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: 5

Answer

(a) Since Option A is favored over Option B and Option C, Option A is favored over all others using a head-to-head comparison. (b) Since Option C received the most points, Option C wins the vote using the Borda count method. (c) The head-to-head criterion is not satisfied.

Work Step by Step

(a) With a head-to-head comparison, each option is compared with every other option. For each pair of options, an option is favored over the other option if the option is ranked higher than the other option on a majority of ballots. We can compare Option A and Option B. Option A: 120 + 30 = 150 Option B: 60 + 30 + 30 = 120 Since Option A is ranked higher than Option B on more ballots, Option A is favored over Option B. We can compare Option A and Option C. Option A: 120 + 30 = 150 Option C: 60 + 30 + 30 = 120 Since Option A is ranked higher than Option C on more ballots, Option A is favored over Option C. We can compare Option C and Option B. Option C: 120 + 60 + 30 = 210 Option B: 30 + 30 = 60 Since Option C is ranked higher than Option B on more ballots, Option C is favored over Option B. Since Option A is favored over Option B and Option C, Option A is favored over all others using a head-to-head comparison. (b) With the Borda count method, each option receives 1 point for each third-place vote, 2 points for each second-place vote, and 3 points for each first-place vote. The option which receives the most points is declared the winner. We can find the total points for each option. Option A: 3(120) + 2(30 + 30) + 1(60 + 30) = 570 points Option B: 3(30 + 30) + 2(60) + 1(120 + 30) = 450 points Option C: 3(60 + 30) + 2(120 + 30) + 1(30) = 600 points Since Option C received the most points, Option C wins the vote using the Borda count method. (c) The head-to-head criterion is not satisfied. Option A is favored over all others using a head-to-head comparison, but Option A did not win the vote using the Borda count method.
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