1.	Explain why majority rule is not a good way to choose between four alternatives. 	

2.	Arrow's Impossibility Theorem states that any voting system can give undesirable outcomes.  Explain what this means. 	

3.	Explain the Condorcet Winner Criterion. 	

4.	Explain how the plurality runoff voting system might not choose a winner.	

5.	Determine the number of ways it is possible to rank six candidates if no ties are allowed. 	

6.	Which of the voting procedures, plurality, the Borda count, sequential pairwise voting, or the Hare system, satisfies the Condorcet winner criterion? 	

7.	Which of the voting procedures, plurality, the Borda count, sequential pairwise voting, or the Hare system, satisfies the Pareto condition? 	

8.	Which of the voting procedures, plurality, the Borda count, sequential pairwise voting, or the Hare system, satisfies monotonicity? 	

9.	In order to choose which type of music to listen to in a student center music lounge, a Condorcet vote is held by the 15 students present.  Below are the preference schedules for the students.  Is there a Condorcet winner and, if so, which music type?

Number of Students	

6	
5	
4	

First choice	
classical	
rock	
rock	

Second choice	
jazz	
jazz	
classical	

Third choice	
rock	
classical	
jazz	

10.	Given the preference schedule of 23 voters below, which candidate, if any, wins in a majority-rule election?

Number of Voters	

8	
5	
6	
4	

First choice	
A	
C	
B	
B	

Second choice	
C	
A	
C	
A	

Third choice	
B	
B	
A	
C	

11.	Given the preference schedule of 23 voters below, if a rank method is used, which candidate, if any, wins in a straight plurality election?

Number of Voters	

8	
5	
6	
4	

First choice	
A	
C	
B	
B	

Second choice	
C	
A	
C	
A	

Third choice	
B	
B	
A	
C	

12.	Given the preference schedule of 23 voters below, which candidate, if any, wins if an election is held between A and C and the winner of that race runs against B? Who wins the final election?

Number of Voters	

8	
5	
6	
4	

First choice	
A	
C	
B	
B	

Second choice	
C	
A	
C	
A	

Third choice	
B	
B	
A	
C	

13.	Given below is the preference schedule of 23 voters. If a Borda count is used that assigns 3 points for a first place vote, 2 points for a second place vote, and 1 point for a third place vote, who wins the election?

Number of Voters	

8	
5	
6	
4	

First choice	
A	
C	
B	
B	

Second choice	
C	
A	
C	
A	

Third choice	
B	
B	
A	
C	

14.	Can the four voters in the last column vote strategically to change the outcome of question 13 to one they would like better?  Why or why not? 	

15.	A poll by 15 sports announcers chooses the best basketball team from among three schools: University of Nevada at Las Vegas (LV), University of North Carolina (NC), and Indiana University (IU).  If the individual rankings are as summarized below, which team wins if they use a rank method that assigns 5, 3, and 1 point(s) to each first, second, and third choice respectively?

Number of Announcers	

5	
6	
2	
2	

First choice	
LV	
NC	
IU	
NC	

Second choice	
IU	
LV	
NC	
IU	

Third choice	
NC	
IU	
LV	
LV	

16.	A poll by 15 sports announcers chooses the best basketball team from among three schools: University of Nevada at Las Vegas (LV), University of North Carolina (NC), and Indiana University (IU).  If the plurality method is used, who is selected?  Who is selected using the plurality runoff method?

Number of Announcers	

5	
6	
2	
2	

First choice	
LV	
NC	
IU	
NC	

Second choice	
IU	
LV	
NC	
IU	

Third choice	
NC	
IU	
LV	
LV	

17.	A 17-member committee must elect one of four candidates: R, S, T, or W.  Their preference schedule is shown below.  Which candidate wins under pairwise sequential voting with the agenda S, T, W, R?

Number of Members	

6	
4	
3	
4	

First choice	
R	
S	
T	
W	

Second choice	
S	
R	
S	
T	

Third choice	
T	
T	
R	
S	

Fourth choice	
W	
W	
W	
R	

18.	Seventeen board members vote on four candidates, A, B, C, or D, for a new position on their board.  Their preference schedules are shown below.  Which candidate will be selected if they use the Hare system?

Number of Members	

7	
6	
4	

First choice	
A	
D	
C	

Second choice	
B	
A	
B	

Third choice	
C	
B	
D	

Fourth choice	
D	
C	
A	

19.	Seventeen board members vote on four candidates, A, B, C, or D, for a new position on their board, using the Hare system according to the preference schedules shown below.  What happens if A rejects the offer before the ranking?

Number of Members	

7	
6	
4	

First choice	
A	
D	
C	

Second choice	
B	
A	
B	

Third choice	
C	
B	
D	

Fourth choice	
D	
C	
A	

20.	There are 18 delegates to a political party's convention at which four people A, B, C, and D have been nominated as the party's candidate for governor.  The delegates' preference schedule is shown below.  What nominee would be elected if the party uses a Borda count?

Number of Delegates	

8	
9	
4	

First choice	
A	
B	
C	

Second choice	
B	
A	
B	

Third choice	
C	
D	
A	

Fourth choice	
D	
C	
D	

21.	There are 18 delegates to a political party's convention at which four people A, B, C, and D have been nominated as the party's candidate for governor.  The delegates' preference schedule is shown below.  What nominee would be elected if the party uses a rank system that assigns 5, 4, 1, and 0 point(s) for a first, second, third, and fourth choice, respectively?  Is this result different from that which results from a Borda count?

Number of Delegates	

8	
9	
4	

First choice	
A	
B	
C	

Second choice	
B	
A	
B	

Third choice	
C	
D	
A	

Fourth choice	
D	
C	
D	

22.	There are 18 delegates to a political party's convention at which four people A, B, C, and D have been nominated as the party's candidate for governor.  The delegates' preference schedule is shown below.  Who wins using the plurality method?  Do the results change if the plurality runoff method is used?

Number of Delegates	

8	
9	
4	

First choice	
A	
B	
C	

Second choice	
B	
A	
B	

Third choice	
C	
D	
A	

Fourth choice	
D	
C	
D	

23.	An 11-member committee must choose one of the four applicants K, L, M, and N for membership on the committee.  The committee members have preferences among the applicants as given below.  Who wins under the plurality method?  Who wins under the plurality runoff method?

Number of Members	

6	
2	
3	

First choice	
K	
M	
M	

Second choice	
L	
L	
N	

Third choice	
N	
K	
L	

Fourth choice	
M	
N	
K	

24.	An 11-member committee must choose one of the four applicants K, L, M, and N for membership on the committee.  The committee members have preferences among the applicants as shown below.  Which applicant will be given the position if the members use the Borda count to choose the new member?

Number of Members	

6	
2	
3	

First choice	
K	
M	
M	

Second choice	
L	
L	
N	

Third choice	
N	
K	
L	

Fourth choice	
M	
N	
K	

25.	In how many ways can one rank three candidates if ties are allowed? 	

26.	Consider the following preference table: 

Number of voters	

3	
2	
2	

First choice	
A	
B	
C	

Second choice	
B	
C	
B	

Third choice	
C	
A	
A	

Which candidate will be chosen if majority rule is used? 	

27.	Consider the following preference table:

Number of voters	

6	
3	
4	
2	

First choice	
A	
C	
C	
B	

Second choice	
B	
B	
D	
A	

Third choice	
C	
D	
A	
C	

Fourth choice	
D	
A	
B	
D	

Which candidate will be chosen if plurality voting is used? 	

28.	Consider the following preference table:

Number of voters	

4	
6	
8	
4	

First choice	
D	
C	
A	
B	

Second choice	
C	
B	
D	
A	

Third choice	
B	
D	
C	
C	

Fourth choice	
A	
A	
B	
D	

Which candidate will be chosen if the Borda count is used? 	

29.	Consider the following preference table: 

Number of voters	

3	
2	
2	

First choice	
A	
B	
C	

Second choice	
B	
C	
B	

Third choice	
C	
A	
A	

Which candidate, if any, is the Condorcet winner? 	

30.	After their star pitcher moved to another town, the eight remaining members of the company baseball team needed to select a new pitcher. They used approval voting on the four prospects, and the results are listed below. An “X” indicates an approval vote.
 
Alan	
X	
X	

	
X	

Bob	
X	
	
X	
X	
X	

X	

Chuck	
	
X	
	
X	

X	
X	

David	
	
X	
X	

	
X	
X	

Which pitcher is chosen if just one is to be selected? 	

Answer Key

1.	Majority rule is not a good way to choose among four alternatives because it is possible that none of the four will get a majority of the vote.	
2.	For any voting system it is possible to find a set of voter's preferences that will cause the voting system to violate a condition deemed desirable for a fair voting system.  These conditions may include the Condorcet Winner Criterion and the Independence of Irrelevant Alternatives condition.	
3.	A voting system satisfies the Condorcet Winner Criterion if the winner of an election is also the Condorcet winner, if a Condorcet winner exists.	
4.	There could be a tie among the candidates in the runoff.	
5.	Six candidates can be ranked in 720 ways if ties are not allowed.	
6.	Only sequential pairwise voting satisfies the Condorcet Winner Criterion.	
7.	Plurality, the Borda count, and the Hare system all satisfy the Pareto condition.	
8.	Plurality, the Borda count, and sequential pairwise voting all satisfy monotonicity.	
9.	Rock is the Condorcet winner.	
10.	There is no majority-rule winner.	
11.	B wins in a straight plurality vote.	
12.	A wins.	
13.	A wins.	
14.	No.  They cannot make B win the election and if they switch any rankings to place C higher, then C would win and this is their least desirable outcome.	
15.	North Carolina wins with 51 points.	
16.	North Carolina wins both plurality method and plurality runoff method selections.	
17.	S wins.	
18.	D wins.	
19.	B wins.	
20.	B wins with 51 points.	
21.	With the given rank system, candidate B wins.  Under a Borda Count, candidate B also wins.	
22.	B wins under both plurality and plurality runoff methods.	
23.	K wins under both plurality and plurality runoff.	
24.	Applicant K is chosen.	
25.	Three candidates can be ranked in 13 ways if ties are allowed.	
26.	A	
27.	C	
28.	C	
29.	B	
30.	Bob