1.  Prove Lemma 3:  If Dxy is decisive for x over y, then Dxy is decisive for any alternative w over any other alternative z.

2.  Given the profile P

8  5  2 4  2
A B B C C
C C A A B
B A C B A

(a)  Find the winner using the Hare (Single Transferrable Vote) method of voting,

(b)  Construct a new profile from P in which some voters’ preferences are changed by moving the winner in (a) up with the result that the former winner loses.

(c)  Find the winner using the Coombs voting method.

(d)  Can one modify profile P so as to show that the Coombs method also violates monotonicity?

3.(Optional, or may be assigned after Prof. Norman's fourth class.)  A challenge:  Find an example to show that the Coombs method violates monotoncity.
 
 
 
 

4.    This question is based on the handout entitled "Paradoxes of Preferential Voting," in particular the data in Figure 2, which describes what would
have happened if the Smiths had succeeded in getting to the polls.
 
a)  Show that one of the candidates would lose in pairwise comparison with each of the others.  (This is what is called a
"Condorcet loser.")
 
b)  Let us suppose that the data in Figure 2 describe the intentions of the voters as they lined up to vote.  But let's assume that exactly one of the voters,
while standing in line and talking with others, changed her mind, and when she voted gave a different ranking of the candidates.  You are asked to tell
which ranking this voter had initially (in Figure 2) and how she changed it if, as a result, Huff became the winner.  (There may be more than one correct
answer.)
 
c)  Suppose the data in Figure 2 represents the preferences of the voters who are using Approval Voting.   Suppose that 1/3 of the voters vote for their
second choice as well as their first choice.  Which candidate would win?  Which candidate would win if 1/2 of the voters vote for their second choice as
well as their first choice?
 
d)  In the story the Smiths were favoring Mrs. Bitt.  Among the voting methods we examined in the course, and using the data of Figure 2,  is there any
that could result in Mrs. Bitt's winning?