1.	Explain the difference between sincere and strategic voting. 	

2.	What is agenda manipulation? 	

3.	According to the Chair's Paradox, what surprising situation can befall the tie-breaking chair of a committee?	

4.	If a voting system has three or more alternatives, satisfies the Pareto condition, always produces a unique winner, and is not a dictatorship, what conclusion follows from the GS theorem?  	

5.	Are there voting methods that are group manipulable though not individual manipulable?	

6.	Are there voting methods that are never manipulable?  	

7.	Are there situations in which the Borda Count method is known to be never manipulable?	

8.	What is purpose of Copeland's Rule with regard to Condorcet's Method of voting?	

9.	A 17-member committee must elect one of four candidates: R, S, T, or W.  According to their preference schedule as shown below, R wins using the plurality method.  Could those members who most prefer W vote strategically in some way to change the outcome in a way that will benefit them?

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	

10.	A 17-member committee must elect one of four candidates: R, S, T, or W.  According to their preference schedule as shown below, R wins using the plurality method.  Could those members who most prefer T vote strategically in some way to change the outcome in a way that will benefit them?

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	

11.	A 17-member committee must elect one of four candidates: R, S, T, or W.  According to their preference schedule as shown below, R wins using the plurality method.  Could those members who most prefer S vote strategically in some way to change the outcome in a way that will benefit them?

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	

12.	A 17-member committee must elect one of four candidates: R, S, T, or W.  Is it possible to manipulate the results of a sequencial pairwise election?

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	

13.	A 17-member committee must elect one of four candidates: R, S, T, or W.  In a plurality runoff election candidate S wins.  What would happen if the four voters who prefer W insincerely voted for T instead?  Is this in their best interests?

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	

14.	A 17-member committee must elect one of four candidates: R, S, T, or W.  In a plurality runoff election candidate S wins.  What would happen if the four voters who prefer T insincerely voted for S instead?  Is this in their best interests?

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	

15.	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.  If the committee uses pairwise sequential voting with the agenda K, L, M, N, applicant K wins.  Can the three voters who least prefer K vote strategically in some way to change the outcome to one they find more favorable?  Why or why not?

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	

16.	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.  If the committee uses pairwise sequential voting with the agenda K, L, M, N, applicant K wins.  Is it possible that another agenda will yield a different winner?

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	

17.	An 11-member committee must choose one of the four applicants K, L, M, and N for membership on the committee using the Borda Count method.  The committee members have preferences among the applicants as given below.  Who wins the election? Can the group of three voters favorably impact the results through insincere voting?

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	

18.	An 11-member committee must choose one of the four applicants K, L, M, and N for membership on the committee using the Borda Count method.  The committee members have preferences among the applicants as given below.  Who wins the election? Can the group of two voters favorably impact the results through insincere voting?

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	

19.	An 11-member committee must choose one of the four applicants K, L, M, and N for membership on the committee using the Borda Count method.  The committee members have preferences among the applicants as given below.  Suppose the group of six suspect that the group of two intends to insincerely exchange M and L in their rankings.  Can the group of six counteract in order to protect K as the winner?  

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	

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.  If the party uses a Borda count, candidate B would be elected.  Can the four voters who most prefer C vote strategically in some way to change this outcome to one they would find more favorable?  Why or why not?

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.	Consider an 11-member committee that must choose one of three alternatives: X, Y, or Z, using the Hare system.  Their schedule of preferences is shown below.  Who wins?  Is it possible for the group of five voters to change the outcome in a way that would benefit them?

Number of Voters	

5	
4	
2	

First choice	
Z	
X	
Y	

Second choice	
Y	
Y	
X	

Third choice	
X	
Z	
Z	

22.	Consider an 11-member committee that must choose one of three alternatives: X, Y, or Z, using the Hare system.  Their schedule of preferences is shown below.  Who wins?  Is it possible for the group of two voters to change the outcome in a way that would benefit them?

Number of Voters	

5	
4	
2	

First choice	
Z	
X	
Y	

Second choice	
Y	
Y	
X	

Third choice	
X	
Z	
Z	

23.	Consider an 11-member committee that must choose one of three alternatives: X, Y, or Z, using the Hare system.  Their schedule of preferences is shown below.  However, the committee suspects that the group of five plans to insincerely reorder their preferences as Y, Z, X.  How can the group of four respond? 

Number of Voters	

5	
4	
2	

First choice	
Z	
X	
Y	

Second choice	
Y	
Y	
X	

Third choice	
X	
Z	
Z	

24.	Twenty-nine voters must choose from among three alternatives: A, B, and C using the Borda Count method.  The voters' preference schedules are shown below. Who wins Borda count? Can the group of six voters change their preference list to produce an outcome they like better?

Number of Voters	

12	
8	
6	
3	

First choice	
B	
C	
A	
C	

Second choice	
C	
A	
B	
B	

Third choice	
A	
B	
C	
A	

25.	Twenty-nine voters must choose from among three alternatives: A, B, and C using the Borda Count method.  The voters' preference schedules are shown below. Suppose the group of voters anticipate that the group of six plan to insincerely rank B above A.  How can the remaining voters respond in their own rankings?

Number of Voters	

12	
8	
6	
3	

First choice	
B	
C	
A	
C	

Second choice	
C	
A	
B	
B	

Third choice	
A	
B	
C	
A	

26.	A group of 22 young people must decide whether to go to the beach (B), the mountain (M), or the zoo (Z) on a field trip.  Their preference rankings are summarized in the table below, and the decision will be made using a Borda count. Who wins the vote?  Can the four voters in the last column change the results of the vote to their favor by changing their preference rankings?

Number of Voters	

10	
8	
4	

First choice	
B	
M	
Z	

Second choice	
M	
B	
M	

Third choice	
Z	
Z	
B	

27.	A group of 22 young people must decide whether to go to the beach (B), the mountain (M), or the zoo (Z) on a field trip.  Their preference rankings are summarized in the table below, and the decision will be made using a Borda count. Who wins the vote?  Can the 10 voters in the first column change the results of the vote to their favor by changing their preference rankings?

Number of Voters	

10	
8	
4	

First choice	
B	
M	
Z	

Second choice	
M	
B	
M	

Third choice	
Z	
Z	
B	

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	

Who wins using plurality?  Could the four voters who most prefer D vote insincerely to change the outcome in a way that would benefit them?	

29.	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	

Who wins using plurality?  Could the four voters who most prefer B vote insincerely to change the outcome in a way that would benefit them?	

30.	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	

Who wins using plurality?  Could the six voters who most prefer C vote insincerely to change the outcome in a way that would benefit them?	

Answer Key

1.	Sincere voting means submitting a ballot that reflects the voter's true preferences.  Strategic voting means submitting a ballot that does not reflect the voter's true preferences but will lead to an outcome the voter likes better than would occur if the voter voted sincerely.	
2.	The ability to control the winner of an election by the selection of a particular agenda.	
3.	The Chair's least favorite candidate can be elected by the group.	
4.	The voting method can be manipulated.	
5.	Yes, plurality for example.	
6.	Yes, Condorcet's Method for example.	
7.	Yes, when there are only 2 or 3 candidates, for example.	
8.	It extends Condorcet's Method to designating a candidate with the best win-loss record the winner when there is no Condorcet's Method winner.	
9.	Yes.  If these four voters ranked candidate T first, T would win with 7 votes.  Since the voters prefer T to R, this outcome would be more desirable.	
10.	Yes.  If these three voters ranked candidate S first, S would win with 7 votes.  Since the voters prefer S to R, this outcome would be more desirable.	
11.	No.	
12.	No.  S will always win.	
13.	T and R would face off in the runoff, and R would win.  In this case, their least favorite candidate would win instead of a higher ranked alternative.	
14.	S and R would face off in the runoff, but S would again win.  They cannot force a win for their first choice, but they can show allegiance to their second choice and eventual winner.	
15.	No.  The six voters who most prefer applicant K represent a majority of the committee.  No matter how the three voters rank the applicants, K will win.	
16.	No.  The six voters who most prefer applicant K represent a majority of the committee.  No matter how the voters are ordered, K will win.	
17.	K currently wins.  Yes.  For example, by exchanging L and N, L will win instead.	
18.	K currently wins.  Yes.  For example, by exchanging L and M, L will win instead.	
19.	Yes.  If the group of two exchange M and L and the group of six exchange L and N, K will still win.	
20.	No.  They do not have enough votes to make C win under any possible ranking.  Their second choice is B, who is the winner.	
21.	X wins.  If the group of five voters exchange their rankings of Y and Z, then Y wins.	
22.	X wins.  The group of two voters cannot change the outcome by insincerely changing their preference ordering.	
23.	If the group of five exchange Y and Z, then Y wins.  The group of four cannot retaliate against this change.	
24.	C wins.   If the group of six exchange A and B in their rankings, then B will win instead.	
25.	B is the winner.  The remaining voters are unable to alter their rankings in order to elect a more favored candidate.	
26.	M wins.  No, they cannot change the results through insincere voting.	
27.	M wins.  Yes; by exchanging M and Z in their rankings, B would win instead.	
28.	A wins.  If the group of four insincerely votes for C instead of D, then C would win.	
29.	A wins.  No.  They cannot force B to win through insincere voting, and A was their second choice.	
30.	A wins.  If the group of six insincerely votes for B instead of C, then B would win.