1. Voting and Apportionment
CHAPTER 14
Voting and Apportionment

2. 14.1
Voting Methods

3. Objective
Understand and use preference tables.
Use the plurality method to determine an election’s winner.
Use the Borda count method to determine an election’s winner.
Use the plurality-with-elimination method to determine an election’s winner.
Use the pairwise comparison method to determine an election’s winner.

4. Preference Tables
Preference ballots are ballots in which a voter is asked to rank all the candidates in order of preference.
Example: Four candidates are running for president of the Student Film Institute: Paul (P), Rita (R), Sarah (S), and Tim (T). Each of the club’s members submits a secret ballot indicating his or her first, second, third, and fourth choice for president.
The 37 ballots are shown to the right.

5. Preference Tables
A preference table shows how often each particular outcome occurred.
Example:
The election ballots are placed into identical stacks.
Then the identical stacks are placed into a preference table.

6. Example 1: Understanding a Preference Table
How many people selected Donna (D) as their first choice?
Solution: We find the number of people who voted for D as their first choice by reading across the row that says First Choice. When you see a D in this row, write the number above it. Then find the sum of the numbers:
= 220
Thus, 220 people selected Donna as their first choice.

7. The Plurality Method
The candidate (or candidates, if there is more than one) with the most first-place votes is the winner.
Example: For the previous preference table, who is declared the winner using the plurality method?
Solution: The candidate with the most first-place votes is the winner. When using the preference table, we only need to look at the row indicating the number of first-choice votes.
Thus, P (Paul) is the winner and the new president of the Student Film Institute.

8. The Borda Count Method
Each voter ranks the candidates from the most favorable to the least favorable.
Each last-place vote is given 1 point, each next-to-last-place vote is given 2 points, each third-from-last-place vote is given 3 points, and so on.
The points are totaled for each candidate separately.
The candidate with the most points is the winner.

9. Example 3: Using the Borda Count Method
Example: In the previous table, who is the winner using the Borda method?
Solution: Because there are four candidates, a first-place vote is worth 4 points, a second place vote is worth 3 points, a third-place vote is worth 2 points, and a fourth-place vote is worth 1 point.
Number of Votes 14 10 8 4 1
First Choice: 4 pts
P: 144=56
S: 104=40
T: 84=32
R: 44=16
S: 14=4
Second Choice: 3 pts
R: 143=42
R: 103=30
S: 83=24
T: 43=12
T: 13=3
Third Choice: 2 pts
S: 142=28
T: 102=20
R: 82=16
S: 42=8
R: 12=2
Fourth Choice: 1 pt
T: 141=14
P: 101=10
P: 81=8
P: 41=4
P: 11=1

10. Example 3: Using the Borda Count Method
Now, we read down each column and total the points for each candidate separately:
P: = 79 points
R: = 106 points
S: = 104 points
T: = 81 points
Because Rita (R) has received the most points, she is the winner and the new president of the Student Film Institute.

11. The Plurality-with-Elimination Method
The candidate with the majority of first-place votes is the winner.
If no candidate receives a majority of first-place votes, eliminate the candidate (candidates, if there is a tie) with the fewest first-place votes from the preference table.
Move the candidates in each column below each eliminated candidate up one place.
The candidate with the majority of first-place votes in the new preference table is the winner.
If no candidate receives a majority of first-place votes, repeat this process until a candidate receives a majority.

12. Example 4: Using the Plurality-with-Elimination Method
Example: In the previous table, who is declared the winner using the plurality-with-elimination method?
Number of Votes 14 10 8 4 1
First Choice P S T R
Second Choice
Third Choice
Fourth Choice
Solution: Recall that there are 37 people voting. In order to receive a majority, a candidate must receive more than 50% of the first-place votes, i.e., meaning 19 or more votes.

13. Example 4: Using the Plurality-with-Elimination Method
The number of first-place votes for each candidate is
P(Paul) = S(Sarah) = = 11 T(Tim) = 8 R(Rita) = 4
We see that no candidate receives a majority of first-place votes. Because Rita received the fewest first-place votes, she is eliminated in the first round. The new preference table is
So, the number of first-place votes for each candidate is
P(Paul) = S(Sarah) = = T(Tim) = = 12.
Number of Votes 14 10 8 4 1
First Choice P S T
Second Choice
Third Choice

14. Example 4: Using the Plurality-with-Elimination Method
Once again, no candidate receives a majority first-place votes. Because Sarah received the fewest first-place votes, she is eliminated from the second round. The new preference table is
The number of first-place votes for each candidate is now
P(Paul) = T(Tim) = = 23.
Because T(Tim) has received the majority of first-place votes, i.e., Tim received more than 19 votes, he is the winner and the new president of the Student Film Institute.
Number of Votes 14 10 8 4 1
First Choice P T
Second Choice

15. The Pairwise Comparison Method
Using the pairwise comparison method, every candidate is compared one-on-one with every other candidate.
The number of comparisons made using the pairwise comparison method is given by
where n is the number of candidates, and C is the number of comparisons that must be made.

16. The Pairwise Comparison Method
Voters rank all the candidates and the results are summarized in a preference table.
The table is used to make a series of comparisons in which each candidate is compared to each of the other candidates.
For each pair of candidates, X and Y, use the table to determine how many voters prefer X to Y and vice versa.
If a majority prefer X to Y, then X receives 1 point.
If a majority prefers Y to X, then Y receives 1 point.
If the candidates tie, then each receives half a point.
After all comparisons have been made, the candidate receiving the most points is the winner.

17. Example 5: Using the Pairwise Comparison Method
For the previous table, who is declared the winner using the pairwise comparison method?
Solution: We first find how many comparisons that must be made. Since there are 4 candidates, then n=4 and
Number of Votes 14 10 8 4 1
First Choice P S T R
Second Choice
Third Choice
Fourth Choice

18. Example 5: Using the Pairwise Comparison Method
With P, R, S, T, the comparisons are P vs. R, P vs. S, P vs. T, R vs. S, R vs. T, and S vs. T.

19. Example 5: Using the Pairwise Comparison Method
Using the six comparisons and conclusion to add points we get
P: no points S: = 3 points
R: = 2 points T: 1 point
After all the comparisons have been made, the candidate receiving the most points is S(Sarah). Sarah is the winner and new president of the Student Film Institute.