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: 17

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 D has a majority of first-place votes, Candidate D is not selected as the winner. (c) The head-to-head criterion is not satisfied. Although Candidate D is favored over every other candidate when we do a head-to-head comparison, Candidate D is not selected as the winner.

Work Step by Step

(a) With the Borda count method, each candidate receives 1 point for each fifth-place vote, 2 points for each fourth-place vote, and so on. The candidate with the most points is declared the winner. We can find the total points for each candidate. Candidate A: 5(16) + 4(4 + 2) + 3(14) + 2(0) + 1(12) = 158 points Candidate B: 5(0) + 4(16 + 14 + 12) + 3(4) + 2(2) + 1(0) = 184 points Candidate C: 5(4) + 4(0) + 3(16) + 2(14 + 12) + 1(2) = 122 points Candidate D: 5(14 + 12) + 4(0) + 3(2) + 2(16 + 4) + 1(0) = 176 points Candidate E: 5(2) + 4(0) + 3(12) + 2(0) + 1(16 + 14 + 4) = 80 points 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. Candidate D has 26 first-place votes which is more than half of the votes. Although Candidate D has a majority of first-place votes, Candidate D is not selected as the winner. Therefore, the majority criterion is not satisfied. (c) The head-to-head criterion is not satisfied. Since Candidate D has a majority of the first-place votes, Candidate D is favored over every other candidate when we do a head-to-head comparison. However, Candidate D is not selected as the winner. Therefore, the head-to-head criterion is not satisfied.
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