Exam Review Suggestions

•	Given a preference schedule, be able to carry out various voting methods: plurality, Borda count, sequential pairwise, Hare system, and Condorcet method.
•	Determine whether there is a Condorcet winner.
•	Describe a Condorcet paradox.
•	Explain Arrow’s Impossibility Theorem.
•	Know how to explain CWC, IIA, Pareto condition, and monotonicity and show how different voting methods satisfy or fail to satisfy these conditions. 
•	How does approval voting work and what are some good uses of it? 

Sample Problems

1. Given the following set of preference lists, calculate the winner using:
	               
							    			Number of voters
Rank	22	22	21	11	10	8	
first	A	B	C	A	D	C	
second	D	D	B	B	B	A	
third	C	C	D	D	A	D	
forth	B	A	A	C	C	B	

a. plurality voting.

b. the Hare system.

c. the Borda count.

d. sequential pairwise voting with the agenda A, B, C, D.

e. sequential pairwise voting with the agenda B, A, D, C.

2. Use the preference lists below to answer the following questions.

                 Number of Voters		
	10	9	3	2	
First choice	A	B	C	C	
Second choice	B	C	A	D	
Third choice	C	D	D	B	
Fourth choice	D	A	B	A	

a. Find the winner(s) under plurality.

b. Use the agenda CDBA to determine the winner under the sequential pairwise system.  

c. Determine the winner under the Hare system.

3. Use the preference lists below to answer the following questions:

Number of Voters	
	11	9	7	3	
First choice	A	B	B	D	
Second choice	C	A	A	A	
Third choice	B	C	D	C	
Fourth choice	D	D	C	B	

a.	 Who wins under the Borda count?

b.	 Suppose the 9 voters with preference list BACD change their preference list to BDCA.  Who wins under the Borda count?  
                 Number of Voters	
Rank	11	9	7	3	
First choice	A	B	B	D	
Second choice	C	D	A	A	
Third choice	B	C	D	C	
Fourth choice	D	A	C	B	

c.	 Does this example show that Independence of Irrelevant Alternatives (IIA) is violated by the Borda count?  Clearly explain your answer.  

4.
a. In an election, every voter preferred candidate Renee French to candidate Joe Smith. Joe Smith ended up winning the election. Which condition does this violate? Name a voting system in which this can happen.

b. Fred Gylys would have defeated each of his other opponents in a “one-on-one” election, and yet he didn’t win the election. Which condition does this violate? Name a voting system in which this can happen.

c. Candidate Julia Suero wins the election. A new election is held and Peter Joseph wins instead. Yet no voter moved Mr. Joseph ahead of Ms. Suero on their preference list. Which condition does this violate? Name a voting system in which this can happen.

d. Another election is held. This time the only change made is that some voter moved Peter Joseph to a place higher on the preference list. This time he doesn’t win the election. Which condition does this violate? Name a voting system in which this can happen.

5. An election for chairman of the club is held by approval voting for 3 candidates. The votes are shown in the following table, where an X indicates and approval vote. Which candidate wins the election?			
Number of Voters	
Candidates	6	4	4	2	2	1	1	
A	X		X			X		
B		X	X	X				
C	X	X			X