Name:
Discrete Mathematics
Date:
Block: _________________
Chapter 1: The Mathematics of Voting Test Review Packet
1. A math class is asked by the instructor to vote among four possible times for the final exam – A
(December 15, 8:00 A.M.), B (December 20, 9:00 P.M.), C (December 21, 7:00 A.M.), and D
(December 23, 11:00 A.M.). The following is the class preference schedule.
# of
voters
1st
choice
2nd
choice
3rd
choice
4th
choice
3
4
9
9
2
5
8
3
12
A
A
A
B
B
B
C
C
D
B
B
C
C
A
C
D
A
C
C
D
B
D
C
A
B
D
A
D
C
D
A
D
D
A
B
B
a. How many students in the class voted?
b. How many first-place votes are needed for a majority?
c. Which alternative(s) had the least first-place votes?
d. Which alternative(s) had the least last-place votes?
e. Which alternative would win under the plurality method?
2. Consider an election with 1025 voters, what is the smallest number of votes needed to be a
majority candidate?
3. The Latin Club is holding an election to choose its presidents. There are three candidates, Arsenio,
Beatrice, and Carlos (A, B, and C for short). Following are the votes of the 11 members of the club
that voted.
Voters
1st
choice
2nd
choice
3rd
choice
Sue
Bill
Tom
Pat
Tina
Mary
Alan
Chris
Paul
Kate
Ron
C
A
C
A
B
C
A
A
C
B
A
A
C
B
B
C
B
C
C
B
C
B
B
B
A
C
A
A
B
B
A
A
C
a. How many first-place votes are needed for a majority?
b. Which candidate has the most first-place votes? Is it a majority or a plurality?
c. Write out the preference schedule for this election.
# of voters
1st choice
2nd choice
3rd choice
4. The student body at Eureka High School is having an election for Homecoming Queen. The
candidates are Alicia, Brandy, Cleo, and Dionne (A, B, C and D). The preference schedule for the
election is as follows:
# of
voters
1st
choice
2nd
choice
3rd
choice
4th
choice
30
25
32
10
15
20
22
14
36
25
A
A
A
B
B
B
C
C
D
D
C
B
D
D
C
C
A
B
A
B
B
D
C
A
D
A
D
A
C
C
D
C
B
C
A
D
B
D
B
A
a. Find the winner of the election under the Borda count method.
A=
B=
C=
D=
Borda winner:
b. Suppose that before the votes are counted, Dionne is found to be ineligible because of her
grades. It is decided that Dionne’s name will be removed from the original preference
schedule. Find the preference schedule when Dionne’s name is removed, and then find the
winner of this new election under the Borda count method.
# of
voters
1st
choice
2nd
choice
3rd
choice
30
25
32
10
15
20
A=
B=
C=
Borda winner:
22
14
36
25
5. The members of the Tasmania State University soccer team are having an election to choose the
captain of the team from among the four seniors – Anderson, Bergman, Chou, and Delgado. The
preference schedule for the election is given in the following table.
# of voters
4
1
9
8
5
1st choice
A
B
C
A
C
2nd choice
B
A
D
D
D
3rd choice
D
D
A
B
B
4th choice
C
C
B
C
A
a. Find the winner under the Borda count method.
A=
B=
C=
D=
Borda winner:
b. Explain why this election shows that the Borda count method violates the majority criterion.
(Hint: you have to find the majority candidate first)
Majority candidate:
6. An election with three candidates and 100 voters is to be determined using the Borda count
method.
a. What is the maximum number of points a candidate can receive?
b. What is the minimum number of points a candidate can receive?
7. An election is to be decided using the Borda count method. There are five candidates (A, B, C, D,
and E) and 40 voters. If candidate A gets 139 points, candidate B gets 121 points, candidate C gets
80 points, and candidate D gets 113 points, who is the winner of the election?
8. Refer to the preference schedule in question #5 with the Tasmania State University soccer captain
election.
9.
a.)
Find the winner under the plurality-with-elimination method.
b.)
Explain why the winner in (a) can be determined in the first round.
An election is to be decided using the Borda count method. There are 5 candidates (Daphne,
Velma, Fred, Shaggy, and Scooby-Doo) in this election. How many points are given out by one
ballot?
10.
# of
voters
1st
choice
2nd
choice
3rd
choice
4th
choice
11.
Refer to the original preference table in question #4 with the Homecoming Queen. Find the
winner under the plurality-with-elimination method.
30
25
32
10
15
20
22
14
36
25
A
A
A
B
B
B
C
C
D
D
C
B
D
D
C
C
A
B
A
B
B
D
C
A
D
A
D
A
C
C
D
C
B
C
A
D
B
D
B
A
Refer to the preference schedule in question #1 with the final exam election. Find the winner of
the election under the method of pairwise comparisons.
# of
voters
1st
choice
2nd
choice
3rd
choice
4th
choice
3
4
9
9
2
5
8
3
12
A
A
A
B
B
B
C
C
D
B
B
C
C
A
C
D
A
C
C
D
B
D
C
A
B
D
A
D
C
D
A
D
D
A
B
B
A vs. B:
A vs. C:
A vs. D:
B vs. C:
B vs. D:
C vs. D:
Winner by pairwise comparison: