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: 24

Answer

New York is selected 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 New York and Chicago. We can find the number of ballots with New York ranked higher than Chicago, and the number of ballots with Chicago ranked higher than New York. New York: 16 + 8 + 6 = 30 Chicago: 4 Since New York is ranked higher than Chicago on more ballots, New York receives 1 point. We can compare New York and San Francisco. We can find the number of ballots with New York ranked higher than San Francisco, and the number of ballots with San Francisco ranked higher than New York. New York: 8 + 6 + 4 = 18 San Francisco: 16 Since New York is ranked higher than San Francisco on more ballots, New York receives 1 point. We can compare San Francisco and Chicago. We can find the number of ballots with San Francisco ranked higher than Chicago, and the number of ballots with Chicago ranked higher than San Francisco. San Francisco: 16 + 8 = 24 Chicago: 6 + 4 = 10 Since San Francisco is ranked higher than Chicago on more ballots, San Francisco receives 1 point. After all the pairwise comparisons have been made, we can add up the total number of points for each candidate. Chicago: 0 points New York: 1 + 1 = 2 points San Francisco: 1 point Since New York received the most points, New York is declared the winner. New York is selected using the pairwise comparison method.
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