Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 13 - Voting and Apportionment - 13.1 Voting Methods - Exercise Set 13.1: 26

Answer

Professor Gates is declared the new president using the pairwise comparison method.

Work Step by Step

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 who receives the most points is declared the winner. We can compare Ford and Sarnoff. We can find the number of ballots with Ford ranked higher than Sarnoff, and the number of ballots with Sarnoff ranked higher than Ford. Ford: 30 Sarnoff: 22 + 18 + 10 + 2 = 52 Since Sarnoff is ranked higher than Ford on more ballots, Sarnoff receives 1 point. We can compare Ford and Gates. We can find the number of ballots with Ford ranked higher than Gates, and the number of ballots with Gates ranked higher than Ford. Ford: 30 Gates: 22 + 18 + 10 + 2 = 52 Since Gates is ranked higher than Ford on more ballots, Gates receives 1 point. We can compare Gates and Sarnoff. We can find the number of ballots with Gates ranked higher than Sarnoff, and the number of ballots with Sarnoff ranked higher than Gates. Gates: 30 + 22 + 2 = 54 Sarnoff: 18 + 10 = 28 Since Gates is ranked higher than Sarnoff on more ballots, Gates receives 1 point. We can compare Ford and Disney. We can find the number of ballots with Ford ranked higher than Disney, and the number of ballots with Disney ranked higher than Ford. Ford: 30 Disney: 22 + 18 + 10 + 2 = 52 Since Disney is ranked higher than Ford on more ballots, Disney receives 1 point. We can compare Disney and Sarnoff. We can find the number of ballots with Disney ranked higher than Sarnoff, and the number of ballots with Sarnoff ranked higher than Disney. Disney: 30 + 22 + 10 = 62 Sarnoff: 18 + 2 = 20 Since Disney is ranked higher than Sarnoff on more ballots, Disney receives 1 point. We can compare Gates and Disney. We can find the number of ballots with Gates ranked higher than Disney, and the number of ballots with Disney ranked higher than Gates. Gates: 22 + 18 + 2 = 42 Disney: 30 + 10 = 40 Since Gates is ranked higher than Disney on more ballots, Gates receives 1 point. After all the pairwise comparisons have been made, we can add up the total number of points for each candidate. Sarnoff: 1 point Ford: 0 points Gates: 1 + 1 + 1 = 3 points Disney: 1 + 1 = 2 points Since Gates received the most points, Gates is declared the winner. Professor Gates is declared the new president using the pairwise comparison method.
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