Review
Show all of your work to receive credit. Guesses will receive no credit, even if they are correct!
Chapter 1)
Number of votes
st
1 choice
2nd choice
3rd choice
3
4
5
6
B
A
C
B
C
A
C
B
A
A
C
B
1) How many people voted in this election?
2) Find the plurality candidate.
3) Find the Borda winner.
4) Find the plurality with elimination candidate.
5) Find the pairwise comparison candidate
6) "If there is a choice that has a majority of the first-place votes in an election, then that
choice should be the winner of the election." This fairness criterion is called the
a) majority criterion
b) monotonicity criterion
c) independence of irrelevant alternatives criterion
d) Condorcet criterion
e) None of the above
7) "If in an election there is a Condorcet candidate, then such a candidate should be the
winner of the election." This statement is another way to phrase the
a) monotonicity criterion
b) independence of irrelevant alternatives criterion
c) Condorcet criterion
d) majority criterion
e) None of the above
8) What does the Independence of Irrelevant Alternatives criterion say?
9) What does the Monotonicity criterion say?
Chapter 2)
1) How many comparisons are there for 10 candidates (use the formula from Chapter 1 to receive
credit)?
2) Find the sum 1+2+3+…+14 (again, use the appropriate formula from Chapter 1 to receive
credit).
3) How many players are there in the weighted voting system [10: 7,3,1,1]?
4) What is the quota in the weighted voting system [10: 7,3,1,1]?
5) In the weighted voting system [9: 4,4,2]
a) P1 is a dictator
b) P1 and P2 are the only players with veto power
c) P3 is a dictator
d) every player has veto power
e) None of the above
6) In the weighted voting system [q: 10,3,2], what is the smallest and largest possible value q can
take?
7) Say you are making a sandwich with 1 of 3 types of bread and 1 of 5 types of meat. How many
possible sandwiches could you make?
8) How many sequential coalitions are possible for players P1 , P2 , P3 (to get credit you must make
use of the appropriate formula from Chapter 2)?
9) How many coalitions are possible for players P1 , P2 , P3 (to get credit you must make use of the
appropriate formula from Chapter 2)?
10) Find the Banzhaf distribution for the weighted voting system [6: 4,2,1].
11) Find the Shapley-Shubik distribution for the weighted voting system [6: 4,2,1].
12) Simplify and evaluate the following
a) 11!/10!
b) 10!/8!
c) 10!/11!
d) 5!
Chapter 3)
1) Jack buys a chocolate and strawberry cake for $20. He values chocolate 4 times as much
as strawberry.
Chocolate
Strawberry
a) What is the value of the chocolate half to Jack?
b) What is the value of the strawberry half to Jack?
c) What is the value of the below piece to Jack (setup, do not simplify)?
Choc.
60
Straw.
40
2) Now suppose Jack and Lisa value a chocolate strawberry cake for $20. They decide to split
the cake using the divider-chooser method. He values the chocolate 4 times more than the
strawberry and she values the strawberry 3 times more than the chocolate.
Strawberry
Chocolate
a) If Jack were the divider, how would he cut it (assuming he does not cut down the middle,
and setup only)?
b) If Lisa were the divider, how would she cut it (assuming she does not cut down the middle,
and setup only)?
c) If Jack is the divider and Lisa is the chooser, give the value (in dollars) of Jack and Lisa’s
fair share from each of their respective values.
d) If Lisa is the divider and Jack is the chooser, give the value (in dollars) of Lisa and Jack’s
fair share from each of their respective values.
3) 3 players must divide a cake that is divided into 3 slices. If the chart represents the value
of the pieces from the perspective of Abby, Bob, and Carol
Abby
Bob
Carol
a)
b)
c)
d)
S1
S2
S3
30%
30%
40%
40%
35%
33.34%
30%
35%
26.66%
find a fair share for A.
find a fair share for B.
find a fair share for C.
describe a fair division of the cake.
4)
$15
Strawberry
$0
Strawberry
$5
Strawberry
Chocolate
$5
Chocolate
$20
Chocolate
$15
Abby
Bob
Carol
Suppose Abby, Bob, and Carol are going to divide a $20 cake using the Lone Chooser method.
Suppose Carol is the chooser. Abby and Bob flip a coin and Abby is the first divider. Describe a
fair division of the cake using the Lone Chooser method and find the respective value of each
person’s fair share.