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: 25

Answer

Professor Hawking is declared the new division chair 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 Freud and Hawking. We can find the number of ballots with Freud ranked higher than Hawking, and the number of ballots with Hawking ranked higher than Freud. Freud: 22 + 20 = 42 Hawking: 30 + 12 + 2 = 44 Since Hawking is ranked higher than Freud on more ballots, Hawking receives 1 point. We can compare Freud and Einstein. We can find the number of ballots with Freud ranked higher than Einstein, and the number of ballots with Einstein ranked higher than Freud. Freud: 30 + 20 + 2 = 52 Einstein: 22 + 12 = 34 Since Freud is ranked higher than Einstein on more ballots, Freud receives 1 point. We can compare Einstein and Hawking. We can find the number of ballots with Einstein ranked higher than Hawking, and the number of ballots with Hawking ranked higher than Einstein. Einstein: 22 + 20 = 42 Hawking: 30 + 12 + 2 = 44 Since Hawking is ranked higher than Einstein on more ballots, Hawking receives 1 point. We can compare Freud and Darwin. We can find the number of ballots with Freud ranked higher than Darwin, and the number of ballots with Darwin ranked higher than Freud. Freud: 22 + 20 + 12 + 2 = 56 Darwin: 30 Since Freud is ranked higher than Darwin on more ballots, Freud receives 1 point. We can compare Darwin and Hawking. We can find the number of ballots with Darwin ranked higher than Hawking, and the number of ballots with Hawking ranked higher than Darwin. Darwin: 30 Hawking: 22 + 20 + 12 + 2 = 56 Since Hawking is ranked higher than Darwin on more ballots, Hawking receives 1 point. We can compare Einstein and Darwin. We can find the number of ballots with Einstein ranked higher than Darwin, and the number of ballots with Darwin ranked higher than Einstein. Einstein: 22 + 20 + 12 = 54 Darwin: 30 + 2 = 32 Since Einstein is ranked higher than Darwin on more ballots, Einstein receives 1 point. After all the pairwise comparisons have been made, we can add up the total number of points for each candidate. Hawking: 1 + 1 + 1 = 3 points Freud: 1 + 1 = 2 points Einstein: 1 point Darwin: 0 points Since Hawking received the most points, Hawking is declared the winner. Professor Hawking is declared the new division chair using the pairwise comparison method.
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