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: 4

Answer

(a) Since Boston is favored over Atlanta and Chicago, Boston is favored over all others using a head-to-head comparison. (b) Atlanta has the highest number of first-place votes, so Atlanta wins the vote using the plurality method. (c) The head-to-head criterion is not satisfied.

Work Step by Step

(a) With a head-to-head comparison, each candidate is compared with every other candidate. For each pair of candidates, a candidate is favored over the other candidate if the candidate is ranked higher than the other candidate on a majority of ballots. We can compare Boston and Atlanta. Boston: 19 + 5 = 24 Atlanta: 20 Since Boston is ranked higher than Atlanta on more ballots, Boston is favored over Atlanta. We can compare Boston and Chicago. Boston: 20 + 19 = 39 Chicago: 5 Since Boston is ranked higher than Chicago on more ballots, Boston is favored over Chicago. We can compare Chicago and Atlanta. Chicago: 19 + 5 = 24 Atlanta: 20 Since Chicago is ranked higher than Atlanta on more ballots, Chicago is favored over Atlanta. Since Boston is favored over Atlanta and Chicago, Boston is favored over all others using a head-to-head comparison. (b) Using the plurality method, the candidate which has the most first-place votes is declared the winner. We can find the number of first-place votes for each city. The number of first-place votes for Boston is 19 votes. The number of first-place votes for Chicago is 5 votes. The number of first-place votes for Atlanta is 20 votes. Atlanta has the highest number of first-place votes, so Atlanta wins the vote using the plurality method. (c) The head-to-head criterion is not satisfied. Boston is favored over all others using a head-to-head comparison, but Boston did not win the vote using the plurality method.
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