PROBLEMS
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From The Mathematics of Elections and Voting by W.D. Wallis
Practice Exercise 2.3
Suppose the preference profile of an election is :
A
B
C
7
A
A
A
A
B
B
B
B
C
C
C
C
5
8
3
4
Assuming that voters stick with their rankings as shown above, what is the result of the election in the following cases?
​
(i) The absolute majority is used.
The total number of votes is 7+5+8+3+4=27 votes. Therefore, in order for a candidate to receive the majority (NOT compared to other voters), he or she needs at least 14 votes to win the election. However, none of the candidates that many, and therefore there is no absolute majority winner.
​
(ii) The plurality method is used.
This method is the most popular in elections today. Plurality doesn't utilize preference profiles, so the candidate with the most first-place votes compared to the others wins, even if they do not receive the absolute majority. Candidate A received 12 votes, Candidate B received 11 votes, and Candidate C received 4 votes. Therefore, Candidate A is the plurality winner.
Practice Exercise 2.4
Suppose the preference profile of an election is :
6
7
7
7
2
C
B
B
D
A
C
A
D
C
B
C
D
A
B
D
C
A
B
7
D
C
B
A
5
B
D
C
A
2
D
B
C
A
Assuming that voters stick with their rankings as shown above, what is the result of the election in the following cases?
​
(i) The plurality method is used.
The candidate with the most votes compared to the other candidates wins. Candidate A received 13 votes, Candidate B received 12 votes, Candidate C received 7 votes, and Candidate D received 11 votes. Therefore, Candidate B is winner using the plurality system.
​
(ii) The Hare (or instant runoff method) is used.
The Hare method indicates that candidates with the least number of first-place votes are eliminated one by one until a winner is left. Candidate C received the least number of first-place votes (7), so he/she is eliminated first. This means that for these seven voters, Candidate D is now their highest ranked candidate. Candidate B would the next person to be eliminated with only 12 first-place votes. Finally, Candidate A would be removed (13 votes compared to Candidate D's 30 votes). Candidate D is the Hare method winner.
​
(iii) The Coombs rule is used.
With the Coombs rule, candidates with the greatest number of last-place votes are eliminated one by one. Candidate B received the greatest number of last-place votes (22), so he/she is eliminated first. Candidate A would be the next person to be eliminated from the race with 23 last-place votes. And finally, Candidate C would be removed from the competition (30 last-place votes vs. Candidate D's 13 votes). Candidate D is the Coombs rule winner.
​
In this case, Candidate D came out on top in two of the three methods used. Let's look at a problem in which three different methods produce three different results!
Sample Problem 2.3
Suppose the preference profile of an election with four candidates and 33 voters is :
A
B
10
8
7
6
C
D
B
D
C
A
C
B
D
A
D
C
B
A
Assuming that voters stick with their rankings as shown above, what is the result of the election in the following cases?
​
(i) The plurality method is used.
Candidate A is the plurality winner with 10 votes.
​
(ii) The Hare method is used.
Candidate D is eliminated first (only 6 first-place votes), then Candidate B, and finally Candidate C is deemed the winner under system (21 first-place votes vs. Candidate A's 10 first-place votes).
​
(iii) The Coombs rule is used.
Candidate A is first eliminated (21 last-place votes), then Candidate C, and Candidate D finally beats Candidate B 16-15.
A
D