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: 10

Answer

(a) Whoopi Goldberg is selected as the speaker. (b) We can see the new preference table below. Whoopi Goldberg is selected as the speaker. (c) The irrelevant alternatives criterion is satisfied.
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Work Step by Step

(a) With the pairwise comparison method, each candidate is compared with every other candidate. For each pair of candidates, if one candidate is ranked higher than the other candidate on a majority of ballots, then the higher-ranked candidate receives 1 point. If the two candidates tie, then they each receive 0.5 points. After all the comparisons have been made, the candidate with the most points is declared the winner. We can compare Whoopi Goldberg and Meryl Streep. Whoopi Goldberg: 8 + 6 = 14 Meryl Streep: 12 Since Whoopi Goldberg is ranked higher than Meryl Streep on more ballots, Whoopi Goldberg receives 1 point. We can compare Whoopi Goldberg and Julia Roberts. Whoopi Goldberg: 12 + 6 = 18 Julia Roberts: 8 Since Whoopi Goldberg is ranked higher than Julia Roberts on more ballots, Whoopi Goldberg receives 1 point. We can compare Julia Roberts and Meryl Streep. Julia Roberts: 8 + 6 = 14 Meryl Streep: 12 Since Julia Roberts is ranked higher than Meryl Streep on more ballots, Julia Roberts receives 1 point. After all the pairwise comparisons have been made, we can add up the total number of points for each candidate. Whoopi Goldberg: 1 + 1 = 2 points Julia Roberts: 1 point Meryl Streep: 0 points Since Whoopi Goldberg received the most points, Whoopi Goldberg is selected as the speaker. (b) Suppose Meryl Streep withdraws before the votes are counted. Then each candidate who was below Meryl Streep on a ballot will move up one spot on that ballot. We can see the new preference table below. To find the selected speaker, we can use the pairwise comparison method with the remaining two candidates. We can compare Whoopi Goldberg and Julia Roberts. Whoopi Goldberg: 12 + 6 = 18 Julia Roberts: 8 Since Whoopi Goldberg is ranked higher than Julia Roberts on more ballots, Whoopi Goldberg receives 1 point. Since Whoopi Goldberg received the most points, Whoopi Goldberg is selected as the speaker. (c) The irrelevant alternatives criterion is satisfied. Initially, Whoopi Goldberg was selected using the pairwise comparison method. After one of the losing candidates withdrew, Whoopi Goldberg was still selected using the pairwise comparison method. Therefore, the irrelevant alternatives criterion is satisfied.
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