5-1 Election Theory
Voting Methods
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 1
WHAT YOU WILL LEARN
• Preference tables
• Voting methods
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 2
Example: Voting

Voting for Math Club President:
Four students are running for president of the
Math Club: Jerry, Thomas, Annette and Becky.
The club members were asked to rank all
candidates. The resulting preference table for
this election is shown on the next slide.
a) How many students voted in the election?
b) How many students selected the candidates
in this order: A, J, B, T?
c) How many students selected A as their first
choice?
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 3
Example: Voting (continued)
# of
Votes
First
Second
Third
14
12
9
4
1
T
J
A
B
A
J
A
J
B
J
B
A
A
B
J
Fourth
B
T
T
T
T
a) How many students voted in the election?
Add the row labeled Number of Votes
14 + 12 + 9 + 4 + 1 = 40
Therefore, 40 students voted in the election.
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 4
Example: Voting (continued)
# of
Votes
First
Second
Third
14
12
9
4
1
T
J
A
B
A
J
A
J
B
J
B
A
A
B
J
Fourth
B
T
T
T
T
b) How many students selected the
candidates in this order: A, J, B, T?
3rd column of numbers, 9 people
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 5
Example: Voting (continued)
# of
Votes
First
Second
Third
14
12
9
4
1
T
J
A
B
A
J
A
J
B
J
B
A
A
B
J
Fourth
B
T
T
T
T
c) How many students selected A as
their first choice?
9 + 1 = 10
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 6
Plurality Method

This is the most commonly used method, and it
is the easiest method to use when there are
more than two candidates.

Each voter votes for one candidate. The
candidate receiving the most votes is declared
the winner.
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 7
Example: Plurality Method


Who is elected math club president using the
plurality method?
We will assume that each member would vote
for the person he or she listed in first place.
# of
Votes
First
Second
Third
14
12
9
4
1
T
J
A
B
A
J
A
J
B
J
B
A
A
B
J
Fourth
B
T
T
T
T
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 8
Example: Plurality Method (continued)





Thomas received 14 votes
Becky received 12 votes
Annette received 10 votes
Jerry received 4 votes
Thomas would be elected since he received the
most votes. Note that Thomas received 14/40,
or 35%, of the first-place votes, which is less
than a majority.
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 9
Borda Count Method

Voters rank candidates from the most favorable
to the least favorable. Each last-place vote is
awarded one point, each next-to-last-place vote
is awarded two points, each third-from-lastplace vote is awarded three points, and so forth.
The candidate receiving the most points is the
winner of the election.
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 10
Example: Borda Count

Use the Borda count method to determine the
winner of the election for math club president.

Since 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.
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 11
Example: Borda Count (continued)






# of
Votes
First
Second
Third
14
12
9
4
1
T
J
A
B
A
J
A
J
B
J
B
A
A
B
J
Fourth
B
T
T
Thomas
14 first place votes
0 second place
0 third place
26 fourth place
14(4) + 0 + 0 + 26(1) = 82
Copyright © 2009 Pearson Education, Inc.






T
T
Annette
10 first place votes
12 second place
18 third place
0 fourth place
10(4) + 12(3) + 18(2) + 0
= 112
Chapter 15 Section 1 - Slide 12
Example: Borda Count (continued)






# of
Votes
First
Second
Third
14
12
9
4
1
T
J
A
B
A
J
A
J
B
J
B
A
A
B
J
Fourth
B
T
T
Betty
12 first place votes
5 second place
9 third place
14 fourth place
12(4) + 5(3) + 9(2) + 14 =
95
Copyright © 2009 Pearson Education, Inc.






T
T
Jerry
4 first place votes
23 second place
13 third place
0 fourth place
4(4) + 23(3) + 13(2) + 0 =
111
Chapter 15 Section 1 - Slide 13
Example: Borda Count (continued)





Thomas - 82
Annette - 112
Betty - 95
Jerry - 111
Annette, with 112 points, receives the most
points and is declared the winner.
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 14
Plurality with Elimination

Each voter votes for one candidate. If a
candidate receives a majority of votes, that
candidate is declared the winner. If no
candidate receives a majority, eliminate the
candidate with the fewest votes and hold
another election. (If there is a tie for the fewest
votes, eliminate all candidates tied for the
fewest votes.) Repeat this process until a
candidate receives a majority.
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 15
Example: Plurality with Elimination






Use the plurality with elimination method to
determine the winner of the election for
president of the math club.
Count the number of first place votes
Annette 10 # of
14
12
9
4
Votes
Betty 12
First
T
B
A
J
Thomas 14 Second
J
A
J
B
Third
A
J
B
A
Jerry 4
Fourth
Copyright © 2009 Pearson Education, Inc.
B
T
T
T
1
A
B
J
T
Chapter 15 Section 1 - Slide 16
Example: Plurality with Elimination
(continued)





Since 40 votes were cast, a candidate must
have 20 first place votes to receive a majority.
Jerry had the fewest number of first place votes,
so he is eliminated.
Redo the table. # of
14 12
9
4
1
Thomas 14
Votes
Annette 10
First
T
B
A
B
A
Betty 16
Second
A
A
B
A
B
Third
Copyright © 2009 Pearson Education, Inc.
B
T
T
T
T
Chapter 15 Section 1 - Slide 17
Example: Plurality with Elimination
(continued)





Still, no candidate received a majority. Annette
has the fewest number of first-place votes, so
she is eliminated.
New preference table
# of
14 12 9
4
1
Betty 26
Votes
Thomas 14
First
T
B
B
B B
Betty is the winner.
Second
Copyright © 2009 Pearson Education, Inc.
B
T
T
T
T
Chapter 15 Section 1 - Slide 18
Pairwise Comparison Method

Voters rank the candidates. A series of
comparisons in which each candidate is
compared with each of the other candidates
follows. If candidate A is preferred to candidate
B, A receives one point. If candidate B is
preferred to candidate A, B receives 1 point. If
the candidates tie, each receives ½ point. After
making all comparisons among the candidates,
the candidate receiving the most points is
declared the winner.
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 19
Example: Pairwise Comparison


Use the pairwise comparison method to
determine the winner of the election for math
club president.
Number of comparisons needed:
n(n  1) 4(3)
c

6
2
2
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 20
Example: Pairwise Comparison
(continued)



Thomas versus Jerry
 T = 14 J = 12 + 9 + 4 + 1 = 26
Jerry = 1
Thomas versus Annette
 T = 14 A = 12 + 9 + 4 + 1 = 26 Annette = 1
Thomas versus Betty
 T = 14 B = 12 + 9 + 4 + 1 = 26 Betty = 1
# of
Votes
First
Second
Third
14
12
9
4
1
T
J
A
B
A
J
A
J
B
J
B
A
A
B
J
Fourth
B
T
T
T
T
Copyright © 2009 Pearson Education, Inc.
Chapter 15 Section 1 - Slide 21
Example: Pairwise Comparison
(continued)
Betty versus Annette
 B = 12 + 4 = 16
A = 14 + 9 + 1 = 24 Annette = 1
 Betty versus Jerry
 B = 12 + 1 = 13
J = 14 + 9 + 4 = 27 Jerry = 1
 Annette versus Jerry
 A = 12 + 9 + 1 = 22
J = 14 + 4 = 18
Annette = 1
# of
14
12
9
4
1
Annette would win
Votes
with 3 total points, the First
T
B
A
J
A
most from the pairwise
Second
J
A
J
B
B
comparison method.
Third
A
J
B
A
J

Fourth
Copyright © 2009 Pearson Education, Inc.
B
T
T
T
T
Chapter 15 Section 1 - Slide 22
The employees at a local law firm are voting on
the entrée for their annual holiday party. Their
choices are chicken (C), salmon (S), and steak
(T). The preference table follows.
Number of Votes
30
18
12
23
First
T
C
T
S
Second
S
T
C
C
Third
C
S
S
T
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 23
Here’s the preference table, again.
How many employees voted?
Number of Votes
30
18
12
23
First
T
C
T
S
Second
S
T
C
C
Third
C
S
S
T
a. 48
b. 58
Copyright © 2009 Pearson Education, Inc.
c. 60
d. 83
Slide 15 - 24
Here’s the preference table, again.
How many employees voted?
Number of Votes
30
18
12
23
First
T
C
T
S
Second
S
T
C
C
Third
C
S
S
T
a. 48
b. 58
Copyright © 2009 Pearson Education, Inc.
c. 60
d. 83
Slide 15 - 25
Here’s the preference table, again.
Does any choice have a majority of votes?
Number of Votes
30
18
12
23
First
T
C
T
S
Second
S
T
C
C
Third
C
S
S
T
a. Yes.
b. No.
Copyright © 2009 Pearson Education, Inc.
c. Can’t determine.
Slide 15 - 26
Here’s the preference table, again.
Does any choice have a majority of votes?
Number of Votes
30
18
12
23
First
T
C
T
S
Second
S
T
C
C
Third
C
S
S
T
a. Yes.
b. No.
Copyright © 2009 Pearson Education, Inc.
c. Can’t determine.
Slide 15 - 27
Here’s the preference table, again.
Determine the winner using the plurality method.
Number of Votes
30
18
12
23
First
T
C
T
S
Second
S
T
C
C
Third
C
S
S
T
a. Chicken
b. Salmon
c. Steak
d. Can’t determine
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 28
Here’s the preference table, again.
Determine the winner using the plurality method.
Number of Votes
30
18
12
23
First
T
C
T
S
Second
S
T
C
C
Third
C
S
S
T
a. Chicken
b. Salmon
c. Steak
d. Can’t determine
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 29
Here’s the preference table, again. Determine the
winner using the Borda count method.
Number of Votes
30
18
12
23
First
T
C
T
S
Second
S
T
C
C
Third
C
S
S
T
a. Chicken
b. Salmon
c. Steak
d. Can’t determine
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 30
Here’s the preference table, again. Determine the
winner using the Borda count method.
Number of Votes
30
18
12
23
First
T
C
T
S
Second
S
T
C
C
Third
C
S
S
T
a. Chicken
b. Salmon
c. Steak
d. Can’t determine
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 31
Here’s the preference table, again. Determine the
winner using the plurality method with elimination
method.
Number of Votes
30
18
12
23
First
T
C
T
S
Second
S
T
C
C
Third
C
S
S
T
a. Chicken
b. Salmon
c. Steak
d. Can’t determine
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 32
Here’s the preference table, again. Determine the
winner using the plurality method with elimination
method.
Number of Votes
30
18
12
23
First
T
C
T
S
Second
S
T
C
C
Third
C
S
S
T
a. Chicken
b. Salmon
c. Steak
d. Can’t determine
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 33
Here’s the preference table, again. Determine the
winner using the pairwise comparison method.
Number of Votes
30
18
12
23
First
T
C
T
S
Second
S
T
C
C
Third
C
S
S
T
a. Chicken
b. Salmon
c. Steak
d. Can’t determine
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 34
Here’s the preference table, again. Determine the
winner using the pairwise comparison method.
Number of Votes
30
18
12
23
First
T
C
T
S
Second
S
T
C
C
Third
C
S
S
T
a. Chicken
b. Salmon
c. Steak
d. Can’t determine
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 35
The employees also must pick the dessert for
their holiday party. The choices are cheese cake
(C), apple tart (A), chocolate mousse cake (M),
or vanilla ice cream (V). Here’s the preference
table.
Number of Votes 19 16 12 22 14
First
A
C
M
V
A
Second
M
A
V
M
M
Third
C
V
A
C
C
Fourth
V
M
C
A
V
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 36
Which dessert wins this election if the plurality
method is used?
Number of Votes
19
16
12
22
14
First
A
C
M
V
A
Second
M
A
V
M
M
Third
C
V
A
C
C
Fourth
V
M
C
A
V
a. Cheese cake
b. Apple tart
c. Chocolate Mousse
d. Vanilla ice cream
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 37
Which dessert wins this election if the plurality
method is used?
Number of Votes
19
16
12
22
14
First
A
C
M
V
A
Second
M
A
V
M
M
Third
C
V
A
C
C
Fourth
V
M
C
A
V
a. Cheese cake
b. Apple tart
c. Chocolate Mousse
d. Vanilla ice cream
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 38
Which dessert wins this election if the Borda
count method is used?
Number of Votes
19
16
12
22
14
First
A
C
M
V
A
Second
M
A
V
M
M
Third
C
V
A
C
C
Fourth
V
M
C
A
V
a. Cheese cake
b. Apple tart
c. Chocolate Mousse
d. Vanilla ice cream
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 39
Which dessert wins this election if the Borda
count method is used?
Number of Votes
19
16
12
22
14
First
A
C
M
V
A
Second
M
A
V
M
M
Third
C
V
A
C
C
Fourth
V
M
C
A
V
a. Cheese cake
b. Apple tart
c. Chocolate Mousse
d. Vanilla ice cream
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 40
Which dessert wins this election if the plurality
with elimination method is used?
Number of Votes
19
16
12
22
14
First
A
C
M
V
A
Second
M
A
V
M
M
Third
C
V
A
C
C
Fourth
V
M
C
A
V
a. Cheese cake
b. Apple tart
c. Chocolate Mousse
d. Vanilla ice cream
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 41
Which dessert wins this election if the plurality
with elimination method is used?
Number of Votes
19
16
12
22
14
First
A
C
M
V
A
Second
M
A
V
M
M
Third
C
V
A
C
C
Fourth
V
M
C
A
V
a. Cheese cake
b. Apple tart
c. Chocolate Mousse
d. Vanilla ice cream
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 42
Which dessert wins this election if the pairwise
comparison method is used?
Number of Votes
19
16
12
22
14
First
A
C
M
V
A
Second
M
A
V
M
M
Third
C
V
A
C
C
Fourth
V
M
C
A
V
a. Cheese cake
b. Apple tart
c. Chocolate Mousse
d. Vanilla ice cream
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 43
Which dessert wins this election if the pairwise
comparison method is used?
Number of Votes
19
16
12
22
14
First
A
C
M
V
A
Second
M
A
V
M
M
Third
C
V
A
C
C
Fourth
V
M
C
A
V
a. Cheese cake
b. Apple tart
c. Chocolate Mousse
d. Vanilla ice cream
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 44
Practice Problems
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 45
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 46
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 47
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 48
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 49
Copyright © 2009 Pearson Education, Inc.
Slide 15 - 50