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?

Accepted Answer

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]

Accepted Answer

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]

Accepted Answer

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.

Accepted Answer

The answer of A committee consists of five members (A,...

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?

Accepted Answer

The answer of There are six different people who live...

Question 6

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

Accepted Answer

The answer of Consider the weighted voting system  [22:...

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?

Accepted Answer

The answer of Consider the weighted voting system  [q:...

Question 8

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

Accepted Answer

The answer of Consider the weighted voting system[25: w1, 7,...

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?

Accepted Answer

The answer of Consider the weighted voting system  [q:...

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.

Accepted Answer

The answer of The Party Planning Team for your local...

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.

Accepted Answer

The answer of Your college's Math Club consists of four...

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?

Accepted Answer

The answer of Consider the weighted voting system  [q:...

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?

Accepted Answer

The answer of On the popular television show Survivor, each...

Question 14

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

Accepted Answer

The answer of Explain, in your own words, what it...

Question 15

many of the four share-holders have veto power?

Accepted Answer

The answer of  many of the four share-holders have...

Question 16

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

Accepted Answer

The answer of Clearly explain why it is not possible...

Question 17

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

Accepted Answer

The answer of Consider the weighted voting system  [16:...

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.

Accepted Answer

The answer of Consider the weighted voting system  [18:...

Question 19

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

Accepted Answer

The answer of Consider the weighted voting system  [22:...

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.

Accepted Answer

The answer of A household consists of the following five...

Quiz 2: The Mathematics of Power: Weighted Voting

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?

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]

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]

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?

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

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?

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

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?

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?

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?

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

many of the four share-holders have veto power?

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

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

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

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

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?

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

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