Question 1

Given the weighted voting system [52: 45, 43, 7, 5], which of the following is a minimal winning coalition? 
$I \{ A , B \}$
II $\{ A , C \}$
III $\{ A , B , C \}$

Accepted Answer

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

Question 2

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

Accepted Answer

The answer of A weighted voting system has five voters....

Question 3

Given the weighted voting system [38: 20, 15, 12, 5], which of the coalitions listed are blocking coalitions? 
$I \left\{ A _ { 2 } \right\}$
II $\{ \mathrm { C } , \mathrm { D } \}$
$\operatorname { III } \{ A , B , C \}$

Accepted Answer

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

Question 4

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:

Accepted Answer

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

Question 5

How many permutations of five voters can be made?

Accepted Answer

A)  14 
B)  25 
C)  32 
D)  120 
A)  14 
B)  25 
C)  32 
D)  120

Question 6

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

Accepted Answer

A critical voter in a winning or blockin

Question 7

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

Accepted Answer

A) True 
 B)False

Question 8

In the weighted voting system [6: 3, 1, 1, 1, 1, 1, 1], find the Banzhaf power index for voter A.

Accepted Answer

A)  6 
B)  20 
C)  40 
D)  82 
A)  6 
B)  20 
C)  40 
D)  82

Question 9

Calculate the Shapley-Shubik power index for each voter in the weighted voting system [8: 5, 2, 2, 2, 2, 2].

Accepted Answer

A)  (7/12, 1/12, 1/12, 1/12, 1/12, 1/12) 
B)  (5/8, 3/40, 3/40, 3/40, 3/40, 3/40) 
C)  (5/6, 1/30, 1/30, 1/30, 1/30, 1/30) 
D)  (1/3, 2/15, 2/15, 2/15, 2/15, 2/15) 
A)  (7/12, 1/12, 1/12, 1/12, 1/12, 1/12) 
B)  (5/8, 3/40, 3/40, 3/40, 3/40, 3/40) 
C)  (5/6, 1/30, 1/30, 1/30, 1/30, 1/30) 
D)  (1/3, 2/15, 2/15, 2/15, 2/15, 2/15)

Question 10

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

Accepted Answer

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

Question 11

Which of the following describe legitimate weighted voting systems? $$\begin{array} { l l } 
\text { I } & { [ 16 : 13,8,6,4 ] } \
\text { II } & { [ 15 : 10,8,7,5 ] }
\end{array}$$

Accepted Answer

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

Question 12

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

Accepted Answer

A) True 
 B)False

Question 13

Which of the following describe legitimate weighted voting systems? $$\begin{array} { l l } 
\text { I } & { [ 20 : 16,12,8 ] } \
\text { II } & { [ 24 : 20,15,10 ] }
\end{array}$$

Accepted Answer

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

Question 14

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?

Accepted Answer

Yes, voter A is a critical voter. Withou

Question 15

Calculate ${ } _ { 10 } C _ { 8 }$

Accepted Answer

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

Question 16

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

Accepted Answer

No. A voter with veto power ha

Question 17

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

Accepted Answer

(7/20, 1/1

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.

Accepted Answer

Answers may vary. One solution is: In th

Question 19

Calculate $32 C _ { 3}$

Accepted Answer

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

Question 20

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

Accepted Answer

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

Exam 11: Weighted Voting Systems

Given the weighted voting system [52: 45, 43, 7, 5], which of the following is a minimal winning coalition? I{A,B}I \{ A , B \}I{A,B} II {A,C}\{ A , C \}{A,C} III {A,B,C}\{ A , B , C \}{A,B,C}

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

Given the weighted voting system [38: 20, 15, 12, 5], which of the coalitions listed are blocking coalitions? I{A2}I \left\{ A _ { 2 } \right\}I{A2​} II {C,D}\{ \mathrm { C } , \mathrm { D } \}{C,D} III⁡{A,B,C}\operatorname { III } \{ A , B , C \}III{A,B,C}

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:

How many permutations of five voters can be made?

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

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

In the weighted voting system [6: 3, 1, 1, 1, 1, 1, 1], find the Banzhaf power index for voter A.

Calculate the Shapley-Shubik power index for each voter in the weighted voting system [8: 5, 2, 2, 2, 2, 2].

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

Which of the following describe legitimate weighted voting systems? I [16:13,8,6,4] II [15:10,8,7,5]

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

Which of the following describe legitimate weighted voting systems? I [20:16,12,8] II [24:20,15,10]

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?

Calculate 10C8{ } _ { 10 } C _ { 8 }10​C8​

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

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

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.

Calculate 32C332 C _ { 3}32C3​

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

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Given the weighted voting system [52: 45, 43, 7, 5], which of the following is a minimal winning coalition? $I \{ A , B \}$ II $\{ A , C \}$ III $\{ A , B , C \}$

Given the weighted voting system [38: 20, 15, 12, 5], which of the coalitions listed are blocking coalitions? $I \left\{ A _ { 2 } \right\}$ II $\{ \mathrm { C } , \mathrm { D } \}$ $\operatorname { III } \{ A , B , C \}$

Calculate ${ } _ { 10 } C _ { 8 }$

Calculate $32 C _ { 3}$