Question 1

In a weighted voting system with 20 voters, how many distinct coalitions would there be in which exactly eight members voted YES?

Accepted Answer

A)  12 
B)  56 
C)  160 
D)  125,970 
A)  12 
B)  56 
C)  160 
D)  125,970

Question 2

Which of the following describe legitimate weighted voting systems? $$\begin{array} { l l } 
\text { I } & { [ 27 : 20,15,12,5 ] } \
\text { II } & { [ 30 : 20,17,10,5 ] }
\end{array}$$

Accepted Answer

A)  I only 
B)  II only 
C)  I and II 
D)  neither I nor II 
A)  I only 
B)  II only 
C)  I and II 
D)  neither I nor II

Question 3

Are the weighted voting systems [11: 10, 9, 2] and [2: 1, 1, 1] equivalent?

Accepted Answer

A) True 
 B)False

Question 4

Given the weighted voting system [9: 6, 4, 2], which of the voters are dummy voters?

Accepted Answer

A)  C only 
B)  B and C 
C)  B only 
D)  none 
A)  C only 
B)  B and C 
C)  B only 
D)  none

Question 5

Determine whether the weighted voting systems [10: 9, 1] and [5: 4, 4] are equivalent.

Accepted Answer

Yes, they are equiva

Question 6

Given the weighted voting system [8: 3, 5, 2, 4], which is a blocking coalition?

Accepted Answer

A)  {A, B} 
B)  {A, C} 
C)  {A} 
D)  {B} 
A)  {A, B} 
B)  {A, C} 
C)  {A} 
D)  {B}

Question 7

Suppose a weighted voting system has five members, but that voter E is a dummy voter. Can you find the Banzhaf power index of voter E?

Accepted Answer

The Banzha

Question 8

A committee has six members with the system [8: 5, 2, 2, 2, 2, 2]. Which voters are critical in the coalition {A, C, D}?

Accepted Answer

A)  A only 
B)  C and D 
C)  A, C, and D 
D)  No voter is critical. 
A)  A only 
B)  C and D 
C)  A, C, and D 
D)  No voter is critical.

Question 9

Given the weighted voting system [38: 20, 15, 12, 5], which voters are dummy voters.

Accepted Answer

A)  D only 
B)  D and C 
C)  D, C, and B 
D)  There are no dummy voters. 
A)  D only 
B)  D and C 
C)  D, C, and B 
D)  There are no dummy voters.

Question 10

A sorority has an executive board consisting of a chair, vice chair, and three other members. Its voting rules indicate that an issue can pass in two ways: if the chair, vice chair, and one other member support the issue or if one of the chair or vice chair and two other members support the issue. Express this as a weighted voting system.

Accepted Answer

Answers may vary. On

Question 11

Given the weighted voting system [30: 20, 17, 10, 5], calculate the Banzhaf power index.

Accepted Answer

A)  (10, 6, 6, 0) 
B)  (8, 6, 6, 2) 
C)  (10, 6, 6, 2) 
D)  (8, 4, 4, 0) 
A)  (10, 6, 6, 0) 
B)  (8, 6, 6, 2) 
C)  (10, 6, 6, 2) 
D)  (8, 4, 4, 0)

Question 12

Given the weighted voting system [6: 4, 3, 2, 1], which of the coalitions listed are winning coalitions? 
$I \{ A , B \}$
II $\{ A , C \}$
III $\{ B , C , D \}$

Accepted Answer

A)  I only 
B)  I and II 
C)  I and III 
D)  I, II, and III 
A)  I only 
B)  I and II 
C)  I and III 
D)  I, II, and III

Question 13

Voter D has veto power if a motion will pass if and only if D is in favor, and the votes of the other members make no difference.

Accepted Answer

A) True 
 B)False

Question 14

Given the weighted voting system [27: 20, 15, 12, 5], calculate the Banzhaf power index.

Accepted Answer

A)  (8, 8, 8, 0) 
B)  (8, 6, 6, 2) 
C)  (6, 6, 6, 1) 
D)  (8, 6, 6, 0) 
A)  (8, 8, 8, 0) 
B)  (8, 6, 6, 2) 
C)  (6, 6, 6, 1) 
D)  (8, 6, 6, 0)

Question 15

Which voters in the system [19: 9, 7, 5, 3, 1] have veto power?

Accepted Answer

A)  A only 
B)  A and B 
C)  A, B, and C 
D)  none 
A)  A only 
B)  A and B 
C)  A, B, and C 
D)  none

Question 16

Calculate the Shapley-Shubik power index for each voter in the system [15: 8, 7, 6].

Accepted Answer

A)  (4/6, 1/6, 1/6) 
B)  (3/6, 3/6, 0/6) 
C)  (2/6, 2/6, 2/6) 
D)  (4/6, 2/6, 2/6) 
A)  (4/6, 1/6, 1/6) 
B)  (3/6, 3/6, 0/6) 
C)  (2/6, 2/6, 2/6) 
D)  (4/6, 2/6, 2/6)

Question 17

Are the weighted voting systems [8: 5, 4, 3] and [11: 10, 9, 2] equivalent?

Accepted Answer

A) True 
 B)False

Question 18

Which of the following describe legitimate weighted voting systems? 
I. [10: 5, 4, 3, 8]
II) [9: 5, 8, 3]

Accepted Answer

A)  I only 
B)  II only 
C)  I and II 
D)  neither I nor II 
A)  I only 
B)  II only 
C)  I and II 
D)  neither I nor II

Question 19

Given the weighted voting system [38: 20, 15, 12, 5], how many winning coalitions are there?

Accepted Answer

A)  1 
B)  2 
C)  3 
D)  4 
A)  1 
B)  2 
C)  3 
D)  4

Question 20

Given the weighted voting system [8: 5, 4, 3], find the Banzhaf power index for each voter.

Accepted Answer

The answer of Given the weighted voting system [8: 5,...

Exam 11: Weighted Voting Systems

In a weighted voting system with 20 voters, how many distinct coalitions would there be in which exactly eight members voted YES?

Which of the following describe legitimate weighted voting systems? I [27:20,15,12,5] II [30:20,17,10,5]

Are the weighted voting systems [11: 10, 9, 2] and [2: 1, 1, 1] equivalent?

Given the weighted voting system [9: 6, 4, 2], which of the voters are dummy voters?

Determine whether the weighted voting systems [10: 9, 1] and [5: 4, 4] are equivalent.

Given the weighted voting system [8: 3, 5, 2, 4], which is a blocking coalition?

Suppose a weighted voting system has five members, but that voter E is a dummy voter. Can you find the Banzhaf power index of voter E?

A committee has six members with the system [8: 5, 2, 2, 2, 2, 2]. Which voters are critical in the coalition {A, C, D}?

Given the weighted voting system [38: 20, 15, 12, 5], which voters are dummy voters.

Given the weighted voting system [30: 20, 17, 10, 5], calculate the Banzhaf power index.

Given the weighted voting system [6: 4, 3, 2, 1], which of the coalitions listed are winning coalitions? I{A,B}I \{ A , B \}I{A,B} II {A,C}\{ A , C \}{A,C} III {B,C,D}\{ B , C , D \}{B,C,D}

Voter D has veto power if a motion will pass if and only if D is in favor, and the votes of the other members make no difference.

Given the weighted voting system [27: 20, 15, 12, 5], calculate the Banzhaf power index.

Which voters in the system [19: 9, 7, 5, 3, 1] have veto power?

Calculate the Shapley-Shubik power index for each voter in the system [15: 8, 7, 6].

Are the weighted voting systems [8: 5, 4, 3] and [11: 10, 9, 2] equivalent?

Which of the following describe legitimate weighted voting systems? I. [10: 5, 4, 3, 8] II) [9: 5, 8, 3]

Given the weighted voting system [38: 20, 15, 12, 5], how many winning coalitions are there?

Given the weighted voting system [8: 5, 4, 3], find the Banzhaf power index for each voter.

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Given the weighted voting system [6: 4, 3, 2, 1], which of the coalitions listed are winning coalitions? $I \{ A , B \}$ II $\{ A , C \}$ III $\{ B , C , D \}$