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 861: 9

Answer

(a) Steven Spielberg is selected as the speaker. (b) We can see the new preference table below. Steven Spielberg 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 Steven Spielberg and Ron Howard. Steven Spielberg: 8 + 5 = 13 Ron Howard: 10 Since Steven Spielberg is ranked higher than Ron Howard on more ballots, Steven Spielberg receives 1 point. We can compare Steven Spielberg and Spike Lee. Steven Spielberg: 10 + 5 = 15 Spike Lee: 8 Since Steven Spielberg is ranked higher than Spike Lee on more ballots, Steven Spielberg receives 1 point. We can compare Spike Lee and Ron Howard. Spike Lee: 8 + 5 = 13 Ron Howard: 10 Since Spike Lee is ranked higher than Ron Howard on more ballots, Spike Lee receives 1 point. After all the pairwise comparisons have been made, we can add up the total number of points for each candidate. Steven Spielberg: 1 + 1 = 2 points Spike Lee: 1 point Ron Howard: 0 points Since Steven Spielberg received the most points, Steven Spielberg is selected as the speaker. (b) Suppose Ron Howard withdraws before the votes are counted. Then each candidate who was below Ron Howard 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 Steven Spielberg and Spike Lee. Steven Spielberg: 10 + 5 = 15 Spike Lee: 8 Since Steven Spielberg is ranked higher than Spike Lee on more ballots, Steven Spielberg receives 1 point. Since Steven Spielberg received the most points, Steven Spielberg is selected as the speaker. (c) The irrelevant alternatives criterion is satisfied. Initially, Steven Spielberg was selected using the pairwise comparison method. After one of the losing candidates withdrew, Steven Spielberg was still selected using the pairwise comparison method. Therefore, the irrelevant alternatives criterion is satisfied.
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