Deck 10: The Manipulability of Voting Systems

Question 1

Use the following information to answer Questions
A 17-member committee must elect one of four candidates: R, S, T, or W. See the preference schedule below.
R wins using the plurality method. Could those members who most prefer T vote strategically in some way to change the outcome in a way that will benefit them?

Answer 1

Yes. If these three voters ranked candidate S first, S would win with seven votes. Since the voters prefer S to R, this outcome would be more desirable.

Question 2

Determine whether plurality voting method is manipulable.

Answer 2

Plurality voting cannot be manipulated by a single individual, but it is group manipulable.

Question 3

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 in the table. If the committee uses pairwise sequential voting with the agenda K, L, M, N, applicant K wins. Is it possible that another agenda will yield a different winner?

Answer 3

No. The six voters who most prefer applicant K represent a majority of the committee. No matter how the voters are ordered, K will win.

Question 4

Use the following information to answer Questions
A 17-member committee must elect one of four candidates: R, S, T, or W. See the preference schedule below.
R wins using the plurality method. Could those members who most prefer W vote strategically in some way to change the outcome in a way that will benefit them?

Question 5

According to the Chair's Paradox, what surprising situation can befall the tie-breaking chair of a committee?

Question 6

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 uses the Borda count method. The committee members have preferences among the applicants as given in the table. Suppose the group of six suspects that the group of two intends to insincerely exchange M and L in their rankings. Can the group of six counteract to protect K as the winner?

Question 7

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 uses the Borda count method. The committee members have preferences among the applicants as given in the table. Who wins the election? Can the group of three voters favorably affect the results through insincere voting?

Question 8

Use the following information to answer Questions
A 17-member committee must elect one of four candidates: R, S, T, or W. See the preference schedule below.
In a plurality runoff election, candidate S wins. What would happen if the four voters who prefer W insincerely voted for T instead? Is this in their best interests?

Question 9

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 uses the Borda count method. The committee members have preferences among the applicants as given in the table. Who wins the election? Can the group of two voters favorably affect the results through insincere voting?

Question 10

If a voting system has three or more alternatives, satisfies the Pareto condition, always produces a unique winner, and is not a dictatorship, what conclusion follows from the GS theorem?

Question 11

Explain the difference between sincere and strategic voting.

Question 12

Use the following information to answer Questions
A 17-member committee must elect one of four candidates: R, S, T, or W. See the preference schedule below.
In a plurality runoff election, candidate S wins. What would happen if the four voters who prefer T insincerely voted for S instead? Is this in their best interests?

Question 13

Determine whether the Borda count method is manipulable. Are there situations in which the Borda count method is known to be never manipulable?

Question 14

What is purpose of Copeland's Rule with regard to Condorcet's method of voting?

Question 15

Use the following information to answer Questions
A 17-member committee must elect one of four candidates: R, S, T, or W. See the preference schedule below.
Is it possible to manipulate the results of a sequential pairwise election?

Question 16

What is agenda manipulation?

Question 17

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.
There are 21 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. If the party uses a Borda count, candidate B would be elected. Can the four voters who most prefer candidate C vote strategically in some way to change this outcome to one they would find more favorable? Why or why not?

Question 18

Determine whether Condorcet's method is manipulable.

Question 19

Use the following information to answer Questions
A 17-member committee must elect one of four candidates: R, S, T, or W. See the preference schedule below.
R wins using the plurality method. Could those members who most prefer S vote strategically in some way to change the outcome in a way that will benefit them?

Question 20

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 in the table. If the committee uses pairwise sequential voting with the agenda K, L, M, N, applicant K wins. Can the three voters who least prefer K vote strategically in some way to change the outcome to one they find more favorable? Why or why not?

Question 21

Use the following information to answer Questions
Consider an 11-member committee that must choose one of three alternatives-X, Y, or Z- using the Hare system. Their schedule of preferences is shown below.
Who wins? Is it possible for the group of five voters to change the outcome in a way that would benefit them?

Question 22

A vote using Condorcet's method is:

A) never manipulable.
B) sometimes manipulable.
C) always manipulable.

Question 23

Use the following information to answer Questions .
A group of 22 young people must decide whether to go to the beach (B), the mountains (M), or the zoo (Z) on a field trip. Their preference rankings are summarized in the table below, and the decision will be made using a Borda count.
Who wins the vote? Can the 10 voters in the first column change the results of the vote to their favor by changing their preference rankings?

Question 24

Among all two-candidate voting systems that never result in a tie, majority rule is the only one that treats all voters equally, treats both candidates equally, and is nonmanipulable.

Question 25

A vote among exactly three candidates using the Borda count method is:

A) never manipulable.
B) sometimes manipulable.
C) always manipulable.

Question 26

Use the following preference table to answer Questions .
Who wins using plurality? Could the six voters who most prefer candidate C vote insincerely to change the outcome in a way that would benefit them?

Question 27

Use the following information to answer Questions
Consider an 11-member committee that must choose one of three alternatives-X, Y, or Z- using the Hare system. Their schedule of preferences is shown below.
The committee suspects that the group of five plans to insincerely reorder their preferences as Y, Z, X. How can the group of four respond?

Question 28

A vote using the Borda count method is:

A) never manipulable.
B) sometimes manipulable.
C) always manipulable.

Question 29

Use the following preference table for an election to be decided by sequential pairwise voting to answer Questions .
Is there an agenda for which candidate B wins? If so, give an example.

Question 30

Use the following preference table to answer Questions .
Who wins using plurality? Could the four voters who most prefer candidate D vote insincerely to change the outcome in a way that would benefit them?

Question 31

Use the following information to answer Questions .
A group of 22 young people must decide whether to go to the beach (B), the mountains (M), or the zoo (Z) on a field trip. Their preference rankings are summarized in the table below, and the decision will be made using a Borda count.
Who wins the vote? Can the four voters in the last column change the results of the vote to their favor by changing their preference rankings?

Question 32

Use the following preference table for an election to be decided by sequential pairwise voting to answer Questions .
Is there an agenda for which candidate E wins? If so, give an example.

Question 33

Use the following preference table for an election to be decided by sequential pairwise voting to answer Questions .
Is there an agenda for which candidate C wins? If so, give an example.

Question 34

When there are only two or three candidates, nonmanipulability and monotonicity are exactly the same thing.

Question 35

Use the following information to answer Questions
Twenty-nine voters must choose from among three alternatives-A, B, and C-using the Borda count method. The voters' preference schedules are shown below.
Suppose the group of voters anticipates that the group of six plans to insincerely rank candidate B above candidate A. How can the remaining voters respond in their own rankings?

Question 36

Use the following information to answer Questions
Twenty-nine voters must choose from among three alternatives-A, B, and C-using the Borda count method. The voters' preference schedules are shown below.
Who wins Borda count? Can the group of six voters change their preference list to produce an outcome they like better?

Question 37

Use the following preference table for an election to be decided by sequential pairwise voting to answer Questions .
Is there an agenda for which candidate A wins? If so, give an example.

Question 38

Use the following preference table to answer Questions .
Who wins using plurality? Could the four voters who most prefer candidate B vote insincerely to change the outcome in a way that would benefit them?

Question 39

Use the following information to answer Questions
Consider an 11-member committee that must choose one of three alternatives-X, Y, or Z- using the Hare system. Their schedule of preferences is shown below.
Who wins? Is it possible for the group of two voters to change the outcome in a way that would benefit them?

Question 40

Use the following preference table for an election to be decided by sequential pairwise voting to answer Questions .
Is there an agenda for which candidate D wins? If so, give an example.

Question 41

The Chair's paradox is the fact that the chair, who can break ties, can:

A) always end up with his or her preferred candidate.
B) sometimes end up with his or her preferred candidate.
C) sometimes end up with his or her least-preferred candidate.
D) never end up with his or her least-preferred candidate.

Question 42

Use the following information to answer the Questions:
Twenty-nine voters must choose from among three alternatives: A, B, and C. The voters' preference schedules are shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c c } & 12 & 8 & 6 & 3 \\\text { First choice } & \mathrm { B } & \mathrm { C } & \mathrm { A } & \mathrm { C } \\\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { B } & \mathrm { B } \\\text { Third choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } & \mathrm { A }\end{array}$

-C wins in a Borda count. Can the six voters in the third column change their preference list to produce an outcome they like better?

A) No, C will always win.
B) Yes, they can rank A, C, B.
C) Yes, they can rank B, A, C.
D) Yes, they can rank C, A, B.

Question 43

Use the following information to answer the Questions
Thirty voters with the preference schedules below are to elect a union spokesman from among five candidates: A, B, C, D, and E. If the Borda count is used, candidate B would win. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l l l l } & 12 & 10 & 8 \\\text { First choice } & \text { A } & \text { B } & \text { D } \\\text { Second choice } & \text { D } & \text { C } & \text { B } \\\text { Third choice } & \text { B } & \text { A } & \text { A } \\\text { Fourth choice } & \text { C } & \text { D } & \text { E } \\\text { Fifth choice } & \text { E } & \text { E } & \text { C }\end{array}$

-Would there be any difference in the result if candidate A withdrew from the race before the ranking?

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

Question 44

Use the following information to answer the Questions
Thirty voters with the preference schedules below are to elect a union spokesman from among five candidates: A, B, C, D, and E. If the Borda count is used, candidate B would win. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l l l l } & 12 & 10 & 8 \\\text { First choice } & \text { A } & \text { B } & \text { D } \\\text { Second choice } & \text { D } & \text { C } & \text { B } \\\text { Third choice } & \text { B } & \text { A } & \text { A } \\\text { Fourth choice } & \text { C } & \text { D } & \text { E } \\\text { Fifth choice } & \text { E } & \text { E } & \text { C }\end{array}$

-Would there be any difference in the result if candidate C withdrew from the race before the ranking?

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

Question 45

Which of the following is NOT a property of the Condorcet method of voting?

A) satisfies the Pareto condition
B) nonmanipulable
C) not a dictatorship
D) always produces a winner

Question 46

Use the following information to answer the Questions
Consider an 11-member committee that must choose one of three alternatives: X, Y, or Z. Their schedule of preferences is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$

$\begin{array} { l c c c } & 5 & 4 & 2 \\\text { First choice } & \mathrm { Z } & \mathrm { X } & \mathrm { Y } \\\text { Second choice } & \mathrm { Y } & \mathrm { Y } & \mathrm { X } \\\text { Third choice } & \mathrm { X } & \mathrm { Z } & \mathrm { Z }\end{array}$

-Is there an agenda for which X would win under sequential pairwise voting?

A) yes, X Y Z
B) yes, Y Z X
C) yes, Z Y X
D) no

Question 47

Use the following information to answer the Questions:
Twenty-nine voters must choose from among three alternatives: A, B, and C. The voters' preference schedules are shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c c } & 12 & 8 & 6 & 3 \\\text { First choice } & \mathrm { B } & \mathrm { C } & \mathrm { A } & \mathrm { C } \\\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { B } & \mathrm { B } \\\text { Third choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } & \mathrm { A }\end{array}$

-C wins in a Borda count. Can the 12 voters in the first column change their preference list to produce an outcome they like better?

A) No, C will always win.
B) Yes, they can rank A, C, B.
C) Yes, they can rank B, A, C.
D) Yes, they can rank C, A, B.

Question 48

Use the following information to answer the Questions
Consider an 11-member committee that must choose one of three alternatives: X, Y, or Z. Their schedule of preferences is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$

$\begin{array} { l c c c } & 5 & 4 & 2 \\\text { First choice } & \mathrm { Z } & \mathrm { X } & \mathrm { Y } \\\text { Second choice } & \mathrm { Y } & \mathrm { Y } & \mathrm { X } \\\text { Third choice } & \mathrm { X } & \mathrm { Z } & \mathrm { Z }\end{array}$

-Is there an agenda for which Z would win under sequential pairwise voting?

A) yes, X Y Z
B) yes, Y Z X
C) yes, Z Y X
D) no

Question 49

Which of the following is also known as an "insincere ballot"?

A) a disingenuous ballot
B) a schematic ballot
C) a preference ballot
D) a dispreference ballot

Question 50

Use the following information to answer the Questions:
Twenty-nine voters must choose from among three alternatives: A, B, and C. The voters' preference schedules are shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c c } & 12 & 8 & 6 & 3 \\\text { First choice } & \mathrm { B } & \mathrm { C } & \mathrm { A } & \mathrm { C } \\\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { B } & \mathrm { B } \\\text { Third choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } & \mathrm { A }\end{array}$

-Using the agenda A, B, C, and sequential pairwise voting, alternative B wins. Is there an agenda that produces C as a winner?

A) No, B always wins.
B) Yes, the agenda A, C, B.
C) Yes, the agenda C, B, A.
D) Yes, the agenda B, A, C.

Question 51

The GS theorem states the nonexistence of a voting system with three or more candidates which satisfies a list of certain properties. Which of the following is not on the list of those properties?

A) The voting system has never ties.
B) The voting system satisfies monotonicity.
C)
D) The voting system is nonmanipulable.
The voting system is not a dictatorship.

Question 52

Which of the following method can be used to break ties, which can extend Condorcet's method?

A) the Hare method
B) Condorcet winner criterion
C) Copeland's rule
D) plurality runoff

Question 53

Use the following information to answer the Questions
Thirty voters with the preference schedules below are to elect a union spokesman from among five candidates: A, B, C, D, and E. If the Borda count is used, candidate B would win. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l l l l } & 12 & 10 & 8 \\\text { First choice } & \text { A } & \text { B } & \text { D } \\\text { Second choice } & \text { D } & \text { C } & \text { B } \\\text { Third choice } & \text { B } & \text { A } & \text { A } \\\text { Fourth choice } & \text { C } & \text { D } & \text { E } \\\text { Fifth choice } & \text { E } & \text { E } & \text { C }\end{array}$

-Would there be any difference in the result if candidate D withdrew from the race before the ranking?

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

Question 54

After considering all possible one-on-one contests, Copeland's rule chooses as the winner:

A) the candidate who wins every contest in which he or she participates.
B) the candidate with the best win-loss record.
C) the candidate who wins the longest sequence of contests.
D) the candidate who wins more contests than he or she loses.

Question 55

When the voter with tie-breaking power can end up with his or her least-preferred candidate as the election winner, this phenomena is called:

A) Condorcet's voting paradox.
B) the Simpson's paradox.
C)
D) the chair's paradox.
The tie-breaking paradox.

Question 56

Use the following information to answer the Questions
Consider an 11-member committee that must choose one of three alternatives: X, Y, or Z. Their schedule of preferences is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$

$\begin{array} { l c c c } & 5 & 4 & 2 \\\text { First choice } & \mathrm { Z } & \mathrm { X } & \mathrm { Y } \\\text { Second choice } & \mathrm { Y } & \mathrm { Y } & \mathrm { X } \\\text { Third choice } & \mathrm { X } & \mathrm { Z } & \mathrm { Z }\end{array}$

-If the Hare system is used, alternative X wins. Could the voters who most prefer Y vote insincerely in some way to change the outcome in a way that would benefit them?

A) Yes, they could switch the ranking of X and Y.
B) Yes, they could switch the ranking of Y and Z.
C) Yes, they could switch the ranking of X and Z.
D) No, X would always win.

Question 57

Use the following information to answer the Questions
Consider an 11-member committee that must choose one of three alternatives: X, Y, or Z. Their schedule of preferences is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$

$\begin{array} { l c c c } & 5 & 4 & 2 \\\text { First choice } & \mathrm { Z } & \mathrm { X } & \mathrm { Y } \\\text { Second choice } & \mathrm { Y } & \mathrm { Y } & \mathrm { X } \\\text { Third choice } & \mathrm { X } & \mathrm { Z } & \mathrm { Z }\end{array}$

-If the Hare system is used, alternative X wins. Could the voters who most prefer Z vote insincerely in some way to change the outcome in a way that would benefit them?

A) Yes, they could switch the ranking of X and Y.
B) Yes, they could switch the ranking of Y and Z.
C) Yes, they could switch the ranking of X and Z.
D) No, X would always win.

Question 58

A vote using plurality can be manipulated by:

A) nothing.
B) individuals only.
C) groups only.
D) both individuals and groups.

Question 59

Agenda manipulation is a technique to affect an outcome using:

A) the Hare method.
B) the plurality-runoff method.
C) the Borda count method.
D) the sequential pairwise method.

Question 60

A person who has tie-breaking power:

A) always ends up with his or her preferred candidate.
B) only votes when there is a tie.
C) may determine the winner when there is a tie on the initial ballot.
D) becomes the winning candidate when there is a tie.

Question 61

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, and the decision will be made using a Borda count. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c } & 10 & 8 & 4 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { Z } \\\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } \\\text { Third choice } & Z & Z & \mathrm {~B}\end{array}$

-Can the four voters in the last column change the results of the vote to their favor by changing their preference rankings?

Question 62

Consider the following preference table for two voters: If a Borda count is used, can the voter on the left manipulate the outcome to his or her benefit with no ties for the winner?

A) No
B) Yes

Question 63

Consider the following preference table for two voters: If a Borda count is used, can the voter on the right manipulate the outcome to his or her benefit with no ties for the winner?

A) No
B) Yes

Question 64

Use the following preference table to answer the Questions $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c c } & 4 & 6 & 8 & 4 \\\text { First choice } & \mathrm { D } & \mathrm { C } & \mathrm { A } & \mathrm { B } \\\text { Second choice } & \mathrm { C } & \mathrm { B } & \mathrm { D } & \mathrm { A } \\\text { Third choice } & \mathrm { B } & \mathrm { D } & \mathrm { C } & \mathrm { C } \\\text { Fourth choice } & \mathrm { A } & \mathrm { A } & \mathrm { B } & \mathrm { D }\end{array}$

-Using plurality, A wins. Could the four voters who most prefer D (and prefer A least) vote insincerely to change the outcome in a way that would benefit them?

A) No, they cannot change the outcome.
B) Yes, they could get B to win.
C) Yes, they could get C to win.
D) Yes, they could get D to win.

Question 65

Consider the following preference table for three voters: If a Borda count is used, can the voter on the left manipulate the outcome to his or her benefit?

A) No
B) Yes

Question 66

Consider the following preference table for three voters to answer Questions
If a Borda count is used, can the voter on the right manipulate the outcome to his or her benefit?

A) No
B) Yes

Question 67

Use the following information to answer the Questions:
Twenty-nine voters must choose from among three alternatives: A, B, and C. The voters' preference schedules are shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c c } & 12 & 8 & 6 & 3 \\\text { First choice } & \mathrm { B } & \mathrm { C } & \mathrm { A } & \mathrm { C } \\\text { Second choice } & \mathrm { C } & \mathrm { A } & \mathrm { B } & \mathrm { B } \\\text { Third choice } & \mathrm { A } & \mathrm { B } & \mathrm { C } & \mathrm { A }\end{array}$

-Using the agenda A, B, C, and sequential pairwise voting, alternative B wins. Is there an agenda that produces A as a winner?

A) No, B always wins.
B) Yes, the agenda A, C, B.
C) Yes, the agenda C, B, A.
D) Yes, the agenda B, A, C.

Question 68

Consider the following preference table for two voters: If a Borda count is used, can the voter on the right manipulate the outcome to his or her benefit?

A) No
B) Yes

Question 69

Use the following preference table to answer the Questions $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c c } & 4 & 6 & 8 & 4 \\\text { First choice } & \mathrm { D } & \mathrm { C } & \mathrm { A } & \mathrm { B } \\\text { Second choice } & \mathrm { C } & \mathrm { B } & \mathrm { D } & \mathrm { A } \\\text { Third choice } & \mathrm { B } & \mathrm { D } & \mathrm { C } & \mathrm { C } \\\text { Fourth choice } & \mathrm { A } & \mathrm { A } & \mathrm { B } & \mathrm { D }\end{array}$

-Using plurality, A wins. Could the six voters who most prefer C (and prefer A least) vote insincerely to change the outcome in a way that would benefit them?

A) No, they cannot change the outcome.
B) Yes, they could get B to win.
C) Yes, they could get C to win.
D) Yes, they could get D to win.

Question 70

Consider the following preference table for three voters to answer Questions
If a Borda count is used, can the voter in the middle manipulate the outcome to his or her benefit?

A) No
B) Yes

Question 71

Consider the following preference table for three voters to answer Questions
If a Borda count is used, can the voter on the right manipulate the outcome to his or her benefit?

A) No
B) Yes

Question 72

Consider the following preference table for three voters to answer Questions
If a Borda count is used, can the voter in the middle manipulate the outcome to his or her benefit?

A) No
B) Yes

Question 73

Consider the following preference table for three voters to answer Questions
If a Borda count is used, can the voter on the left manipulate the outcome to his or her benefit?

A) No
B) Yes

Question 74

Consider the following preference table for two voters: If a Borda count is used, can the voter on the left manipulate the outcome to his or her benefit?

A) No
B) Yes

Question 75

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, and the decision will be made using a Borda count. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c } & 10 & 8 & 4 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { Z } \\\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } \\\text { Third choice } & Z & Z & \mathrm {~B}\end{array}$

-Can the 10 voters in the first column change the results of the vote to their favor by changing their preference rankings?

Question 76

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, and the decision will be made using a Borda count. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c } & 10 & 8 & 4 \\\text { First choice } & \mathrm { B } & \mathrm { M } & \mathrm { Z } \\\text { Second choice } & \mathrm { M } & \mathrm { B } & \mathrm { M } \\\text { Third choice } & Z & Z & \mathrm {~B}\end{array}$

-Can the eight voters in the middle column change the results of the vote to their favor by changing their preference rankings?

Question 77

Consider the following preference table for three voters to answer Questions
If a Borda count is used, can the voter on the left manipulate the outcome to his or her benefit?

A) No
B) Yes

Question 78

Consider the following preference table for two voters: If a Borda count is used, can the voter on the left manipulate the outcome to his or her benefit?

A) No
B) Yes

Question 79

Use the following preference table to answer the Questions $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$
$\begin{array} { l c c c c } & 4 & 6 & 8 & 4 \\\text { First choice } & \mathrm { D } & \mathrm { C } & \mathrm { A } & \mathrm { B } \\\text { Second choice } & \mathrm { C } & \mathrm { B } & \mathrm { D } & \mathrm { A } \\\text { Third choice } & \mathrm { B } & \mathrm { D } & \mathrm { C } & \mathrm { C } \\\text { Fourth choice } & \mathrm { A } & \mathrm { A } & \mathrm { B } & \mathrm { D }\end{array}$

-Using plurality, A wins. Could the four voters who most prefer B vote insincerely to change the outcome in a way that would benefit them?

A) No, they cannot change the outcome.
B) Yes, they could get B to win.
C) Yes, they could get C to win.
D) Yes, they could get D to win.

Question 80

Consider the following preference table for two voters: If a Borda count is used, can the voter on the left manipulate the outcome to his or her benefit?

A) No
B) Yes