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: 33

Answer

Rent 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 which receives the most points is declared the winner. We can compare Cabaret and Rent. We can find the number of ballots with Cabaret ranked higher than Rent, and the number of ballots with Rent ranked higher than Cabaret. Cabaret: 5 + 4 + 2 = 11 Rent: 5 + 3 + 3 = 11 Since Rent and Cabaret are tied, both Rent and Cabaret receive 0.5 points. We can compare Cabaret and The Producers. We can find the number of ballots with Cabaret ranked higher than The Producers, and the number of ballots with The Producers ranked higher than Cabaret. Cabaret: 5 + 4 + 3 = 12 The Producers: 5 + 3 + 2 = 10 Since Cabaret is ranked higher than The Producers on more ballots, Cabaret receives 1 point. We can compare The Producers and Rent. We can find the number of ballots with The Producers ranked higher than Rent, and the number of ballots with Rent ranked higher than The Producers. The Producers: 4 + 3 + 2 = 9 Rent: 5 + 5 + 3 = 13 Since Rent is ranked higher than The Producers on more ballots, Rent receives 1 point. We can compare Cabaret and Sweeney Todd. We can find the number of ballots with Cabaret ranked higher than Sweeney Todd, and the number of ballots with Sweeney Todd ranked higher than Cabaret. Cabaret: 5 + 4 = 9 Sweeney Todd: 5 + 3 + 3 + 2 = 13 Since Sweeney Todd is ranked higher than Cabaret on more ballots, Sweeney Todd receives 1 point. We can compare Sweeney Todd and Rent. We can find the number of ballots with Sweeney Todd ranked higher than Rent, and the number of ballots with Rent ranked higher than Sweeney Todd. Sweeney Todd: 5 + 2 = 7 Rent: 5 + 4 + 3 + 3 = 15 Since Rent is ranked higher than Sweeney Todd on more ballots, Rent receives 1 point. We can compare The Producers and Sweeney Todd. We can find the number of ballots with The Producers ranked higher than Sweeney Todd, and the number of ballots with Sweeney Todd ranked higher than The Producers. The Producers: 5 + 4 + 3 + 2 = 14 Sweeney Todd: 5 + 3 = 8 Since The Producers is ranked higher than Sweeney Todd on more ballots, The Producers receives 1 point. We can compare Cabaret and West Side Story. We can find the number of ballots with Cabaret ranked higher than West Side Story, and the number of ballots with West Side Story ranked higher than Cabaret. Cabaret: 5 + 4 + 2 = 11 West Side Story: 5 + 3 + 3 = 11 Since West Side Story and Cabaret are tied, West Side Story and Cabaret both receive 0.5 points. We can compare The Producers and West Side Story. We can find the number of ballots with The Producers ranked higher than West Side Story, and the number of ballots with West Side Story ranked higher than The Producers. The Producers: 5 + 4 + 2 = 11 West Side Story: 5 + 3 + 3 = 11 Since West Side Story and The Producers are tied, West Side Story and The Producers both receive 0.5 points. We can compare Sweeney Todd and West Side Story. We can find the number of ballots with Sweeney Todd ranked higher than West Side Story, and the number of ballots with West Side Story ranked higher than Sweeney Todd. Sweeney Todd: 5 + 4 + 2 = 11 West Side Story: 5 + 3 + 3 = 11 Since West Side Story and Sweeney Todd are tied, West Side Story and Sweeney Todd both receive 0.5 points. We can compare West Side Story and Rent. We can find the number of ballots with West Side Story ranked higher than Rent, and the number of ballots with Rent ranked higher than West Side Story. West Side Story: 3 + 3 = 6 Rent: 5 + 5 + 4 + 2 = 16 Since Rent is ranked higher than West Side Story on more ballots, Rent receives 1 point. After all the pairwise comparisons have been made, we can add up the total number of points for each candidate. Cabaret: 0.5 + 1 + 0.5 = 2 points The Producers: 1 + 0.5 = 1.5 points Rent: 0.5 + 1 + 1 + 1 = 3.5 points Sweeney Todd: 1 + 0.5 = 1.5 points West Side Story: 0.5 + 0.5 + 0.5 = 1.5 points Since Rent received the most points, Rent is declared the winner. Rent is selected using the pairwise comparison method.
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