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

2.	Explain why the weighted voting system [13: 10, 6, 5, 3, 2] is not a legitimate weighted voting system. 	

3.	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. 	

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

5.	Given the weighted voting system [51: 45, 43, 7, 5], list all blocking coalitions. 	

6.	Given the weighted voting system [51: 45, 43, 7, 5], list all minimal winning coalitions. 	

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

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

9.	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?	

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

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

12.	A weighted voting system has 12 members.  How many distinct coalitions are there in which exactly seven members vote YES? 	

13.	Given the weighted voting system [5: 3, 2, 1, 1, 1], find which voters of the coalition {A, C, D, E} are critical.	

14.	Given the weighted voting system [8: 5, 4, 3], find the Banzhaf power index for each voter. 	

15.	Given the weighted voting system [14: 10, 6, 5, 3], find the Banzhaf power index for each voter. 	

16.	Given the weighted voting system [7: 4, 1, 1, 1, 1, 1], find the Banzhaf power index for each voter. 	

17.	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? 	

18.	Give an example of a weighted voting that is equivalent to the [8: 5, 4, 3]. 	

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

20.	Is voter C a critical voter in the coalition {A, B, C} of the weighted voting [15: 10, 6, 5, 3, 2]?  Why or why not? 	

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

22.	Evaluate C83.	

23.	Calculate the Shapely-Shubik power index for the weighted voting system [30: 20, 17, 10, 5]. 	

24.	Calculate the Shapely-Shubik power index for the weighted voting system [8: 6, 1, 1, 1, 1, 1].	

25.	There are five distinct three-member voting systems.  Give an example of three of the five.	

26.	A sorority has an executive board consisting of a chair, vice chair, and three other members.  Its voting rules indicate that an issue can pass in two ways: if the chair, vice chair, and one other member support the issue or if one of the chair or vice chair and two other members support the issue.  Express this as a weighted voting system. 	

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

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

29.	A weighted voting system has 10 members. How many distinct coalitions are there in which exactly six members vote yes? 	

30.	Given the weighted voting system [16: 3, 9, 4, 5, 10], calculate the Banzhaf power index for each voter. 	

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

Answer Key

1.	Answers may vary.  One solution is [9: 6, 5, 2]	
2.	The system given is not a legitimate weighted voting system because the quota is exactly half of the total vote weight.  Two different complementary coalitions exist with vote weight total of 13, (A, D) and (B, C, E).	
3.	Answers may vary.  One solution is:  In the [14: 10, 6, 5, 3, 2], the coalition (A, D) is a blocking coalition because (B, C, E) has only 13 votes.  (A, D) would not be a winning coalition by voting "yes" because (A, D) has only 13 votes.	
4.	(A, D)  (A, C)  (B, C, D)  (A, B, C)  (A, B, D)  (A, C, D)  (A, B, C, D)	
5.	(A, B)  (A, C)  (A, B, C)  (A, B, D)  (A, C, D)  (A, D)  (B, C)  (B, C, D)  (A, B, C, D)	
6.	(A, B)  (A, C)  (B, C, D)	
7.	(A, B)  (A, C)  (B, C, D)	
8.	(A, B)  (A, C)  (A, B, C)  (A, B, D)  (A, C, D)  (B, C, D)  (A, B, C, D)  (A, D)  (B, C)	
9.	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)	
10.	No.  A voter with veto power has enough votes to block any measure, but not necessarily enough to pass any issue.  A dictator has enough votes to pass any issue on his or her own.	
11.	10	
12.	792	
13.	Since the coalition {A, C, D, E} has one extra vote, the only critical member is voter A with weight 3.	
14.	(6, 2, 2)	
15.	(10, 6, 6, 2)	
16.	(32, 20, 20, 20, 20, 20)	
17.	The Banzhaf power index of any dummy voter is 0.	
18.	Answers may vary.  One solution is: [15: 10, 8, 6]	
19.	Answers may vary.  One solution is: [32: 20, 15, 10]	
20.	Yes, A is a critical voter.  Without A, the coalition becomes (B, C) which has only 11 votes; no longer enough to win.  The number of extra votes of the coalition is 6.  Voter A has more than this number of votes so voter A is critical.	
21.	A critical voter in a winning or blocking coalition is any voter who has sufficient weight so that the coalition would no longer be winning or blocking the remaining vote.  The order of voters in the coalition does not matter.  There can be more than one critical voter in a coalition.  A pivotal voter is the first voter who joins a coalition and gives that coalition enough votes to win.  Each permutation has exactly one pivotal voter.	
22.	
	
23.	(5/12, 1/4, 1/4, 1/12)	
24.	(2/3, 1/15, 1/15, 1/15, 1/15, 1/15)	
25.	Answers may vary.  One example of each of the five distinct voting systems is:
[3: 3, 1, 1]
[4: 2, 2, 1]
[2: 1, 1, 1]
[3: 2, 1, 1]
[3: 1, 1, 1]	
26.	Answers may vary.  One solution is: [7: 3, 3, 2, 2, 2]	
27.	{A, C}, {B, C}, {A, B, C}	
28.	{C}, {A, B}, {A, C}, {B, C}, {A, B, C}	
29.	210	
30.	(4, 8, 4, 4, 8)	
31.	(7/20, 1/10, 11/60, 4/15, 1/10)