Deck 9: Social Choice: the Impossible Dream

Question 1

Use the following information to answer Questions
There are 18 delegates to a political party's convention at which four people-A, B, C, and D- have been nominated as the party's candidate for governor. The delegates' preference schedule is shown below.
What nominee would be elected if the party uses a rank system that assigns 5, 4, 1, and 0 point(s) for a first, second, third, and fourth choice, respectively? Is this result different from that which results from a Borda count?

Answer 1

With the given rank system, Candidate B wins. Under a Borda count, Candidate B also wins.

Question 2

Use the following information to answer Questions
Seventeen board members vote on four candidates, A, B, C, or D, for a new position on their board. Their preference schedules are shown below.
Which candidate will be selected if they use the Hare system?

Answer 2

Candidate D wins.

Question 3

In order to choose which type of music to listen to in a student center music lounge, a Condorcet vote is held by the 15 students present. Below are the preference schedules for the students. Is there a Condorcet winner and, if so, which is it?

Answer 3

Rock is the Condorcet winner.
Use the preference schedule of 23 voters for three candidates (A, B, and C) shown below to answer Questions 9-13.
Number of voters
8
5
6
4
First choice
A
C
B
B
Second choice
C
A
C
A
Third choice
B
B
A
C

Question 4

Use the following information to answer Questions
Seventeen board members vote on four candidates, A, B, C, or D, for a new position on their board. Their preference schedules are shown below.
Using the Hare system according to the preference schedules shown, what happens if candidate A rejects the offer before the ranking?

Question 5

If a rank method is used, which candidate, if any, wins in a straight plurality election?

Question 6

Explain the Condorcet Winner Criterion.

Question 7

Which candidate wins if an election is held between A and C and the winner of that race runs against B?

Question 8

Explain when a voting system is subject to manipulability.

Question 9

Explain how the plurality runoff voting system might not choose a winner.

Question 10

Explain why majority rule is not a good way to choose among four alternatives.

Question 11

If a Borda count is used that assigns 3 points for a first-place vote, 2 points for a second-place vote, and 1 point for a third-place vote, who wins the election?

Question 12

Explain how a candidate is chosen to be a winner using Condorcet's method.

Question 13

Explain May's Theorem.

Question 14

A 17-member committee must elect one of four candidates: R, S, T, or W. Their preference schedule is shown below. Which candidate wins under pairwise sequential voting with the agenda S, T, W, R?

Question 15

Can the four voters in the last column vote strategically to change the outcome of Question 12 to one they would like better? Why or why not?

Question 16

Use the following information to answer Questions
There are 18 delegates to a political party's convention at which four people-A, B, C, and D- have been nominated as the party's candidate for governor. The delegates' preference schedule is shown below.
What nominee would be elected if the party uses a Borda count?

Question 17

Use the following information to answer Questions
There are 18 delegates to a political party's convention at which four people-A, B, C, and D- have been nominated as the party's candidate for governor. The delegates' preference schedule is shown below.
Who wins using the plurality method? Do the results change if the plurality runoff method is used?

Question 18

Use the following information to answer Questions
An 11-member committee must choose one of the four applicants-K, L, M, and N-for membership on the committee. The committee members have preferences among the applicants as given below.
Who wins under the plurality method? Who wins under the plurality runoff method?

Question 19

Which candidate, if any, wins in a majority rule election?

Question 20

Arrow's Impossibility Theorem states that any voting system can give undesirable outcomes. Explain what this means.

Question 21

After their star pitcher moved to another town, the eight remaining members of the company baseball team needed to select a new pitcher. They used approval voting on the four prospects, and the results are listed below. An "X" indicates an approval vote. Which pitcher is chosen if just one is to be selected?

Question 22

Which candidate will be chosen if sequential pairwise voting with agenda B, C, D, A is used?

Question 23

Consider the following preference table: Which candidate, if any, is the Condorcet winner?

Question 24

Majority rule is a good way to choose between two alternatives.

Question 25

Use the following preference table to answer Questions
Which candidate wins using Condorcet's system?

Question 26

The Borda count method of voting satisfies the independence of irrelevant alternatives criterion.

Question 27

Plurality voting fails to satisfy the Condorcet winner criterion (CWC).

Question 28

Sequential pairwise voting satisfies the Pareto condition.

Question 29

Use the following information to answer Questions
An 11-member committee must choose one of the four applicants-K, L, M, and N-for membership on the committee. The committee members have preferences among the applicants as given below.
Which applicant will be given the position if the members use the Borda count to choose the new member?

Question 30

Which candidate will be chosen if plurality voting is used?

Question 31

For a given set of voters' preference lists, different voting procedures may produce different winners.

Question 32

Consider the following preference table: Which candidate will be chosen if majority rule is used?

Question 33

Use the following preference table to answer Questions
Which candidate wins using plurality voting?

Question 34

Which candidate will be chosen if Borda count voting is used?

Question 35

Use the following preference table to answer Questions
Which candidate wins using Borda count voting?

Question 36

Sequential pairwise voting satisfies the Condorcet criterion.

Question 37

Explain why approval voting is not a perfect voting system.

Question 38

Use the following preference table to answer Questions
Which candidate wins using sequential pairwise voting with agenda C, A, B, D?

Question 39

In how many ways can you rank three candidates if ties are allowed?

Question 40

Consider the following preference table: Which candidate will be chosen if the Borda count is used?

Question 41

A group of 12 students have to decide among three types of pizza: sausage (S), sushroom (M), and seef (B). Their preference rankings are shown below. Which choice will the group make if they use sequential pairwise voting with agenda B, M, S? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of students }$
$\begin{array} { l l l l l l } & 3 & 3 & 2 & 2 & 2 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } & \mathrm { S } \\\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } \\\text { Third choice } & \mathrm { S } & \mathrm { S } & \mathrm { B } & \mathrm { M } & \mathrm { M }\end{array}$

A) S
B) M
C) B
D) No winner can be chosen.

Question 42

Use the following information to answer Questions
Fifty voters who elect one of the five candidates
-A, B, C, D, or E-have the preference schedule shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c c } & 20 & 14 & 10 & 6 \\\text { First choice } & \mathrm { A } & \mathrm { B } & \mathrm { B } & \mathrm { C } \\\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { A } & \mathrm { D } \\\text { Third choice } & \mathrm { E } & \mathrm { D } & \mathrm { C } & \mathrm { B } \\\text { Fourth choice } & \mathrm { B } & \mathrm { C } & \mathrm { D } & \mathrm { A } \\\text { Fifth choice } & \mathrm { D } & \mathrm { E } & \mathrm { E } & \mathrm { E }\end{array}$

-Which candidate will be elected using the plurality-runoff voting method?

A) A
B) B
C) C
D) No winner is determined.

Question 43

A group of 12 students have to decide among three types of pizza: sausage (S), mushroom (M), and beef (B). Their preference rankings are shown below. Which choice will the group make if they use the Borda count? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of students }$
$\begin{array} { l l l l l l } & 3 & 3 & 2 & 2 & 2 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } & \mathrm { S } \\\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } \\\text { Third choice } & \mathrm { S } & \mathrm { S } & \mathrm { B } & \mathrm { M } & \mathrm { M }\end{array}$

A) S
B) M
C) B
D) No winner can be chosen.

Question 44

A group of 12 students have to decide among three types of pizza: sausage (S), mushroom (M), and beef (B). Their preference rankings are shown below. What is the Borda score of the beef topping? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of students }$
$\begin{array} { l l l l l l } & 3 & 3 & 2 & 2 & 2 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } & \mathrm { S } \\\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } \\\text { Third choice } & \mathrm { S } & \mathrm { S } & \mathrm { B } & \mathrm { M } & \mathrm { M }\end{array}$

A) 6
B) 10
C) 13
D) 15

Question 45

A group of 12 students have to decide among three types of pizza: sausage (S), mushroom (M), and beef (B). Their preference rankings are shown below. Which choice will the group make if they use plurality voting? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of students }$
$\begin{array} { l l l l l l } & 3 & 3 & 2 & 2 & 2 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } & \mathrm { S } \\\text { Second choice } & \mathrm { M } & \mathrm { S } & \mathrm { M } & \mathrm { S } & \mathrm { B } \\\text { Third choice } & \mathrm { S } & \mathrm { B } & \mathrm { B } & \mathrm { M } & \mathrm { M }\end{array}$

A) S
B) M
C) B
D) No winner can be chosen.

Question 46

Use the following information to answer Questions
Fifty voters who elect one of the five candidates
-A, B, C, D, or E-have the preference schedule shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c c } & 20 & 14 & 10 & 6 \\\text { First choice } & \mathrm { A } & \mathrm { B } & \mathrm { B } & \mathrm { C } \\\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { A } & \mathrm { D } \\\text { Third choice } & \mathrm { E } & \mathrm { D } & \mathrm { C } & \mathrm { B } \\\text { Fourth choice } & \mathrm { B } & \mathrm { C } & \mathrm { D } & \mathrm { A } \\\text { Fifth choice } & \mathrm { D } & \mathrm { E } & \mathrm { E } & \mathrm { E }\end{array}$

-Which candidate will be elected using sequential pairwise voting with the agenda A, B, C, D, E?

A) A
B) B
C) C
D) No winner is determined.

Question 47

Use the following information to answer Questions
Fifty voters who elect one of the five candidates
-A, B, C, D, or E-have the preference schedule shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c c } & 20 & 14 & 10 & 6 \\\text { First choice } & \mathrm { A } & \mathrm { B } & \mathrm { B } & \mathrm { C } \\\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { A } & \mathrm { D } \\\text { Third choice } & \mathrm { E } & \mathrm { D } & \mathrm { C } & \mathrm { B } \\\text { Fourth choice } & \mathrm { B } & \mathrm { C } & \mathrm { D } & \mathrm { A } \\\text { Fifth choice } & \mathrm { D } & \mathrm { E } & \mathrm { E } & \mathrm { E }\end{array}$

-Which candidate will be elected using the Hare system?

A) A
B) B
C) C
D) No winner is determined.

Question 48

In how many ways can a voter rank five candidates, without allowing ties?

A) 5
B) 32
C) 60
D) 120

Question 49

Use the following information to answer Questions
Fifty voters who elect one of the five candidates
-A, B, C, D, or E-have the preference schedule shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c c } & 20 & 14 & 10 & 6 \\\text { First choice } & \mathrm { A } & \mathrm { B } & \mathrm { B } & \mathrm { C } \\\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { A } & \mathrm { D } \\\text { Third choice } & \mathrm { E } & \mathrm { D } & \mathrm { C } & \mathrm { B } \\\text { Fourth choice } & \mathrm { B } & \mathrm { C } & \mathrm { D } & \mathrm { A } \\\text { Fifth choice } & \mathrm { D } & \mathrm { E } & \mathrm { E } & \mathrm { E }\end{array}$

-Which candidate will be elected using the Borda count?

A) A
B) B
C) C
D) No winner is determined.

Question 50

A group of 12 students have to decide among three types of pizza: sausage (S), mushroom (M), and beef (B). Their preference rankings are shown below. Is there a Condorcet winner among the pizza types? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of students }$
$\begin{array} { l l l l l l } & 3 & 3 & 2 & 2 & 2 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } & \mathrm { S } \\\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } \\\text { Third choice } & \mathrm { S } & \mathrm { S } & \mathrm { B } & \mathrm { M } & \mathrm { M }\end{array}$

A) S
B) M
C) B
D) No winner can be chosen.

Question 51

Use the following information to answer the Questions
Suppose that a nine-member committee needs to elect one of the four alternatives-A, B, C, or D. Their preference schedule is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of members }$
$\begin{array} { l c c c } & 4 & 3 & 2 \\\text { First choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } \\\text { Second choice } & \mathrm { B } & \mathrm { D } & \mathrm { D } \\\text { Third choice } & \mathrm { C } & \mathrm { A } & \mathrm { A } \\\text { Fourth choice } & \mathrm { D } & \mathrm { C } & \mathrm { B }\end{array}$

-What is the Borda score of the alternative A?

A) 5
B) 10
C) 12
D) 17

Question 52

Use the following information to answer Questions
Fifty voters who elect one of the five candidates
-A, B, C, D, or E-have the preference schedule shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c c } & 20 & 14 & 10 & 6 \\\text { First choice } & \mathrm { A } & \mathrm { B } & \mathrm { B } & \mathrm { C } \\\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { A } & \mathrm { D } \\\text { Third choice } & \mathrm { E } & \mathrm { D } & \mathrm { C } & \mathrm { B } \\\text { Fourth choice } & \mathrm { B } & \mathrm { C } & \mathrm { D } & \mathrm { A } \\\text { Fifth choice } & \mathrm { D } & \mathrm { E } & \mathrm { E } & \mathrm { E }\end{array}$

-Which candidate will be elected using plurality voting?

A) A
B) B
C) C
D) No winner is determined.

Question 53

If there are an odd number of voters, there is always a Condorcet winner.

Question 54

Use the following information to answer the Questions
Thirty board members must vote on five candidates: X, Y, Z, U, and V. Their preference rankings are summarized in the table below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of students }$
$\begin{array} { l c c c } & 12 & 10 & 8 \\\text { First choice } & \mathrm { X } & \mathrm { Y } & \mathrm { Z } \\\text { Second choice } & \mathrm { U } & \mathrm { Z } & \mathrm { U } \\\text { Third choice } & \mathrm { Y } & \mathrm { X } & \mathrm { X } \\\text { Fourth choice } & \mathrm { Z } & \mathrm { U } & \mathrm { V } \\\text { Fifth choice } & \mathrm { V } & \mathrm { V } & \mathrm { Y }\end{array}$

-Find the winner using the Borda count.

A) X
B) Y
C) Z
D) No winner is chosen.

Question 55

A group of 12 students have to decide among three types of pizza: sausage (S), mushroom (M), and beef (B). Their preference rankings are shown below. Which choice will the group make if they use the Hare system? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of students }$
$\begin{array} { l l l l l l } & 3 & 3 & 2 & 2 & 2 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } & \mathrm { S } \\\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } \\\text { Third choice } & \mathrm { S } & \mathrm { S } & \mathrm { B } & \mathrm { M } & \mathrm { M }\end{array}$

A) S
B) M
C) B
D) No winner can be chosen.

Question 56

A group of 12 students have to decide among three types of pizza: sausage (S), mushroom (M), and beef (B). Their preference rankings are shown below. Which choice will the group make if they use the plurality-runoff voting method? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of students }$
$\begin{array} { l l l l l l } & 3 & 3 & 2 & 2 & 2 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { S } & \mathrm { B } & \mathrm { S } \\\text { Second choice } & \mathrm { M } & \mathrm { S } & \mathrm { M } & \mathrm { S } & \mathrm { B } \\\text { Third choice } & \mathrm { S } & \mathrm { B } & \mathrm { B } & \mathrm { M } & \mathrm { M }\end{array}$

A) S
B) M
C) B
D) No winner can be chosen.

Question 57

How many votes are needed for a majority winner if there are 25 voters?

A) 12
B) 12.5
C) 13
D) 25

Question 58

Use the following information to answer the Questions
Thirty board members must vote on five candidates: X, Y, Z, U, and V. Their preference rankings are summarized in the table below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of students }$
$\begin{array} { l c c c } & 12 & 10 & 8 \\\text { First choice } & \mathrm { X } & \mathrm { Y } & \mathrm { Z } \\\text { Second choice } & \mathrm { U } & \mathrm { Z } & \mathrm { U } \\\text { Third choice } & \mathrm { Y } & \mathrm { X } & \mathrm { X } \\\text { Fourth choice } & \mathrm { Z } & \mathrm { U } & \mathrm { V } \\\text { Fifth choice } & \mathrm { V } & \mathrm { V } & \mathrm { Y }\end{array}$

-Find the winner using sequential pairwise voting with the agenda X, Y, Z, U, V.

A) X
B) Z
C) U
D) V

Question 59

Use the following information to answer the Questions
Thirty board members must vote on five candidates: X, Y, Z, U, and V. Their preference rankings are summarized in the table below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of students }$
$\begin{array} { l c c c } & 12 & 10 & 8 \\\text { First choice } & \mathrm { X } & \mathrm { Y } & \mathrm { Z } \\\text { Second choice } & \mathrm { U } & \mathrm { Z } & \mathrm { U } \\\text { Third choice } & \mathrm { Y } & \mathrm { X } & \mathrm { X } \\\text { Fourth choice } & \mathrm { Z } & \mathrm { U } & \mathrm { V } \\\text { Fifth choice } & \mathrm { V } & \mathrm { V } & \mathrm { Y }\end{array}$

-Find the winner using the Hare system.

A) X
B) Y
C) Z
D) No winner is chosen.

Question 60

If the winner of an election is not affected by moving that winner higher on some voting ballets with no other changes, the voting system is satisfying:

A) Condorcet winner criterion.
B) monotonicity.
C) Pareto condition.
D) Borda condition.

Question 61

Twenty-nine voters must choose from among three alternatives: A, B, and C. The voters' preference schedules are shown below. Which alternative wins using the Hare system? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters}$
$\begin{array} { l c c c c } & 12 & 8 & 6 & 3 \\\text { First choice } & \mathrm { B } & \mathrm { C } & \mathrm { A } & \mathrm { C } \\\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { B } & \mathrm { B } \\\text { Third choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } & \mathrm { A }\end{array}$

A) A
B) B
C) C
D) No winner is determined.

Question 62

Use the following information to answer the Questions
One hundred voters are to elect one of the four candidates: A, B, C, or D. Their preference schedule is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters}$
$\begin{array} { l c c c c } & 40 & 32 & 18 & 10 \\\text { First choice } & \text { A } & \text { B } & \text { D } & \text { C } \\\text { Second choice } & \text { C } & \text { C } & \text { C } & \text { D } \\\text { Third choice } & \text { B } & \text { A } & \text { B } & \text { A } \\\text { Fourth choice } & \text { D } & \text { D } & \text { A } & \text { B }\end{array}$

-Which candidate wins using sequential pairwise voting with the agenda A, C, B, D?

A) A wins.
B) B wins.
C) C wins.
D) D wins.

Question 63

Use the following information to answer the Questions
A group of 22 young people must decide whether to go to the beach (B), the mountain (M), or the zoo (Z) on a field trip. Their preference rankings are summarized in the table below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters}$
$\begin{array} { l c c c } & 10 & 8 & 4 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { Z } \\\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } \\\text { Third choice } & \mathrm { Z } & \mathrm { Z } & \mathrm { B }\end{array}$

-Which choice wins using plurality voting?

A) M
B) B
C) Z
D) No winner is determined.

Question 64

Use the following information to answer the Questions
Eight board members vote by approval voting on four candidates-A, B, C, and D-for new positions on their board as indicated in the following table. An X indicates an approval vote. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Voters }$
$\begin{array} { l l l l l l l l l } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\mathrm {A} & \mathrm { X } & \mathrm { X } & \mathrm { X } & \mathrm { X } & & \mathrm { X } & \mathrm { X } & \mathrm { X } \\\mathrm { B } & & \mathrm { X } & \mathrm { X } & & \mathrm { X } & \mathrm { X } & & \mathrm { X } \\\mathrm { C } & & \mathrm { X } & & \mathrm { X } & \mathrm { X } & & \mathrm { X } & \mathrm { X } \\\mathrm { D } & \mathrm { X } & & \mathrm { X } & \mathrm { X } & \mathrm { X } & \mathrm { X } & \mathrm { X } &\end{array}$

-Which candidate(s) is(are) elected if 80% approval is necessary and at most two are elected?

A) No candidates are elected.
B) Only candidate A is elected.
C) Only candidate D is elected.
D) Both candidates A and D are elected.

Question 65

Twenty-nine voters must choose from among three alternatives: A, B, and C. The voters' preference schedules are shown below. C wins in a Borda count. Is the outcome different in a rank system that assigns five points for first choice, two points for second, and one point for third? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters}$
$\begin{array} { l c c c c } & 12 & 8 & 6 & 3 \\\text { First choice } & \mathrm { B } & \mathrm { C } & \mathrm { A } & \mathrm { C } \\\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { B } & \mathrm { B } \\\text { Third choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } & \mathrm { A }\end{array}$

A) No, C still wins.
B) Yes, A now wins.
C) Yes, B now wins.
D) Yes, now there is no winner.

Question 66

Use the following information to answer the Questions
Suppose that a nine-member committee needs to elect one of the four alternatives-A, B, C, or D. Their preference schedule is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of members }$
$\begin{array} { l c c c } & 4 & 3 & 2 \\\text { First choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } \\\text { Second choice } & \mathrm { B } & \mathrm { D } & \mathrm { D } \\\text { Third choice } & \mathrm { C } & \mathrm { A } & \mathrm { A } \\\text { Fourth choice } & \mathrm { D } & \mathrm { C } & \mathrm { B }\end{array}$

-Is there a Condorcet winner?

A) Yes, B wins.
B) Yes, C wins.
C) Yes, D wins.
D) No Condorcet winner is determined.

Question 67

Use the following information to answer the Questions
A group of 22 young people must decide whether to go to the beach (B), the mountain (M), or the zoo (Z) on a field trip. Their preference rankings are summarized in the table below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters}$
$\begin{array} { l c c c } & 10 & 8 & 4 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { Z } \\\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } \\\text { Third choice } & \mathrm { Z } & \mathrm { Z } & \mathrm { B }\end{array}$

-Which choice wins using the Hare system?

A) M
B) B
C) Z
D) No winner is determined.

Question 68

Use the following information to answer the Questions
A group of 22 young people must decide whether to go to the beach (B), the mountain (M), or the zoo (Z) on a field trip. Their preference rankings are summarized in the table below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters}$
$\begin{array} { l c c c } & 10 & 8 & 4 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { Z } \\\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } \\\text { Third choice } & \mathrm { Z } & \mathrm { Z } & \mathrm { B }\end{array}$

-Which choice wins using the plurality-runoff voting method?

A) M
B) B
C) Z
D) No winner is determined.

Question 69

Use the following information to answer the Questions
A group of 22 young people must decide whether to go to the beach (B), the mountain (M), or the zoo (Z) on a field trip. Their preference rankings are summarized in the table below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters}$
$\begin{array} { l c c c } & 10 & 8 & 4 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { Z } \\\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } \\\text { Third choice } & \mathrm { Z } & \mathrm { Z } & \mathrm { B }\end{array}$

-Which choice wins using sequential pairwise voting with the agenda Z, B, M?

A) M
B) B
C) Z
D) No winner is determined.

Question 70

Use the following information to answer the Questions
One hundred voters are to elect one of the four candidates: A, B, C, or D. Their preference schedule is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters}$
$\begin{array} { l c c c c } & 40 & 32 & 18 & 10 \\\text { First choice } & \text { A } & \text { B } & \text { D } & \text { C } \\\text { Second choice } & \text { C } & \text { C } & \text { C } & \text { D } \\\text { Third choice } & \text { B } & \text { A } & \text { B } & \text { A } \\\text { Fourth choice } & \text { D } & \text { D } & \text { A } & \text { B }\end{array}$

-Is there a winner using the plurality-runoff method?

A) Yes, A wins.
B) Yes, B wins.
C) Yes, C wins.
D) No winner is determined.

Question 71

Use the following information to answer the Questions
A group of 22 young people must decide whether to go to the beach (B), the mountain (M), or the zoo (Z) on a field trip. Their preference rankings are summarized in the table below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters}$
$\begin{array} { l c c c } & 10 & 8 & 4 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { Z } \\\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } \\\text { Third choice } & \mathrm { Z } & \mathrm { Z } & \mathrm { B }\end{array}$

-Is there a Condorcet winner?

A) M
B) B
C) Z
D) No winner is determined.

Question 72

Use the following information to answer the Questions
One hundred voters are to elect one of the four candidates: A, B, C, or D. Their preference schedule is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters}$
$\begin{array} { l c c c c } & 40 & 32 & 18 & 10 \\\text { First choice } & \text { A } & \text { B } & \text { D } & \text { C } \\\text { Second choice } & \text { C } & \text { C } & \text { C } & \text { D } \\\text { Third choice } & \text { B } & \text { A } & \text { B } & \text { A } \\\text { Fourth choice } & \text { D } & \text { D } & \text { A } & \text { B }\end{array}$

-Is there a Condorcet winner?

A) Yes, A wins.
B) Yes, B wins.
C) Yes, C wins.
D) No winner is determined.

Question 73

Use the following information to answer the Questions
Eight board members vote by approval voting on four candidates-A, B, C, and D-for new positions on their board as indicated in the following table. An X indicates an approval vote. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Voters }$
$\begin{array} { l l l l l l l l l } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\mathrm {A} & \mathrm { X } & \mathrm { X } & \mathrm { X } & \mathrm { X } & & \mathrm { X } & \mathrm { X } & \mathrm { X } \\\mathrm { B } & & \mathrm { X } & \mathrm { X } & & \mathrm { X } & \mathrm { X } & & \mathrm { X } \\\mathrm { C } & & \mathrm { X } & & \mathrm { X } & \mathrm { X } & & \mathrm { X } & \mathrm { X } \\\mathrm { D } & \mathrm { X } & & \mathrm { X } & \mathrm { X } & \mathrm { X } & \mathrm { X } & \mathrm { X } &\end{array}$

-Which candidate will be chosen for the board if just one of them is to be elected?

A) A
B) B
C) C
D) D

Question 74

Use the following information to answer the Questions
Suppose that a nine-member committee needs to elect one of the four alternatives-A, B, C, or D. Their preference schedule is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of members }$
$\begin{array} { l c c c } & 4 & 3 & 2 \\\text { First choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } \\\text { Second choice } & \mathrm { B } & \mathrm { D } & \mathrm { D } \\\text { Third choice } & \mathrm { C } & \mathrm { A } & \mathrm { A } \\\text { Fourth choice } & \mathrm { D } & \mathrm { C } & \mathrm { B }\end{array}$

-Which alternative wins using the Hare system?

A) A
B) B
C) C
D) D

Question 75

Use the following information to answer the Questions
Suppose that a nine-member committee needs to elect one of the four alternatives-A, B, C, or D. Their preference schedule is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of members }$
$\begin{array} { l c c c } & 4 & 3 & 2 \\\text { First choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } \\\text { Second choice } & \mathrm { B } & \mathrm { D } & \mathrm { D } \\\text { Third choice } & \mathrm { C } & \mathrm { A } & \mathrm { A } \\\text { Fourth choice } & \mathrm { D } & \mathrm { C } & \mathrm { B }\end{array}$

-Which alternative wins using sequential pairwise voting with the agenda A, B, C, D?

A) A
B) B
C) C
D) D

Question 76

Consider a 13-person committee that is considering three applicants-A, B, and C-for an opening. The individual rankings are summarized in the table below. Which applicant would be accepted if the committee used the Borda count? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of members }$
$\begin{array} { l l c c c } & 5 & 4 & 2 & 2 \\\text { First choice } & \mathrm { B } & \mathrm { C } & \mathrm { A } & \mathrm { A } \\\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { B } & \mathrm { C } \\\text { Third choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } & \mathrm { B }\end{array}$

A) A
B) B
C) C
D) There is no Borda count winner.

Question 77

Use the following information to answer Questions
Consider a 13-person committee that is considering three applicants, A, B, and C, for an opening. The individual rankings are summarized in the table below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of members }$
$\begin{array} { l l c c c } & 5 & 4 & 2 & 2 \\\text { First choice } & \mathrm { B } & \mathrm { C } & \mathrm { A } & \mathrm { A } \\\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { B } & \mathrm { C } \\\text { Third choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } & \mathrm { B }\end{array}$

-Which applicant would be accepted if the committee used the plurality-runoff method?

A) A
B) B
C) C
D) There is no plurality-runoff winner.

Question 78

Use the following information to answer the Questions
A group of 22 young people must decide whether to go to the beach (B), the mountain (M), or the zoo (Z) on a field trip. Their preference rankings are summarized in the table below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters}$
$\begin{array} { l c c c } & 10 & 8 & 4 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { Z } \\\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } \\\text { Third choice } & \mathrm { Z } & \mathrm { Z } & \mathrm { B }\end{array}$

-Which choice wins using the Borda count?

A) M
B) B
C) Z
D) No winner is determined.

Question 79

Use the following information to answer Questions
Consider a 13-person committee that is considering three applicants, A, B, and C, for an opening. The individual rankings are summarized in the table below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of members }$
$\begin{array} { l l c c c } & 5 & 4 & 2 & 2 \\\text { First choice } & \mathrm { B } & \mathrm { C } & \mathrm { A } & \mathrm { A } \\\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { B } & \mathrm { C } \\\text { Third choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } & \mathrm { B }\end{array}$

-Which applicant would be accepted if the committee used the plurality method?

A) A
B) B
C) C
D) There is no plurality winner.

Question 80

Use the following information to answer the Questions
One hundred voters are to elect one of the four candidates: A, B, C, or D. Their preference schedule is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters}$
$\begin{array} { l c c c c } & 40 & 32 & 18 & 10 \\\text { First choice } & \text { A } & \text { B } & \text { D } & \text { C } \\\text { Second choice } & \text { C } & \text { C } & \text { C } & \text { D } \\\text { Third choice } & \text { B } & \text { A } & \text { B } & \text { A } \\\text { Fourth choice } & \text { D } & \text { D } & \text { A } & \text { B }\end{array}$

-Which candidate wins using the Hare system?

A) A wins.
B) B wins.
C) D wins.
D) No winner is determined.