MGF 1107 Spring 12 Ref: 14344
Exam 1 Review
Mr. Guillen
Exam 1 will be on 02/02/12 and covers CH 9,11 & Set Theory.
1. Given the weighted voting system [30: 20, 17, 10, 5], list all winning coalitions.
Ans: (A, B) (A, C) (B, C, D) (A, B, C) (A, B, D) (A, C, D) (A, B, C, D)
Format: Short Answer
2. Given the weighted voting system [51: 45, 43, 7, 5], list all blocking coalitions.
Ans: (A, B) (A, C) (A, B, C) (A, B, D) (A, C, D) (A, D) (B, C) (B, C, D) (A, B,
C, D)
Format: Short Answer
3. Given the weighted voting system [51: 45, 43, 7, 5], list all minimal winning coalitions.
Ans: (A, B) (A, C) (B, C, D)
Format: Short Answer
4. Given the weighted voting system [30: 20, 17, 10, 5], list all minimal winning
coalitions.
Ans: (A, B) (A, C) (B, C, D)
Format: Short Answer
5. Given the weighted voting system [30: 20, 17, 10, 5], list all blocking coalitions.
Ans: (A, B) (A, C) (A, B, C) (A, B, D) (A, C, D) (B, C, D) (A, B, C, D) (A, D)
(B, C)
Format: Short Answer
6. A weighted voting system has four voters, A, B, C, and D. List all possible coalitions
of these voters. How many such coalitions are there?
Ans: There are 16 coalitions possible from four voters:
Ø, (A) (B) (C) (D) (A, B) (A, C) (A, D) (B, C) (B, D) (C, D) (A, B, C) (A,
B, D) (A, C, D) (B, C, D) (A, B, C, D)
7. Given the weighted voting system [5: 3, 2, 1, 1, 1], find which voters of the coalition
{A, C, D, E} are critical.
Ans: Since the coalition {A, C, D, E} has one extra vote, the only member who is
critical is voter A with weight 3.
Format: Short Answer
8. Given the weighted voting system [8: 5, 4, 3], find the Banzhaf power index for each
voter.
Ans: (6, 2, 2)
Format: Short Answer
9. Given the weighted voting system [14: 10, 6, 5, 3], find the Banzhaf power index for
each voter.
Ans: (10, 6, 6, 2)
Format: Short Answer
10. Given the weighted voting system [7: 4, 1, 1, 1, 1, 1], find the Banzhaf power index for
each voter.
Ans: (32, 20, 20, 20, 20, 20)
Format: Short Answer
11. Suppose a weighted voting system has five members, but that voter E is a dummy
voter. Can you find the Banzhaf power index of voter E?
Ans: The Banzhaf power index of any dummy voter is 0.
12. In the weighted voting system [9: 10, 5, 3], voter A is a dictator.
A) True
B) False
Ans: A
Format: Multiple-Choice
13. Which voters in the system [30: 20, 17, 10, 5] have veto power?
A) A only
B) A and B
C) A, B, and C
D) None
Ans: D
Format: Multiple-Choice
14. Which voters in the system [38: 20, 15, 12, 5] have veto power?
A) A only
B) A and B
C) A, B, and C
D) None
Ans: B
Format: Multiple-Choice
15. Given the weighted voting system [38: 20, 15, 12, 5], which is a winning coalition?
A) {A, B}
B) {B, C, D}
C) {A, B, D}
D) {A}
Ans: C
Format: Multiple-Choice
16. Given the weighted voting system [38: 20, 15, 12, 5], which of the coalitions listed
is/are blocking coalitions?
I
II {C, D}
III {A, B, C}
{A}
A) III only
B) I and II only
C) I, II, and III
D) None
Ans: C
Format: Multiple-Choice
17. Given the weighted voting system [38: 20, 15, 12, 5], which voter(s) is/are dummy
voters.
A) D only
B) D and C
C) D, C, and B
D) There are no dummy voters.
Ans: D
Format: Multiple-Choice
18. Given the weighted voting system [51: 45, 43, 7, 5], which of the following is a
winning coalition?
A) {A}
B) {A, D}
C) {A, C}
D) {B, D}
Ans: C
19. Given the weighted voting system [27: 20, 15, 12, 5], calculate the Banzhaf power
index.
A) (8, 8, 8, 0)
B) (8, 6, 6, 2)
C) (6, 6, 6, 1)
D) (8, 6, 6, 0)
Ans: A
Format: Multiple-Choice
20. Given the weighted voting system [34: 30, 18, 6], calculate the Banzhaf power index.
A) (6, 4, 2)
B) (6, 2, 2)
C) (6, 6, 6)
D) (6, 4, 0)
Ans: B
Format: Multiple-Choice
21. Given the weighted voting system [30: 20, 17, 10, 5], calculate the Banzhaf power
index.
A) (10, 6, 6, 0)
B) (8, 6, 6, 2)
C) (10, 6, 6, 2)
D) (8, 4, 4, 0)
Ans: C
22. Are the weighted voting systems [8: 5, 4, 3] and [34: 30, 18, 6] equivalent?
A) Yes
B) No
Ans: A
Format: Multiple-Choice
23. Are the weighted voting systems [8: 5, 4, 3] and [11: 10, 9, 2] equivalent?
A) Yes
B) No
Ans: B
Format: Multiple-Choice
24. Are the weighted voting systems [18: 10, 9, 8] and [15: 8, 7, 6] equivalent?
A) Yes
B) No
Ans: B
Format: Multiple-Choice
25. Are the weighted voting systems [11: 10, 9, 2] and [2: 1, 1, 1] equivalent?
A) Yes
B) No
Ans: A
26. Calculate the Shapley-Shubik power index for each voter in the system [8: 5, 4, 3].
A) (4/6, 1/6, 1/6)
B) (3/6, 3/6, 0/6)
C) (2/6, 2/6, 2/6)
D) (4/6, 2/6, 2/6)
Ans: A
Format: Multiple-Choice
27. Calculate the Shapley-Shubik power index for each voter in the system [15: 8, 7, 6].
A) (4/6, 1/6, 1/6)
B) (3/6, 3/6, 0/6)
C) (2/6, 2/6, 2/6)
D) (4/6, 2/6, 2/6)
Ans: B
Format: Multiple-Choice
28. Calculate the Shapley-Shubik power index for each voter in the weighted voting system
[8: 5, 2, 2, 2, 2, 2].
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)
Ans: D
Format: Multiple-Choice
29. Calculate the Shapely-Shubik power index for each voter in the weighted voting system
[6: 3, 1, 1, 1, 1, 1, 1].
A) (3/7, 2/21, 2/21, 2/21, 2/21, 2/21, 2/21)
B) (1/2, 1/12, 1/12, 1/12, 1/12, 1/12, 1/12)
C) (1/3, 1/9, 1/9, 1/9, 1/9, 1/9, 1/9)
D) (2/3, 1/18, 1/18, 1/18, 1/18, 1/18, 1/18)
Ans: A
30. Given the weighted voting system [6: 4, 3, 2, 1], calculate the Banzhaf power index.
A) (11, 6, 2, 2)
B) (5, 4, 4, 1)
C) (10, 6, 2, 2)
D) (10, 6, 6, 2)
Ans: D
Format: Multiple-Choice
31. Calculate the Shapley-Shubik power index for each voter in the weighted voting system
[6: 5, 3, 1].
A) (1, 0, 0)
B) (2/3, 1/6, 1/6)
C) (1/3, 1/3, 1/3)
D) (1/2, 1/2, 0)
Ans: B