MGF 1107
SUPPLEMENTAL HOMEWORK/ CHAPTER 9
1. What is the minimum number of votes that constitutes a majority if the number of votes cast is:
a) 98
b) 111
c) x, where x is odd.
Express your answer as a single fraction.
d) x, where x is even.
Express your answer as a single fraction.
2. Given the following preference lists, find the winner using plurality with runoff.
10
8
6
2
A
B
C
B
C
C
B
A
B
A A
C
3. What other voting method would follow exactly the same procedure in problem 2 as plurality with runoff?
4. The 1980 U.S. Senate race in New York used the plurality method.
The results were
D’Amato (Rep.)
45%
Holtzman (Dem.)
44%
Javits (Liberal party)
11%
so D’Amato won.
Exit polls showed that twothirds of Javits supporters preferred Holtzman to D’Amato.
If
the plurality with runoff method were used instead, what percentage of the votes would D’Amato and
Holtzman have received in the runoff?
Round off your answer to the nearest tenth of a percent.
5. In an election with 4 alternatives and 25 voters, what is the sum of the Borda points?
6. Find all values of x that make C the Borda count winner but not the Condorcet winner.
10
5
4
x
A
B
C
C
B
C
A
B
C
A
B
A
7. Using the same preference lists as in #6, find all values of x that make B the Borda count winner.
8. Find all values of x for which B is the Borda count winner and D is the plurality with runoff winner.
10
8
6
4
x
A
B
D
D
C
B
C
C
B
D
C
A
A
A
B
D
D
B
C
A
9. You are on a committee that will use sequential pairwise voting to choose from among 4 alternatives: A,
B, C, and D.
Your staff has researched the positions of the other voters so you are familiar with everyone’s
preference lists.
As committee chairperson, you get to decide the agenda.
You are the first of the preference
lists below, and hence want D to win.
Find an agenda that will accomplish your goal.
(Hint: There are 12
possible agendas but only 3 that you should bother trying.)
1
2
1
1
2
D
D
A
C
B
A
C
B
A
A
C
B
C
B
D
B
A
D
D
C
10. An election with 4 candidates (A, B, C, and D) and 150 voters is to be decided using the plurality
method.
After 120 ballots have been recorded, A has 26 votes, B has 18 votes, C has 42 votes, and D has 34
MGF 1107Supplemental Homework Problems Chapter 12
http://www2.fiu.edu/~rosentha/MGF1107/1107HW11.htm
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a) What is the smallest number of the remaining 30 votes that A must receive to guarantee a win (without a
tie) for A?
b) What is the smallest number of the remaining 30 votes that C must receive to guarantee a win (without a
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 Fall '08
 Storfer
 Voting system, Sequential Pairwise Voting, Homework Problems Chapter, MGF 1107Supplemental Homework, 1107Supplemental Homework Problems, C B B D B B C B C

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