## Thinking Mathematically (6th Edition)

Published by Pearson

# Chapter 13 - Voting and Apportionment - 13.2 Flaws of Voting Methods - Exercise Set 13.2: 7

#### Answer

(a) Candidate C is declared the winner using the plurality-with-elimination method. (b) Candidate B is declared the winner using the plurality-with-elimination method. (c) The monotonicity criterion is not satisfied.

#### Work Step by Step

(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 which 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: 7 + 4 = 11 Candidate B: 8 Candidate C: 10 In round 1, Candidate B has the fewest number of first-place votes, so Candidate B is eliminated. After Candidate B is eliminated, the other candidates which were ranked below Candidate B 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: 7 + 4 = 11 Candidate C: 10 + 8 = 18 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) Suppose the four voters who voted A, C, B change their votes to C, A, B. We can go through the plurality-with-elimination method again to see who wins the vote. In round 1, we can count the number of first-place votes for each candidate. Candidate A: 7 Candidate B: 8 Candidate C: 10 + 4 = 14 In round 1, Candidate A has the fewest number of first-place votes, so Candidate A is eliminated. After Candidate A is eliminated, the other candidates which were ranked below Candidate A 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 B: 8 + 7 = 15 Candidate C: 10 + 4 = 14 In round 2, Candidate C has the fewest number of first-place votes, so Candidate C is eliminated. After Candidate C is eliminated, Candidate B is the only candidate remaining, so Candidate B is declared the winner. Candidate B is declared the winner using the plurality-with-elimination method. (c) The monotonicity criterion is not satisfied. Initially, Candidate C wins the vote using the plurality-with-elimination method. After this, four voters changed their votes with Candidate C getting a higher ranking than Candidate A on the ballot. That is, the change was in favor of Candidate C. After this, Candidate B won the vote using the plurality-with-elimination method. Therefore, the monotonicity criterion is not satisfied.

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