Deck 16: Voting and Apportionment

Question 1

Solve the problem.
Four members are running for president of the Greater New Orleans Stamp Club: Andy (a), Brett (b), Carol (c), and Denise (d). The voter profile is summarized in the table. Use the Borda Method to determine the winner.

A)Denise
B)Brett
C)Carol
D)Andy

Answer 1

D

Question 2

Solve the problem.
The Mathematics Department is holding an election for department chair. Each member ranks the candidates from first to third. The voter profile below shows the results of the ballots with
Candidates Clark (C), Jones (J), and Smith (S). Determine the winner using the pairwise comparison method and state whether the winner
Received a majority of votes.

A)Jones; no
B)Jones; yes
C)Smith; yes
D)Smith; no

Answer 2

D

Question 3

Solve the problem.
Computer Specialists is planning a group vacation to one of the following locations: Alaska (A), Florida (F), San Antonio (S), or Hawaii (H). The employees rank the four possible sites according to
The following voter profile. Determine the winner using the Borda count method and state whether the winner received a
Majority of votes.

A)Hawaii; no
B)Florida; no
C)Alaska; no
D)San Antonio; no

Answer 3

B

Question 4

Eight voters are asked to rank 3 brands of automobiles: A, B, and C. The eight voters turn in the following ballots showing their preferences in order:

Question 5

Solve the problem.
The Mathematics Department is holding an election for department chair. Each member ranks the candidates from first to third. The voter profile below shows the results of the ballots with
Candidates Clark (C), Jones (J), and Smith (S). Determine the winner using the Hare method and state whether the winner received a majority of
Votes.

A)Smith, yes
B)Jones; no
C)Smith; no
D)Jones; yes

Question 6

Solve the problem.
Four members are running for president of the Greater New Orleans Stamp Club: Andy (a), Brett (b), Carol (c), and Denise (d). The voter profile is summarized in the table. Use the Hare Method to determine the winner.

A)Carol
B)Andy
C)Denise
D)Brett

Question 7

Solve the problem.
Four members are running for president of the Greater New Orleans Stamp Club: Andy (a), Brett (b), Carol (c), and Denise (d). The voter profile is summarized in the table. Use the Plurality Method to determine the winner.

A)Andy
B)Carol
C)Brett
D)Denise

Question 8

Question 9

Solve the problem.
A condominium association is holding an election for president of the board of directors. Each member ranks the candidates from first to third. The voter profile below shows the results of the
Ballots with candidates Abbott(A), Blake (B), Cleary (C), and Downs (D). Determine the winner using the Borda count method and state whether the winner received a
Majority of votes.

A)Downs; no
B)Abbott; no
C)Blake; no
D)Cleary; no

Question 10

Computer Specialists is planning a group vacation to one of the following locations: Alaska (A), Florida (F), San Antonio (S), or Hawaii (H). The employees rank the four possible sites according to
The following voter profile.

Question 11

Eight voters are asked to rank 4 brands of ice cream: A, B, C, and D. The eight voters turn in the following ballots showing their preferences in order:

Question 12

Solve the problem.
Four members are running for president of the Greater New Orleans Stamp Club: Andy (a), Brett (b), Carol (c), and Denise (d). The voter profile is summarized in the table. Use the Pairwise Comparison Method to determine the winner.

A)Denise
B)Andy
C)Brett
D)Carol

Question 13

Question 14

Solve the problem.
Computer Specialists is planning a group vacation to one of the following locations: Alaska (A), Florida (F), San Antonio (S), or Hawaii (H). The employees rank the four possible sites according to
The following voter profile. Determine the winner using the Hare method and state whether the winner received a majority of
Votes.

A)Hawaii; yes
B)Alaska; yes
C)Alaska; no
D)Hawaii; no

Question 15

Solve the problem.
The Mathematics Department is holding an election for department chair. Each member ranks the candidates from first to third. The voter profile below shows the results of the ballots with
Candidates Clark (C), Jones (J), and Smith (S). Determine the winner using the Borda count method and state whether the winner received a
Majority of votes.

A)Smith; yes
B)Jones; no
C)Smith; no
D)Jones; yes

Question 16

Ten voters are asked to rank 4 brands of cell phones: A, B, C, and D. The ten voters turn in the following ballots showing their preferences in order:

Question 17

Solve the problem.
The Mathematics Department is holding an election for department chair. Each member ranks the candidates from first to third. The voter profile below shows the results of the ballots with
Candidates Clark (C), Jones (J), and Smith (S). Determine the winner using the plurality method and state whether the winner received a majority
Of votes.

A)Clark; yes
B)Jones; no
C)Smith; yes
D)Clark; no

Question 18

Solve the problem.
Computer Specialists is planning a group vacation to one of the following locations: Alaska (A), Florida (F), San Antonio (S), or Hawaii (H). The employees rank the four possible sites according to
The following voter profile. Determine the winner using the plurality method and state whether the winner received a majority
Of votes.

A)Hawaii; no
B)Alaska; no
C)Hawaii; yes
D)Alaska; yes

Question 19

Ten voters are asked to rank 3 brands of shoes: A, B, and C. The ten voters turn in the following ballots showing their preferences in order:

Question 20

Solve the problem.
A condominium association is holding an election for president of the board of directors. Each member ranks the candidates from first to third. The voter profile below shows the results of the
Ballots with candidates Abbott(A), Blake (B), Cleary (C), and Downs (D). Determine the winner using the plurality method and state whether the winner received a majority
Of votes.

A)Abbott; no
B)Cleary; no
C)Abbott; yes
D)Cleary; yes

Question 21

Question 22

Given the following voter profile, determine which candidate (if any)is a Condorcet Candidate and who will be selected using the Borda Method. Does the Borda Method violate the Condorcet
Criterion for the voter profile?

Question 23

Given the following voter profile, determine who has the majority of first place votes and then use the Borda Method to determine the outcome. Does the Borda method violate the majority criterion
For the voter profile?

Question 24

Given the following voter profile, determine which candidate (if any)is a Condorcet Candidate and who will be selected using the Plurality Method. Does the Plurality Method violate the Condorcet
Criterion for the voter profile?

Question 25

For the voter profile given below, who (or what)is selected if a second runoff election is held between the two candidates with the most first place votes?

Question 26

Given the following voter profile, determine which candidate (if any)is a Condorcet Candidate and who will be selected using the Borda Method. Does the Borda Method violate the Condorcet
Criterion for the voter profile?

Question 27

Suppose 8 candidates a, b, c, d, e, f, g, h are involved in a Pairwise Comparison Method election. The number of pairwise comparisons each candidate wins is shown in the table. How many comparisons does candidate c win?

A)5
B)4
C)3
D)6

Question 28

Given the following voter profile, determine which candidate (if any)is a Condorcet Candidate and who will be selected using the Plurality Method. Does the Plurality Method violate the Condorcet
Criterion for the voter profile?

Question 29

Given the following voter profile, determine who has the majority of first place votes and then use the Borda Method to determine the outcome. Does the Borda method violate the majority criterion
For the voter profile?

Question 30

For the voter profile given below, who (or what)is selected if a runoff election is held between the candidates that rank second and third; and the winner of that election faces the candidate with the
Most first place votes, to decide the final outcome of the election?

Question 31

Given the following voter profile, determine which candidate (if any)is a Condorcet Candidate and who will be selected using the Hare Method. Does the Hare Method violate the Condorcet criterion
For the voter profile?

Question 32

How many different pairwise comparisons are needed to learn the outcome of an election involving n = 5 candidates?

A)30
B)40
C)10
D)20

Question 33

Suppose 8 candidates a, b, c, d, e, f, g, h are involved in a Pairwise Comparison Method election. The number of pairwise comparisons each candidate wins is shown in the table. Which candidate wins the election and how many pairwise points does the winner have?

A)c, with 7 pairwise points
B)c, with 5 pairwise points
C)c, with 8 pairwise points
D)c, with 6 pairwise points

Question 34

Given the following voter profile, determine who has the majority of first place votes and then use the Borda Method to determine the outcome. Does the Borda method violate the majority criterion
For the voter profile?

Question 35

Question 36

Twenty-five voters are using the Borda method to select one of five candidates: a, b, c, d, or e. If Candidate a receives 32 Borda points, Candidate b receives 58 Borda points, Candidate c receives 46
Borda points, and Candidate d receives 55 Borda points, how many Borda points does Candidate e
Receive? Who wins the Borda election?

A)48; b
B)55; b
C)184; e
D)59; e

Question 37

Given the following voter profile, determine who has the majority of first place votes and then use the Borda Method to determine the outcome. Does the Borda method violate the majority criterion
For the voter profile?

Question 38

Eighteen voters are using the Borda method to select one of four candidates: a, b, c, or d. If Candidate a receives 35 Borda points, Candidate b receives 28 Borda points, and Candidate c
Receives 20 Borda points, how many Borda points does Candidate d receive? Who wins the Borda
Election?

A)37; d
B)97; d
C)28; a
D)25; a

Question 39

Given the following voter profile, determine which candidate (if any)is a Condorcet Candidate and who will be selected using the Hare Method. Does the Hare Method violate the Condorcet criterion
For the voter profile?

Question 40

For the voter profile given below, who (or what)is selected if a runoff election is held between the candidates that rank second and third; and the winner of that election faces the candidate with the
Most first place votes, to decide the final outcome of the election?

Question 41

The Mathematics Department is holding an election for department chair. Each member ranks the candidates from first to third. The voter profile below shows the results of the preliminary
Nonbinding ballots with candidates Clark (C), Jones (J), and Smith (S).

Question 42

$\begin{array} { c | l } \text { Number of Voters } & \text { Ranking } \\\hline 16 & b > c > a \\8 & c > b > a \\6 & c > a > b\end{array}$

-Given that there are 30 voters and 4 candidates, complete the following voter profile so that the Borda Method violates the Majority Criterion.
$\begin{array} { c | l } \text { Number of Voters } & \text { Ranking } \\\hline 16 & \mathrm {~b} > \mathrm { d } > \mathrm { c } > \mathrm { a } \\? & \mathrm { c } > \mathrm { d } > \mathrm { b } > \mathrm { a } \\? & \mathrm { c } > \mathrm { d } > \mathrm { a } > \mathrm { b }\end{array}$

Question 43

The preference table shows the results of an election among three candidates, A, B, and C. (a)Using the plurality method, who is the winner? (b)The voters in the two columns on the right move their last-place candidates from last place to
First place. Construct a new preference table for the election. Using the table and the plurality
Method, who is the winner?
(c)Suppose that candidate C drops out of the new table, but the winner is still chosen by the
Plurality method. Is the irrelevant alternatives criterion satisfied?

A)A; A; no
B)A; A; yes
C)B; A; no
D)A; B; yes

Question 44

In the following voter profile, use the Borda Method to determine the winner. If candidate d drops out, and a second election is held, who is the winner if the Borda Method is again used? Does the
Borda Method violate the Independence of Irrelevant Alternatives Criterion in this election
Process?

Question 45

Solve the problem.
West End school district is purchasing 50 new computers to be distributed among the five schools in the district. The computers will be apportioned based on the school population of each school as
Given in the table below. Use the Jefferson Method to apportion the computers.

Question 46

Solve the problem.
A country has five states with populations as given in the table below and needs to apportion 250 seats in the legislature. Use the Webster Method to apportion the seats.

Question 47

Find the quantity requested. Round to the nearest whole number.
A total of 156 seats were apportioned to 19 states in a certain country which has a total population of 8,977,808. Find the average number of people represented per seat in this country.

A)57,550
B)472,516
C)3029
D)1,093,451

Question 48

In the following voter profile, use the Pairwise Comparison Method to determine the winner. If candidate c drops out, and a second election is held, who is the winner if the Pairwise Comparison
Method is again used? Does the Pairwise Comparison Method violate the Independence of
Irrelevant Alternatives Criterion in this election process?

Question 49

Find the quantity requested. Round to the nearest whole number.
A total of 280 seats were apportioned to 43 states in a certain country which has a total population of 24,853,749. Find the average number of seats assigned per state.

A)577,994
B)9
C)7
D)88,763

Question 50

Construct a voter profile for 41 voters and 4 candidates that has a Condorcet Candidate
that fails to be elected by both the Borda Method and the Hare Method.

Question 51

Computer Specialists is planning a group vacation to one of the following locations: Alaska (A), Florida (F), San Antonio (S), or Hawaii (H). The employees rank the four possible sites according to
The following voter profile:

Question 52

Solve the problem.
A country has five states with populations as given in the table below and needs to apportion 250 seats in the legislature. Use the Jefferson Method to apportion the seats.

Question 53

Solve the problem.
West End school district is purchasing 50 new computers to be distributed among the five schools in the district. The computers will be apportioned based on the school population of each school as
Given in the table below. Use the Hamilton Method to apportion the computers.

Question 54

Suppose that one of the losing candidates is to be deleted from the given voter profile. Which choices would result in the plurality method violating the independence of irrelevant alternatives
Criterion in a second election?

Question 55

Construct a voter profile for 41 voters and 4 candidates that has a Condorcet Candidate
that fails to be elected by both the Borda Method and the Plurality Method.

Question 56

Given that there are 30 voters and 3 candidates, complete the following voter profile so
that the Borda Method violates the Majority Criterion.

Question 57

Construct a voter profile of 41 voters and 5 candidates where the Majority Candidate wins
4 pairwise comparison points and another candidate wins 3 pairwise points.

Question 58

Computer Specialists is planning a group vacation to one of the following locations: Alaska (A), Florida (F), San Antonio (S), or Hawaii (H). The employees initially rank the four possible sites
According to the following voter profile:

Question 59

Solve the problem.
A country has five states with populations as given in the table below and needs to apportion 250 seats in the legislature. Use the Hamilton Method to apportion the seats.

Question 60

In the following voter profile, use the Hare Method to determine the winner. If candidate d drops out, and a second election is held, who is the winner if the Hare Method is again used? Does the
Hare Method violate the Independence of Irrelevant Alternatives Criterion in this election process?

Question 61

Solve the problem.

Question 62

If the Jefferson Method is used to apportion 200 legislative seats to four states with the populations given in the table, then a violation of the Quota Rule occurs. The apportionment for which state violates the Quota Rule?

A)e
B)d
C)b
D)a

Question 63

If the Hamilton Method is used to apportion legislative seats to two original states and a new state with the populations given in the table (in thousands), then one of the states loses a seat to another
If the original number of seats being apportioned is 48. Which state loses a seat to which state, and does the New States paradox occur?

A)b loses a seat to a, yes
B)b loses a seat to a, no
C)a loses a seat to b, no
D)a loses a seat to b, yes

Question 64

Solve the problem.

Question 65

If the Hamilton Method is used to apportion legislative seats to three states with the initial and revised populations given in the table (in thousands), then one of the states loses a seat to another if
The number of seats being apportioned is 13. Which state loses a seat to which state, and does the Population Paradox occur?

A)a loses a seat to c, no
B)b loses a seat to c, yes
C)c loses a seat to a, no
D)c loses a seat to b, yes

Question 66

Solve the problem.
West End school district is purchasing 50 new computers to be distributed among the five schools in the district. The computers will be apportioned based on the school population of each school as
Given in the table below. Use the Webster Method to apportion the computers.

Question 67

If the Hamilton Method is used to apportion legislative seats to four states with the populations given in the table, then the Alabama Paradox occurs when the number of seats being apportioned
Increases from 204 to 205. In which state does the paradox show up?

A)c
B)d
C)b
D)a

Question 68

Solve the problem.

Question 69

Create a population profile for 5 states with a total population of 8,000,000 for which the
Hamilton Method and Webster Method apportionments of 250 legislative seats are the
same, and the Jefferson Method apportionment is different.

Question 70

If the Hamilton Method is used to apportion legislative seats to five states with the populations given in the table, then the Alabama Paradox occurs when the number of seats being apportioned
Increases from 126 to 127. In which state does the paradox show up?

A)d
B)c
C)b
D)a

Question 71

Solve the problem.

Question 72

If the Hamilton Method is used to apportion legislative seats to three states with the initial and revised populations given in the table (in thousands), then one of the states loses a seat to another if
The number of seats being apportioned is 11. Which state loses a seat to which state, and does the Population Paradox occur?

A)c loses a seat to a, no
B)c loses a seat to a, yes
C)c loses a seat to b, yes
D)c loses a seat to b, no

Question 73

If the Hamilton Method is used to apportion legislative seats to two original states and a new state with the populations given in the table (in thousands), then one of the states loses a seat to another
If the original number of seats being apportioned is 100. Which state loses a seat to which state, and does the New States paradox occur?

A)b loses a seat to a, no
B)a loses a seat to b, yes
C)a loses a seat to b, no
D)b loses a seat to a, yes

Question 74

Create a population profile for 5 states with a total population of 950 for which the
Hamilton Method, Jefferson Method, and Webster Method apportionments of 16
legislative seats are all different.

Question 75

If the Jefferson Method is used to apportion 250 legislative seats to five states with the populations given in the table, then a violation of the Quota Rule occurs. The apportionment for which state violates the Quota Rule?

A)b
B)e
C)c
D)a

Question 76

If the Hamilton Method is used to apportion legislative seats to three states with the initial and revised populations given in the table, then one of the states loses a seat to another if the number of
Seats being apportioned is 100. Which state loses a seat to which state, and does the Population Paradox occur?

A)a loses a seat to b, yes
B)b loses a seat to a, no
C)b loses a seat to a, yes
D)a loses a seat to b, no

Question 77

If the Jefferson Method is used to apportion 132 legislative seats to four states with the populations given in the table, then a violation of the Quota Rule occurs. The apportionment for which state violates the Quota Rule?

A)c
B)b
C)a
D)d

Question 78

Solve the problem.
Create a population profile for 5 states with a total population of 900 for which the
Hamilton Method, Jefferson Method, and Webster Method apportionments of 50
legislative seats all agree.

Question 79

If the Jefferson Method is used to apportion 131 legislative seats to six states with the populations given in the table, then a violation of the Quota Rule occurs. The apportionment for which state violates the Quota Rule?

A)e
B)b
C)d
D)f

Question 80

If the Hamilton Method is used to apportion legislative seats to three states with the populations given in the table, then the Alabama Paradox occurs when the number of seats being apportioned
Increases from 150 to 151. In which state does the paradox show up?

A)a
B)c
C)b