate A: 4(14) + 3(4) + 2(0) + 1(8) = 76 points Candidate B: 4(8) + 3(14) + 2(0) + 1(4) = 78 points Candidate C: 4(0) + 3(0) + 2(14 + 8 + 4) + 1(0) = 52 points Candidate D: 4(4) + 3(8) + 2(0) + 1(14) = 54 points Since Candidate B received the most points, Candidate B is declared the winner using the Borda count method. (b) Candidate A has 14 first-place votes, which is more than half of the first-place votes. Although Candidate A has a majority of the first-place votes, Candidate A was not declared the winner. Therefore, the majority criterion is not satisfied. (c) With a head-to-head comparison, each candidate is compared with every other candidate. For each pair of candidates, a candidate is favored over the other candidate if the candidate is ranked higher than the other candidate on a majority of ballots. We can compare Candidate A and Candidate B. Candidate A: 14 + 4 = 18 Candidate B: 8 Since Candidate A is ranked higher than Candidate B on more ballots, Candidate A is favored over Candidate B. We can compare Candidate A and Candidate C. Candidate A: 14 + 4 = 18 Candidate C: 8 Since Candidate A is ranked higher than Candidate C on more ballots, Candidate A is favored over Candidate C. We can compare Candidate A and Candidate D. Candidate A: 14 Candidate D: 8 + 4 = 12 Since Candidate A is ranked higher than Candidate D on more ballots, Candidate A is favored over Candidate D. We can see that Candidate A is favored over all other candidates so there is no need to compare the other candidates with each other. Candidate A is favored over all other candidates in head-to-head comparison. However, Candidate A was not declared the winner. Therefore, the head-to-head criterion is not satisfied. (d) After Candidate C drops out, the other candidates below Candidate C on each ballot move up one spot on that ballot. We can go through the Borda count method again to determine the winner. Candidate A: 3(14) + 2(4) + 1(8) = 58 points Candidate B: 3(8) + 2(14) + 1(4) = 56 points Candidate D: 3(4) + 2(8) + 1(14) = 42 points Since Candidate A received the most points, Candidate A is declared the winner using the Borda count method. Initially, Candidate B was declared the winner. After Candidate C dropped out, Candidate B was not declared the winner. Instead, Candidate A was declared the winner. We can see that after one of the losing candidates dropped out, a different candidate was selected as the winner. Therefore, the irrelevant alternatives criterion is not satisfied.

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