Chapter 13 - Voting and Apportionment - 13.2 Flaws of Voting Methods - Exercise Set 13.2 - Page 862: 14

(a) Candidate C is declared the winner using the plurality-with-elimination method. (b) The head-to-head criterion is not satisfied. Candidate D is favored over all other candidates in head-to-head comparison. However, Candidate D was not declared the winner.

(a) With the plurality-with-elimination method, the candidate with the fewest number of first-place votes is eliminated in each round. After that candidate is eliminated, the other candidates that were ranked below that candidate on each ballot move up one spot on that ballot. The rounds continue in this way until only one candidate remains, and this candidate is declared the winner. In round 1, we can count the number of first-place votes for each candidate. Candidate A: 18 Candidate B: 7 Candidate C: 10 + 9 = 19 Candidate D: 4 + 3 = 7 In round 1, Candidate B and Candidate D tie for the fewest number of first-place votes, so Candidate B and Candidate D are eliminated. After Candidate B and Candidate D are eliminated, the other candidates that were ranked below Candidate B and Candidate D on each ballot move up one spot on that ballot. In round 2, we can count the number of first-place votes for each candidate. Candidate A: 18 + 4 = 22 Candidate C: 10 + 9 + 7 + 3 = 29 In round 2, Candidate A has the fewest number of first-place votes, so Candidate A is eliminated. After Candidate A is eliminated, Candidate C is the only candidate remaining, so Candidate C is declared the winner. Candidate C is declared the winner using the plurality-with-elimination method. (b) 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 Candidate A and Candidate D. Candidate A: 18 Candidate D: 10 + 9 + 7 + 4 + 3 = 33 Since Candidate D is ranked higher than Candidate A on more ballots, Candidate D is favored over Candidate A. We can compare Candidate B and Candidate D. Candidate B: 10 + 7 = 17 Candidate D: 18 + 9 + 4 + 3 = 34 Since Candidate D is ranked higher than Candidate B on more ballots, Candidate D is favored over Candidate B. We can compare Candidate C and Candidate D. Candidate C: 10 + 9 = 19 Candidate D: 18 + 7 + 4 + 3 = 32 Since Candidate D is ranked higher than Candidate C on more ballots, Candidate D is favored over Candidate C. We can see that Candidate D is favored over all other candidates so there is no need to compare the other candidates with each other. Candidate D is favored over all other candidates in head-to-head comparison. However, Candidate D was not declared the winner. Therefore, the head-to-head criterion is not satisfied.