MGF 1107 Summer B 10 Ref: 57524
Mr. Guillen
Review for Exam 2 (part 2)
Exam 2 will be on 07/14/10 and coves the following: Combinatorics, Chapter 11.
1. Given the weighted voting system [30: 20, 17, 10, 5], list all winning coalitions.
2. Given the weighted voting system [51: 45, 43, 7, 5], list all blocking coalitions.
3. Given the weighted voting system [51: 45, 43, 7, 5], list all minimal winning coalitions.
4. Given the weighted voting system [30: 20, 17, 10, 5], list all minimal winning coalitions.
5. Given the weighted voting system [30: 20, 17, 10, 5], list all blocking coalitions.
6. Given the weighted voting system [5: 3, 2, 1, 1, 1], find which voters of the coalition {
A, C, D, E} are critical.
7. Given the weighted voting system [8: 5, 4, 3], find the Banzhaf power index for each
voter.
8. Given the weighted voting system [14: 10, 6, 5, 3], find the Banzhaf power index for
each voter.
9. Given the weighted voting system [7: 4, 1, 1, 1, 1, 1], find the Banzhaf power index for
each voter.
10. Give an example of a weighted voting that is equivalent to the [8: 5, 4, 3].
11. Give an example of a weighted voting system that is equivalent to the [15: 8, 7, 6].
12. Calculate the Shapley-Shubik power index for the weighted voting system [30: 20, 17,
10, 5].
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13. Calculate the Shapley-Shubik power index for the weighted voting system [8: 6, 1, 1, 1,
1, 1].
14. Given the weighted voting system [4: 1, 2, 3], list all winning coalitions.
15. Given the weighted voting system [4: 1, 2, 3], list all blocking coalitions.
16. Given the weighted voting system [16: 3, 9, 4, 5, 10], calculate the Banzhaf power index
for each voter.
17. Given the weighted voting system [14: 8, 2, 5, 7, 4], calculate the Shapley-Shubik power
index for each voter.
18. In the weighted voting system [9: 10, 5, 3], voter A is a dictator.
A) True
B) False
19. 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
20. 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
21. 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.
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22. Given the weighted voting system [9: 6, 4, 2], which of the voters are dummy voters?
A) C only
B) B and C
C) B only
D) None
23. 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)
24. 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)
25. 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)
26. Are the weighted voting systems [8: 5, 4, 3] and [34: 30, 18, 6] equivalent?
A) Yes
B) No
27. Are the weighted voting systems [8: 5, 4, 3] and [11: 10, 9, 2] equivalent?
A) Yes
B) No
28. Are the weighted voting systems [18: 10, 9, 8] and [15: 8, 7, 6] equivalent?
A) Yes
B) No
29. Are the weighted voting systems [11: 10, 9, 2] and [2: 1, 1, 1] equivalent?
A) Yes
B) No
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30. 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)
31. 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)
32. 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)
33. 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)
34. A committee has four voters with the system [30: 20, 17, 10, 5]. Which voters are
critical in the coalition { A, C, D}?
A) A only
B) A and C
C) C and D
D) D only
35. A committee has six members with the system [8: 5, 2, 2, 2, 2, 2]. Which voters are
critical in the coalition { A, C, D}?
A) A only
B) C and D
C) A, C, and D
D) No voter is critical.
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36. Given the weighted voting system [19: 9, 7, 5, 3, 1], which of the voters are dummy
voters?
A) D only
B) D and E
C) E only
D) None
37. A committee has three members with the system [10: 5, 4, 3, 2, 1]. Which voters are
critical in the coalition { A, B, C, E}?
A) A only
B) A and B
C) A and E
D) E only
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Answer Key
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31.
32.
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36.
37.
(A, D) (A, C) (B, C, D) (A, B, C) (A, B, D) (A, C, D) (A, B, C, D)
(A, B) (A, C) (A, B, C) (A, B, D) (A, C, D) (A, D) (B, C) (B, C, D) (A, B, C, D)
(A, B) (A, C) (B, C, D)
(A, B) (A, C) (B, C, D)
(A, B) (A, C) (A, B, C) (A, B, D) (A, C, D) (B, C, D) (A, B, C, D) (A, D) (B, C)
Since the coalition { A, C, D, E} has one extra vote, the only member who is critical is
voter A with weight 3.
(6, 2, 2)
(10, 6, 6, 2)
(32, 20, 20, 20, 20, 20)
Answers may vary. One solution is: [15: 10, 8, 6]
Answers may vary. One solution is: [32: 20, 15, 10]
(5/12, 1/4, 1/4, 1/12)
(2/3, 1/15, 1/15, 1/15, 1/15, 1/15)
{ A, C}, {B, C}, {A, B, C}
{ C}, {A, B}, {A, C}, {B, C}, {A, B, C}
(4, 8, 4, 4, 8)
(7/20, 1/10, 11/60, 4/15, 1/10)
A
D
B
D
A
A
B
C
A
B
B
A
A
B
D
A
B
C
C
B
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