C Roettger, Spring 16
Math 105 – Practice Exam 1
Best do these problems only after you are comfortable with the material,
not to learn it. This is to test yourself, so take a stopwatch – you should be
able to do all of these in 60 minutes.
Problem 1 From the preference schedule shown below, find the full ranking
of the candidates A, B, C, D, E, using (please show work)
a) Plurality:
b) Borda Count:
c) Plurality with Elimination:
d) Pairwise Comparison:
Place\ Votes 20 15 7
3
2
1st
A
C
E B D
B
E
D E C
2nd
3rd
C
B
B A B
D
D
C C A
4th
5th
E
A
A D E
Problem 2 Is there a Condorcet winner for the election in Problem 1? If
so, who? If not, why not?
Problem 3 A candidate who loses in head-to-head comparison against every
other candidate is called a Condorcet Loser. Is there a Condorcet loser in
the election in Problem 1? If so, who? If not, why not?
Problem 4 In the preference schedule below, determine the winner using
the Borda Count system.
a) Why is the IIA fairness criterion violated in this example?
b) Why is none of the other three fairness criteria we discussed violated here?
Place\ Votes 10 8
3
1st
B
C A
2nd
A
B C
3rd
C
A B
Problem 5 The sequence of squares 1, 4, 9, 16, 25 . . . has colors Onyx, Zinnia, Onyx, Zinnia, . . . (colors determined by the remainder after division by
4). Use this to find the remainder of 201720172 after division by 4.
Problem 6 Below are some weighted voting systems in generic description.
a) Circle every weight belonging to a dictator.
b) Underline every weight of a player with veto power.
[19 : 7, 6, 6, 5]
[18 : 7, 6, 6, 5]
[15 : 7, 6, 6, 5]
[20 : 25, 11, 3, 2]
[20 : 15, 11, 3, 2]
[16 : 15, 11, 3, 2]