Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

Calculate 9C3.

Accepted Answer

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

Question 6

Given the weighted voting system [6: 4, 3, 2, 1], calculate the Banzhaf power index.

Accepted Answer

A)  (11, 6, 2, 2) 
B)  (5, 4, 4, 1) 
C)  (10, 6, 2, 2) 
D)  (10, 6, 6, 2) 
A)  (11, 6, 2, 2) 
B)  (5, 4, 4, 1) 
C)  (10, 6, 2, 2) 
D)  (10, 6, 6, 2)

Question 7

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

Accepted Answer

(A, B) (A,

Question 8

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

Accepted Answer

Answers may vary. On

Question 9

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

Accepted Answer

{C}, {A, B

Question 10

Use the weighted voting system [12: 8, 4, 4, 3, 3] to answer the following questions
-Which of the following is a blocking coalition?

Accepted Answer

A)  {A, B} 
B)  {A, D} 
C)  both (
A)  and (
B)  
D)  neither (
A)  nor (
B)  
A)  {A, B} 
B)  {A, D} 
C)  both (
A)  and (
B)  
D)  neither (
A)  nor (
B)

Question 11

Use the weighted voting system [10: 7, 4, 3, 3, 1] to answer the following Questions
-Calculate the Banzhaf power index for this voting system.

Accepted Answer

A)  (26, 8, 8, 0, 0) 
B)  (24, 8, 8, 4, 4) 
C)  (18, 18, 3, 3, 1, 1) 
D)  (24, 8, 8, 8, 0) 
A)  (26, 8, 8, 0, 0) 
B)  (24, 8, 8, 4, 4) 
C)  (18, 18, 3, 3, 1, 1) 
D)  (24, 8, 8, 8, 0)

Question 12

Calculate the Shapley-Shubik power index for each voter in the weighted voting system [6: 5, 3, 1].

Accepted Answer

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

Question 13

Are the weighted voting systems [8: 5, 4, 3] and [34: 30, 18, 6] equivalent?

Accepted Answer

A) True 
 B)False

Question 14

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

Accepted Answer

Answer may

Question 15

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

Accepted Answer

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

Question 16

Calculate ${ } _ { 10 } C _ { 10 } $

Accepted Answer

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

Question 17

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

Accepted Answer

Answers ma

Question 18

Use the weighted voting system [12: 8, 4, 4, 3, 3] to answer the following questions
-Which of the voters are dummy voters?

Accepted Answer

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

Question 19

Which voters in the system [38: 20, 15, 12, 5] 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 20

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

Accepted Answer

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

Exam 11: Weighted Voting Systems

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

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

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

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

Calculate 9C3.

Given the weighted voting system [6: 4, 3, 2, 1], calculate the Banzhaf power index.

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

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

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

Use the weighted voting system [12: 8, 4, 4, 3, 3] to answer the following questions -Which of the following is a blocking coalition?

Use the weighted voting system [10: 7, 4, 3, 3, 1] to answer the following Questions -Calculate the Banzhaf power index for this voting system.

Calculate the Shapley-Shubik power index for each voter in the weighted voting system [6: 5, 3, 1].

Are the weighted voting systems [8: 5, 4, 3] and [34: 30, 18, 6] equivalent?

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

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

Calculate 10C10{ } _ { 10 } C _ { 10 } 10​C10​

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

Use the weighted voting system [12: 8, 4, 4, 3, 3] to answer the following questions -Which of the voters are dummy voters?

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

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

Urban Services

Business Efficiency

Planning and Scheduling

Linear Programming

Exploring Data: Distributions

Exploring Data: Relationships

Data for Decisions

Probability: the Mathematics of Chance

Social Choice: the Impossible Dream

The Manipulability of Voting Systems

Electing the President

Fair Division

Apportionment

Game Theory: the Mathematics of Competition

Identification Numbers

Information Science

Growth and Form

Symmetry and Patterns

Tilings

Savings Models

Borrowing Models

The Economics of Resources

Filters

Calculate ₉C₃.

Calculate ${ } _ { 10 } C _ { 10 }$

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