Deck 2: The Mathematics of Power: Weighted Voting

Question 1

Consider the weighted voting system [25: 18,14, w₃] . What is the largest possible value for the weight w₃ which guarantees P₃ is the only player who is a dummy?

Answer 1

6

Question 2

Find the Shapley-Shibik Power for each player in the following weighted voting system. Give your answers as fractions written in reduced form.
[16: 12,10,4,2]

Answer 2

α1 = 5/12,α2 = 1/ 4,α3 =1/ 4,α4 =1/12

Question 3

Find the Banzhaf Power for each player in the following weighted voting system. Give your answers as fractions written in reduced form.
[16: 12,10,4,2]

Answer 3

β1 = 5/12, β2 = 1/ 4, β3 =1/ 4, β4 =1/12

Question 4

A committee consists of five members (A, B, C, D, and E). A and B both have veto power while C, D, and E each have one vote. For a motion to pass it must have the support of both A and B plus at least one additional member. Find the Banzhaf Power of each member of the committee. Give your answers as fractions written in reduced form.

Question 5

There are six different people who live on your dorm floor. Each person is able to vote on dorm issues. How many different sequential coalitions are possible?

Question 6

Consider the weighted voting system [22: 12,10,7,2] . Which player(s), if any, have veto power?

Question 7

Consider the weighted voting system [q: 8,7,6,2,1,1] . What is the largest value that the quota q can be to ensure that no player has veto power?

Question 8

Consider the weighted voting system[25: w1, 7, 6, 2]. What is the smallest possible value for the

Question 9

Consider the weighted voting system [q: 25,18,12] . What is the largest value of the quota q so that the weighted voting system would be equivalent to a weighted voting system where each member has one vote and a simple majority is needed to pass a motion?

Question 10

The Party Planning Team for your local Student Union is organizing a Spring Fair. The team consists of three members: Kathy, Tim, and Sara. Each member gets to vote on one of two theme ideas; however, Kathy's vote counts twice as much as Tim's vote and three times as much as Sara's vote. If at least 75 % of the counted votes are needed in order to choose a theme, then find the Shapley-Shubik Power for each member. Give your answers as percentages rounded to the nearest whole number.

Question 11

Your college's Math Club consists of four voting members: the president, vice president, secretary, and treasurer. The president's vote counts four times as much as the secretary and treasurer's votes and two times as much as the vice-president's vote. If the secretary and treasurer each have the exactly one vote and any motion needs at least six favorable votes in order to be passed, then give the weighted voting system which models this scenario.

Question 12

Consider the weighted voting system [q: 8,7,6,2,1,1] . What is the smallest value that the quota q can be to ensure that Player 1 and Player 2 are the only players to have veto power?

Question 13

On the popular television show Survivor, each tribe usually begins with 8 members or players. Alliances between the tribe members are basically the same idea as coalitions within a voting system. Using this fact, how many different alliances are possible within a tribe?

Question 14

Explain, in your own words, what it means for a player in a weighted voting system to be a dictator.

Question 15

many of the four share-holders have veto power?

Question 16

Clearly explain why it is not possible for a weighted voting system to have two dictators.

Question 17

Consider the weighted voting system [16: 12,10,4,2] . In how many sequential coalitions is player P₃ pivotal?

Question 18

Consider the weighted voting system [18: 8,7,6,2,1] . There is only one coalition in which P₅ is critical. Give this coalition.

Question 19

Consider the weighted voting system [22: 12,10,7,2] . Which player(s), if any, are dummies?

Question 20

A household consists of the following five individuals who each pay different amounts of the rent. Due to this, the system to make household decisions is based on the following voting system: [15: 12,10,4,2] . Looking at all sequential coalition where Player 2 is listed 3^rd , in what proportion of these sequences is Player 1 considered pivotal? Give your answer as a fraction.

Question 21

Consider the weighted voting system [18: 10,7,4,2,2,1] . Which player(s) are critical in the coalition ?

A) P₁
B) P₃
C) P₁ and P₃
D) All players are critical
E) None of the above.

Question 22

Your friend's family consists of two parents and three children. How many different sequential coalitions are possible?

A) 6
B) 24
C) 31
D) 120
E) None of the above.

Question 23

Consider the weighted voting system [16: 12,10,4] . What is the Shapely-Shubik Power of player one?

A) 3 / 5
B) 2 / 3
C) 3 / 4
D) 5 / 6
E) None of the above.

Question 24

Consider the weighted voting system[16 : 12, 10, 4] . What is the Banzhaf Power of player two?

A) 1/5
B) 2/5
C) 3/5
D) 4/5
E) None of the above.

Question 25

Which of the weighted voting systems given below contains a dictator?

A) [101: 99,99,1,1]
B) [8: 7,1,1,1]
C) [21: 22,18,1,1]
D) [16: 7,6,2,1]
E) None of the above.

Question 26

Consider the weighted voting system [q: 8,7,6,2,1,1] . What is the smallest value that the quota q can take?

A) 13
B) 14
C) 15
D) 16
E) None of the above.

Question 27

A company is run by two senior executives and two junior executives. Suppose that these four executives vote on all decisions made. In order for a decision to pass both senior executives and one junior executive must vote in favor of it. Which weighted voting system models this scenario?

A) [5: 2,2,1,1]
B) [5: 4,4,1,1]
C) [16: 6,6,5,5]
D) [12: 6,6,2,2]
E) None of the above.

Question 28

Tim and his three friends Beth ,Jane, Sam pool their money to buy a pizza . Tim gives $4; Beth gives 43; while both Jane and Sam each gives $2. if there are two choices for pizza and each person gets to vote proportionally to the amont of money he or she gave , then which weighted voting system models the senario if the quota is a simple majority of votes?

A) [6:3,2,1,1]
B) [12:5,3,2,2]
C) [6:4,3,2,2]
D) All of the above .
E) None of the above .

Question 29

Consider the weighted voting system [22: 12,10,4,2] . Which player(s), if any, have veto power?

A) P₁
B) P₂
C) Both P₁ and P₂
D) There are no players with veto power.
E) None of the above.

Question 30

Your friend's family consists of two parents and three children. How many different coalitions are possible?

A) 6
B) 240
C) 32
D) 120
E) None of the above.

Question 31

A company is run by one senior executive and three junior executives. Suppose that these four
executives vote on all decisions made. In order for a decision to pass the senior executive and two
junior executives must vote in favor of it. What is the Shapely-Shubik Power of the senior executive?

A) 40%
B) 50%
C) 60%
D) 70%
E) None of the above.

Question 32

Consider the weighted voting system [q: 8,7,6,2,1,1] . What is the smallest value that the quota q can take so that P 1, P 2 and P 3 are the only players to have veto power?

A) 20
B) 21
C) 22
D) 23
E) None of the above.

Question 33

Consider the weighted voting system [16: 12,10,4,2] . Which player(s), if any, are dummies?

A) P₃
B) P₄
C) Both P₃ and P₄
D) There are no dummies in this system.
E) None of the above.

Question 34

Consider the weighted voting system [16: 12,10,4] . What is the Banzhaf Power of player three?

A) 1 / 5
B) 1 / 6
C) 1 / 4
D) 1 / 3
E) None of the above.

Question 35

Consider the weighted voting system [22: 12,10,4,2,1] . How many of the sequential coalitions have the second individual in the sequence being the pivotal player?

A) 2
B) 6
C) 12
D) 24
E) None of the above.

Question 36

Consider the weighted voting system [37: 18,13,11, w₄, 1,1] . What is the smallest possible value for the weight w₄ which makes P₄ pivotal in the sequential coalition ?

A) 2
B) 5
C) 8
D) 10
E) None of the above.

Question 37

Consider the weighted voting system [25: w₁, 7,6,2] . What is the smallest possible value for the weight w₁ which guarantees P 1 is the only player with veto power.

A) 18
B) 17
C) 15
D) 13
E) None of the above.

Question 38

Which of the weighted voting systems given below contains exactly two dummies?

A) [4: 4,1,1,1]
B) [200: 100,100,98,1]
C) [8: 6,2,2,2]
D) All of the above.
E) None of the above.

Question 39

A company is run by two senior executives and three junior executives. Suppose that these five executives vote on all decisions made. In order for a decision to pass both senior executives and one junior executive must vote in favor of it. What is the Banzhaf Power of a senior executive?

A) 1 / 17
B) 6 / 17
C) 7 / 17
D) 8 / 17
E) None of the above.

Question 40

Consider the weighted voting system [26: 12,10,4] . What is the Shapely-Shubik Power of player two?

A) 1 / 6
B) 1 / 3
C) 2 / 3
D) 5 / 6
E) None of the above.