Deck 11: Weighted Voting Systems

Question 1

Given the weighted voting system [30: 20, 17, 10, 5], list all minimal winning coalitions.

Answer 1

(A, B) (A, C) (B, C, D)

Question 2

Is voter A a critical voter in the coalition {A, B, C} of the weighted voting [15: 10, 6, 5, 3, 2]? Why or why not?

Answer 2

Yes, voter A is a critical voter. Without A, the coalition becomes (B, C), which has only 11 votes, no longer enough to win. The number of extra votes of the coalition is six. Voter A has more than this number of votes so voter A is critical.

Question 3

Given the weighted voting system [51: 45, 43, 7, 5], list all blocking coalitions.

Answer 3

(A, B) (A, C) (A, B, C) (A, B, D) (A, C, D) (A, D) (B, C) (B, C, D) (A, B, C, D)

Question 4

Given the weighted voting system [5: 3, 2, 1, 1, 1], find which voters of the coalition {A, C, D, E} are critical.

Question 5

Give an example of a weighted voting system that is equivalent to the [15: 8, 7, 6].

Question 6

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

Question 7

Give an example of a weighted voting system that has a dummy voter but no dictator.

Question 8

Given the weighted voting system [51: 45, 43, 7, 5], list all minimal winning coalitions.

Question 9

Given the weighted voting system [30: 20, 17, 10, 5], list all winning coalitions.

Question 10

Give an example of a weighted voting that is equivalent to the [8: 5, 4, 3].

Question 11

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?

Question 12

Explain why the weighted voting system [13: 10, 6, 5, 3, 2] is not a legitimate weighted voting system.

Question 13

Given the weighted voting system [14: 10, 6, 5, 3], find the Banzhaf power index for each voter.

Question 14

Given the weighted voting system [30: 20, 17, 10, 5], find the pivotal voter in the permutation {B, D, C, A}.

Question 15

For the weighted voting system [30: 20, 17, 10, 5], give a permutation such that the voter D is a pivotal voter.

Question 16

A weighted voting system has five voters. How many distinct coalitions are there in which exactly three members vote YES?

Question 17

A weighted voting system has 12 members. How many distinct coalitions are there in which exactly seven members vote YES?

Question 18

Give an example of a weighted voting system that has a blocking coalition that would not be a winning coalition if all its members voted YES.

Question 19

Given the weighted voting system [7: 4, 1, 1, 1, 1, 1], find the Banzhaf power index for each voter.

Question 20

In a weighted voting system, is a voter with veto power the same as a dictator? Explain why or why not.

Question 21

Given the weighted voting system [14: 8, 2, 5, 7, 4], calculate the Shapley-Shubik power index for each voter.

Question 22

Give two blocking coalitions.

Question 23

What is the difference between a "critical" voter in a coalition and a "pivotal" voter in a permutation?

Question 24

Calculate the Shapley-Shubik power index for the weighted voting system [8: 6, 1, 1, 1, 1, 1].

Question 25

A weighted voting system has 10 members. How many distinct coalitions are there in which exactly six members vote yes?

Question 26

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

Question 27

Given the weighted voting system [4: 1, 2, 3], list all blocking coalitions.

Question 28

Is any voter a dummy voter?

Question 29

Which of the following describe legitimate weighted voting systems?

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

Question 30

Evaluate ₈C₇.

Question 31

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.

Question 32

For a weighted voting system, if the sum of the voting weights of voters who favor a motion is at least a certain number, then YES wins. The number is called the:

A) critical value.
B) quota.
C) majority value.
D) pivot.

Question 33

Calculate the Banzhaf power index for each voter.

Question 34

Given the weighted voting system [4: 1, 2, 3], list all winning coalitions.

Question 35

Given the weighted voting system [16: 3, 9, 4, 5, 10], calculate the Banzhaf power index for each voter.

Question 36

Give a winning coalition that is not a minimal winning coalition.

Question 37

What would be the quota for a voting system that has a total of 30 votes and uses a simple majority quota?

A) 14
B) 15
C) 16
D) 30

Question 38

Given weighted voting system of seven voters, the Banzhaf power index for voter A is 5. Find the probability that the voter A is a critical voter.

Question 39

Calculate the Shapley-Shubik power index for the weighted voting system [30: 20, 17, 10, 5].

Question 40

What would be the quota for a voting system that has a total of 16 votes and uses a simple majority quota?

A) 7
B) 8
C) 9
D) 16

Question 41

Which of the following describe legitimate weighted voting systems?

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

Question 42

Given the weighted voting system [51: 45, 43, 7, 5], which of the following is a blocking coalition?

A) II only
B) I and II
C) I, II, and III
D) none

Question 43

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.

Question 44

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

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

Question 45

For the system [q: 30, 10, 6], which of the following cannot be the quota q?

A) 24
B) 32
C) 40
D) 47

Question 46

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

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

Question 47

In a weighted voting system, all blocking coalitions would become winning coalitions if each voter switched his or her vote from NO to YES.

Question 48

Given the weighted voting system [6: 4, 3, 2, 1], which of the coalitions listed are winning coalitions?

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

Question 49

A weighted voting system can have a dictator without dummy voters.

Question 50

Given the weighted voting system [52: 45, 43, 7, 5], which of the following is a minimal winning coalition?

A) I only
B) I and II
C) I, II, and III
D) none

Question 51

Which of the following describe legitimate weighted voting systems?

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

Question 52

In the weighted voting system [9: 10, 5, 3], voter A is a dictator.

Question 53

Given the weighted voting system [8: 5, 4, 3], which of the coalitions given are minimal winning coalitions?

A) {A, B, C}
B) {A, B}
C) {B, C}
D) There are no minimal winning coalitions.

Question 54

In a weighted voting system, all winning coalitions would become blocking coalitions if each voter switched his or her vote from YES to NO.

Question 55

Given the weighted voting system [38: 20, 15, 12, 5], which of the coalitions listed are blocking coalitions?

A) III only
B) I and II only
C) I, II, and III
D) none

Question 56

In a weighted voting system, any voter with veto power is a dictator.

Question 57

Which voters in the system [38: 20, 15, 12, 5] have veto power?

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

Question 58

Given the weighted voting system [51: 45, 43, 7, 5], which of the following is a winning coalition?

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

Question 59

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

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

Question 60

Which voters in the system [30: 20, 17, 10, 5] have veto power?

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

Question 61

Given the weighted voting system [8: 4, 3, 2, 1], who is the pivotal voter for the permutation {A, B, C, D}?

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

Question 62

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

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

Question 63

Given the weighted voting system [34: 30, 18, 6], calculate the Banzhaf power index.

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

Question 64

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}?

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

Question 65

A committee has four voters and the system [30: 20, 17, 10, 5]. How many extra votes does the coalition {A, C, D} have?

A) 1
B) 5
C) 10
D) 35

Question 66

Given the weighted voting system [6: 4, 3, 2, 1], which of the coalitions listed are minimal winning coalitions?

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

Question 67

Calculate

A) 0
B) 1
C) 10
D) 15

Question 68

A committee has four voters with the system [30: 20, 17, 10, 5]. Which voters are critical in the coalition {A, C, D}?

A) A only
B) A and C
C) C and D
D) D only

Question 69

If there are five voters in a weighted voting system, how many different combinations of YES and NO votes can there be?

A) 10
B) 25
C) 32
D) 120

Question 70

If there are four voters in a weighted voting system, how many distinct coalitions of voters can be formed?

A) 8
B) 16
C) 24
D) 30

Question 71

Calculate

A) 0
B) 1
C) 10
D) 15

Question 72

Calculate ₉C₃.

A) 27
B) 84
C) 504
D) 729

Question 73

Calculate

A) 45
B) 80
C) 90
D) 180

Question 74

Given the weighted voting system [4: 2, 2, 2, 1], who is the pivotal voter for the permutation {A, C, B, D}?

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

Question 75

Given the weighted voting system [6: 4, 3, 2, 1], which of the coalitions listed are blocking coalitions?

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

Question 76

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

A) 32
B) 40
C) 56
D) 256

Question 77

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

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

Question 78

Calculate

A) 10
B) 96
C) 4960
D) 29,760

Question 79

A committee has six voters with the system [8: 5, 2, 2, 2, 2, 2]. How many extra votes does the coalition {B, C, E, F} have?

A) 0
B) 1
C) 2
D) 8

Question 80

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

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