Answer
(a) Since Candidate B received the most points, Candidate B is declared the winner using the Borda count method.
(b) The majority criterion is not satisfied. Although Candidate A has a majority of the first-place votes, Candidate A was not declared the winner.
(c) The head-to-head criterion is not satisfied. Candidate A is favored over all other candidates in head-to-head comparison. However, Candidate A was not declared the winner.
(d) The irrelevant alternatives criterion is not satisfied. Initially, Candidate B was declared the winner. After Candidate C dropped out, Candidate B was not declared the winner. Instead, Candidate A was declared the winner.
Work Step by Step
(a) With the Borda count method, each candidate receives 1 point for each fourth-place vote, 2 points for each third-place vote, 3 points for each second-place vote, and 4 points for each first-place vote. The candidate with the most points is declared the winner.
We can find the total points for each candidate.
Candidate A:
4(14) + 3(4) + 2(0) + 1(8) = 76 points
Candidate B:
4(8) + 3(14) + 2(0) + 1(4) = 78 points
Candidate C:
4(0) + 3(0) + 2(14 + 8 + 4) + 1(0) = 52 points
Candidate D:
4(4) + 3(8) + 2(0) + 1(14) = 54 points
Since Candidate B received the most points, Candidate B is declared the winner using the Borda count method.
(b) Candidate A has 14 first-place votes, which is more than half of the first-place votes. Although Candidate A has a majority of the first-place votes, Candidate A was not declared the winner. Therefore, the majority criterion is not satisfied.
(c) 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 B.
Candidate A: 14 + 4 = 18
Candidate B: 8
Since Candidate A is ranked higher than Candidate B on more ballots, Candidate A is favored over Candidate B.
We can compare Candidate A and Candidate C.
Candidate A: 14 + 4 = 18
Candidate C: 8
Since Candidate A is ranked higher than Candidate C on more ballots, Candidate A is favored over Candidate C.
We can compare Candidate A and Candidate D.
Candidate A: 14
Candidate D: 8 + 4 = 12
Since Candidate A is ranked higher than Candidate D on more ballots, Candidate A is favored over Candidate D.
We can see that Candidate A is favored over all other candidates so there is no need to compare the other candidates with each other.
Candidate A is favored over all other candidates in head-to-head comparison. However, Candidate A was not declared the winner. Therefore, the head-to-head criterion is not satisfied.
(d) After Candidate C drops out, the other candidates below Candidate C on each ballot move up one spot on that ballot. We can go through the Borda count method again to determine the winner.
Candidate A:
3(14) + 2(4) + 1(8) = 58 points
Candidate B:
3(8) + 2(14) + 1(4) = 56 points
Candidate D:
3(4) + 2(8) + 1(14) = 42 points
Since Candidate A received the most points, Candidate A is declared the winner using the Borda count method.
Initially, Candidate B was declared the winner. After Candidate C dropped out, Candidate B was not declared the winner. Instead, Candidate A was declared the winner. We can see that after one of the losing candidates dropped out, a different candidate was selected as the winner. Therefore, the irrelevant alternatives criterion is not satisfied.