CHAPTER 14
Voting and
Apportionment
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14.1
Voting Methods
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2
1.
2.
3.
4.
5.
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.
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3
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.
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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.
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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:
120 + 100 = 220
Thus, 220 people selected Donna as their first choice.
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6
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.
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7
The Borda Count Method
• Each voter ranks the candidates from the most
favorable to the least favorable.
1. Each last-place vote is given 1 point, each nextto-last-place vote is given 2 points, each thirdfrom-last-place vote is given 3 points, and so on.
2. The points are totaled for each candidate
separately.
3. The candidate with the most points is the winner.
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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
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
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4
1
9
Example 3: Using the Borda Count Method
Now, we read down each column and total the points
for each candidate separately:
P: 56 + 10 + 8 + 4 + 1 = 79 points
R: 42 + 30 + 16 + 16 + 2 = 106 points
S: 28 + 40 + 24 + 8 + 4 = 104 points
T: 14 + 20 + 32 + 12+ 3 = 81 points
Because Rita (R) has received the most points, she is the
winner and the new president of the Student Film
Institute.
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10
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.
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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
S
Second Choice
R
R
S
T
T
Third Choice
S
T
R
S
R
Fourth Choice
T
P
P
P
P
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.
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Example 4: Using the Plurality-with-Elimination
Method
The number of first-place votes for each candidate is
P(Paul) = 14
S(Sarah) = 10 + 1 = 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
Number of
Votes
14 10
8
4
1
First Choice
P
S
T
T
S
Second Choice
S
T
S
S
T
Third Choice
T
P
P
P
P
So, the number of first-place votes for each candidate is
P(Paul) = 14
S(Sarah) = 10 + 1 = 11
T(Tim) = 8 + 4 = 12.
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13
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
Number of
Votes
14 10
8
4
1
First Choice
P
T
T
T
T
Second Choice
T
P
P
P
P
The number of first-place votes for each candidate is now
P(Paul) = 14
T(Tim) = 10+ 8 + 4 + 1 = 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.
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14
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
nn  1
C
2
where n is the number of candidates, and C is the
number of comparisons that must be made.
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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.
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Example 5: Using the Pairwise Comparison
Method
For the previous table, who is declared the winner using
the pairwise comparison method?
Number of
Votes
14
10
8
4
1
First Choice
P
S
T
R
S
Second Choice
R
R
S
T
T
Third Choice
S
T
R
S
R
Fourth Choice
T
P
P
P
P
Solution: We first find how many comparisons that
must be made. Since there are 4 candidates, then n=4
and
nn  1 44  1
C

 6.
2
2
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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.
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Example 5: Using the Pairwise Comparison
Method
Using the six comparisons and conclusion to add points we get
P: no points
R: 1 + 1 = 2 points
S: 1 + 1 + 1 = 3 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.
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Homework
Pg 777 – 778, #1 – 6, 8, 9, 12, 13, 16, 17, 21, 22,
24, 25, 27 – 30.
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