Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 13 - Voting and Apportionment - 13.2 Flaws of Voting Methods - Exercise Set 13.2 - Page 862: 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.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.