Question 1

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

Accepted Answer

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. 
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. B

Question 2

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

βA =7 / 17,βB=7/17,βC = 1/17,βD=1/17,βE=1/17

Question 3

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

Accepted Answer

P1 and P2

Question 4

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?

Accepted Answer

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

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

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,

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

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  3rd , in what proportion of these sequences is Player 1 considered pivotal? Give your answer as a fraction.

Accepted Answer

Question 10

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

Accepted Answer

A player is a dictat

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

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.

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

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?

Accepted Answer

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

Question 17

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?

Accepted Answer

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

Question 18

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?

Accepted Answer

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. 
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 19

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

Accepted Answer

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

Question 20

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:...

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

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

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?

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?

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]

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

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

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

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

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

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.

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

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

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?

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 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?

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

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?

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The Mathematics of Touring: Traveling Salesman Problems

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The Mathematics of Scheduling: Chasing the Critical Path

Population Growth Models: There Is Strength in Numbers

Financial Mathematics: Money Matters

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Graphs, Charts, and Numbers: the Data Show and Tell

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The Mathematics of Normality: the Call of the Bell

Filters

Exam 2: The Mathematics of Power: Weighted Voting

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

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

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?

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?

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]

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

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

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

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

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

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.

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

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

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?

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 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?

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

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?

The Mathematics of Elections: the Paradoxes of Democracy

The Mathematics of Sharing: Fair-Division Games

The Mathematics of Apportionment: Making the Rounds

The Mathematics of Getting Around: Euler Paths and Circuits

The Mathematics of Touring: Traveling Salesman Problems

The Cost of Being Connected

The Mathematics of Scheduling: Chasing the Critical Path

Population Growth Models: There Is Strength in Numbers

Financial Mathematics: Money Matters

The Mathematics of Symmetry: Beyond Reflection

Fractal Geometry: the Kinky Nature of Nature

Fibonacci Numbers and the Golden Ratio: Tales of Rabbits and Gnomons

Censuses, Surveys, Polls, and Studies: the Joys of Collecting Data

Graphs, Charts, and Numbers: the Data Show and Tell

Probabilities, Odds, and Expectations: Measuring Uncertainty and Risk

The Mathematics of Normality: the Call of the Bell

Filters

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.