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: 12

Answer

(a) Candidate B is declared the winner using the Borda count method. (b) The majority criterion is not satisfied. Candidate A has a majority of the first-place votes but Candidate A was not declared the winner of the election.

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(20) + 3(0) + 2(0) + 1(15 + 3 + 1) = 99 points Candidate B: 4(15) + 3(20 + 1) + 2(3) + 1(0) = 129 points Candidate C: 4(3) + 3(15) + 2(20 + 1) + 1(0) = 99 points Candidate D: 4(1) + 3(3) + 2(15) + 1(20) = 63 points Since Candidate B received the most points, Candidate B is declared the winner using the Borda count method. (b) The total number of votes in this election is 20 + 15 + 3 + 1 which is 39. Candidate A has 20 first-place votes which is more than half of the total votes in this election. Therefore, Candidate A has a majority of first-place votes. The majority criterion is not satisfied. Candidate A has a majority of the first-place votes but Candidate A was not declared the winner of the election.
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