1.	Consider the following distribution of 33 voters at eight different positions over the interval [0, 1].
 
Position i	
1	
2	
3	
4	
5	
6	
7	
8	

Location li 
  of position i	
0.1	
0.3	
0.4	
0.5	
0.6	
0.7	
0.8	
0.9	

Number of voters
  at position i	

3	

4	

7	

2	

3	

6	

4	

4	

What is the equilibrium position? 	

2.	Consider the following distribution of 33 voters at eight different positions over the interval [0, 1].
 
Position i	
1	
2	
3	
4	
5	
6	
7	
8	

Location li 
  of position i	
0.1	
0.3	
0.4	
0.5	
0.6	
0.7	
0.8	
0.9	

Number of voters
  at position i	

3	

4	

7	

2	

3	

6	

4	

4	

What is the mean? 	

3.	Consider the following distribution of 33 voters at 8 different positions over the interval [0, 1].
 
Position i	
1	
2	
3	
4	
5	
6	
7	
8	

Location li 
  of position i	
0.1	
0.3	
0.4	
0.5	
0.6	
0.7	
0.8	
0.9	

Number of voters
  at position i	

3	

4	

7	

2	

3	

6	

4	

4	

What is the median? 	

4.	Suppose there are three classes of voters that rank four candidates as shown below. For each class of voters, the preferred subset of candidates is enclosed in the first set of parentheses and the non-preferred subset in the second set of parentheses.  Thus, the seven class I voters prefer B and D, between whom they are indifferent, to A and C, between whom they are also indifferent: 
 
I.  7:  (B D) (A C)
II.  6:  (A D) (B C)
III.  5:  (B C) (A D)
 
Assuming that each class of voters chooses its dominant strategy, who wins? 	

5.	Suppose there are four classes of voters that rank four candidates as shown below. For each class of voters, the preferred subset of candidates is enclosed in the first set of parentheses and the non-preferred subset in the second set of parentheses.  Thus, the four class I voters prefer A and C, between whom they are indifferent, to B and D, between whom they are also indifferent: 
 
I.  4:  (A C) (B D) 
II.  3:  (A B D) (C)
III. 7:  (B) (A C D)
IV. 5: (C) (A B D)
 
Assuming that each class of voters chooses its dominant strategy, who wins? 	

6.	Give an example of a voter distribution over the interval [0, 1] that is skewed to the left.	

7.	Explain the justification for the electoral college. 	

8.	Describe the composition of the electoral college. 	

9.	Assume there are three states with 6, 8, and 23 voters, and they are all toss-up states.  If both the Democratic and Republican candidates choose strategies that maximize their expected electoral vote (the 3/2's rule), and they have the same total resources (D = R = 100), what resources should be allocated for each state? 	

10.	Assume there are three states with 6, 8, and 23 voters, and they are all toss-up states. If both the Democratic and Republican candidates choose strategies that maximize their expected popular vote (the proportional rule), and they have the same total resources (D = R = 100), what resources should be allocated for each state? 	

11.	Assume there are three classes of voters that rank three candidates as follows:
 
I.  3:  A B C
II.  12:  C A B
III.  10:  B A C 

Which candidate is the Condorcet winner? According to the poll assumption, who will win this election? How does this compare to the desires of the class I and III voters who, together, are a majority? 	

12.	Assume there are three classes of voters that rank three candidates as follows:
 
I.  3:  A B C
II.  7:  C A B
III.  5:  B C A

Which candidate is the Condorcet winner? According to the poll assumption, who will win this election? How does this compare to the desires of the class I and III voters who, together, are a majority? 	

13.	Assume there are four classes of voters that rank four candidates as follows:
 
I.  5:  C B A D
II.  3:  B C A D
III.  6:  D C B A
IV.  2:  B C D A

Which candidate is the plurality winner? 	

14.	Assume there are four classes of voters that rank four candidates as follows:
 
I.  5:  C B A D
II.  3:  B C A D
III.  6:  D C B A
IV.  2:  B C D A

Which candidate is the Condorcet winner? 	

15.	Assume there are four classes of voters that rank four candidates as follows:
 
I.  5:  C B A D
II.  3:  B C A D
III.  6:  D C B A
IV.  2:  B C D A

Suppose there is a poll that differentiates the top two candidates.  Which candidate will win the election after the poll? 	

16.	Describe the spoiler problem. 	

17.	What is the bandwagon effect? 	

18.	What makes a candidate a Condorcet candidate? 	

19.	Describe the expected popular vote (EPV). 	

20.	Describe the expected electoral vote (EEV). 	

21.	Define an equilibrium position. 	

22.	What is approval voting? 	

23.	Describe the 2/3-separation opportunity. 	

24.	Describe the 1/3-separation obstacle. 	

25.	Explain the median-voter theorem. 	

26.	Explain the poll assumption. 	

27.	How is strategic voting different from sincere voting?	

28.	Give an example of a set of five data that is bimodal. 	

29.	Give an example of five data that is unimodal.. 	

30.	Suppose voters are distributed so that they are only located at certain positions along the left-right continuum.  What is this type of distribution called?	

Answer Key

1.	0.5	
2.	0.55	
3.	0.5	
4.	D	
5.	B	
6.	Answers will vary. One example is shown below:
 
Position i	
1	
2	
3	
4	
5	
6	
7	
8	

Location li 
  of position i	
0.2	
0.3	
0.4	
0.5	
0.6	
0.7	
0.8	
0.9	

Number of voters
  at position i	

3	

6	

7	

4	

3	

4	

4	

3	

7.	The electoral college was designed to place the selection of a president in the hands of a body that, while its members would be chosen by the people, would be sufficiently removed from them that it could make more deliberative choices.	
8.	Each state gets one electoral vote for each of its two senators. Each state also receives one electoral vote for each of its representatives in the House of Representatives, the number of which is based on population. The District of Columbia is given three electoral votes.	
9.	10, 15, and 75	
10.	16, 22, and 62	
11.	The Condorcet winner is A. But A is not one of the top two candidates, so the plurality winner C will win the election. However, the class I and III voters together picked C as their least favorite candidate.	
12.	There is no Condorcet winner, so C will win the election. However, the class I and III voters together prefer B over C.	
13.	D	
14.	C	
15.	C	
16.	The spoiler problem is caused by a candidate who cannot win but “spoils” the election for a candidate who otherwise would win.	
17.	Voting for a candidate not on the basis of merit but, instead, because of the expectation that he or she will win.	
18.	A Condorcet candidate is the candidate who can defeat each of the other candidates in pairwise contests.	
19.	In toss-up states, the number of voters in each toss-up state, multiplied by the probability that that voter votes for the Democratic (or Republican) candidate, summed across all toss-up states.	
20.	In toss-up states, the number of electoral votes of each toss-up state, multiplied by the probability that the Democratic (or Republican) candidate wins more than 50% of the popular votes in that state, summed across all toss-up states.	
21.	A position is in equilibrium if no candidate has an incentive to depart from it unilaterally.	
22.	Approval voting allows voters to vote for as many candidates as they like or find acceptable.  Each candidate approved of receives one vote, and the candidate with the most approval votes wins.	
23.	If candidates A and B choose distinct positions that are equidistant from the median of a symmetric unimodal distribution and at least 2/3 of the voters lie between A and B, C can defeat both A and B by taking a position exactly between them.	
24.	If candidates A and B choose distinct positions that are equidistant from the median of a symmetric distribution and no more than 1/3 of the voters lie between A and B, C can take no position that will displace both A and B and enable C to win.	
25.	In a two-candidate election, the median is the unique equilibrium position.	
26.	Voters adjust, if necessary, their sincere voting strategies to differentiate between the top two candidates (as revealed in the poll) by voting for the one they prefer.	
27.	Strategic voting might not be sincere, but instead aims to elect an acceptable candidate if the first choice is not viable.	
28.	Answers will vary.  For example, {3, 3, 5, 8, 8}.	
29.	Answers will vary.  For example, {3, 3, 3, 6, 7}.	
30.	A discrete distribution of voters.