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

A) 3 B) 6 C) 8 D) 12

12.

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 majority rule?

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?

Number of Students

3

3

2

2

2

First choice

B

M

S

B

S

Second choice

M

S

M

S

B

Third choice

S

B

B

M

M

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

14.

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?

Number of Students

3

3

2

2

2

First choice

B

M

S

B

S

Second choice

M

S

M

S

B

Third choice

S

B

B

M

M

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

15.

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?

Number of Students

3

3

2

2

2

First choice

B

M

S

B

S

Second choice

M

B

M

S

B

Third choice

S

S

B

M

M

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

16.

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?

Number of Students

3

3

2

2

2

First choice

B

M

S

B

S

Second choice

M

B

M

S

B

Third choice

S

S

B

M

M

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

17.

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 sequential pairwise voting with agenda B, M, S?

Number of Students

3

3

2

2

2

First choice

B

M

S

B

S

Second choice

M

B

M

S

B

Third choice

S

S

B

M

M

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

18.

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?

Number of Students

3

3

2

2

2

First choice

B

M

S

B

S

Second choice

M

B

M

S

B

Third choice

S

S

B

M

M

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

19.

Thirty board members must vote on five candidates: X, Y, Z, U, and V. Their preference rankings are summarized in the table below. Find the winner using the Borda count.

Thirty board members must vote on five candidates: X, Y, Z, U, and V. Their preference rankings are summarized in the table below. Find the winner using the Hare system.

Number of Members

12

10

8

First choice

X

Y

Z

Second choice

U

Z

U

Third choice

Y

X

X

Fourth choice

Z

U

V

Fifth choice

V

V

Y

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

21.

Thirty board members must vote on five candidates: X, Y, Z, U, and V. Their preference rankings are summarized in the table below. Find the winner using sequential pairwise voting with the agenda X, Y, Z, U, V.