Math 115

Quiz 1

September 4, 2013 Name: __ Key __

3 |
4 |
2 |

a |
d |
b |

c |
c |
a |

b |
b |
c |

d |
a |
d |

__ You must ____show all
work ____to receive credit!__

9 People were asked to rank their preferences among 4 candidates. Their preferences are shown in this schedule:

1. Find the winner
using the Borda count method

*A gets 3*4 + 4*1 + 2*3
= 22 points*

*B gets 3*2 + 4*2 + 2*4
= 22 points*

*C gets 3*3 + 4*3 + 2*2
= 25 points*

*D gets 3*1 + 4*4 + 2*1
= 21 points*

*So C is the winner.*

*Note: to quickly check on the arithmetic: There are 10 points on each ballot (1+2+3+4 =
4(5)/2 = 10), and 9 voters, for a total of 90 points.*

*22+22+25+21 = 90, so
the totals have the correct sum, making an error unlikely (there would have to
be multiple errors that cancelled each other out if this turned out to not be
correct)*

2. Find the winner
using the __Plurality with Elimination__ method

First eliminate C, as they have no 1^{st} place
votes. Then eliminate b, who has 2 1^{st} place votes. At this point, A has the support of 5 voters,
and d has the support of 4, so d is eliminated and A is the winner.

3. a. Rank all 4 of the candidates using the __extended____ Plurality__ method.

*D
has the most 1 ^{st} place votes, with 4, and so wins.*

*A has the 2 ^{nd}
most 1^{st} place votes, with 3, and so gets 2^{nd} place.*

*B has
the 3 ^{rd} most 1^{st} place votes, 2, and gets 3^{rd}
place.*

*C
has the fewest 1 ^{st} place votes, 0, and gets 4^{th} place.*

D-A-B-C

b. Again rank the
candidates, this time using the __recursive____
Plurality__ method.

*D has the most 1 ^{st}
place votes, and so wins.*

*If D weren’t present,
C would have 4 1 ^{st} place votes to 3 for a and
2 for b, so C gets 2^{nd} place.*

*If D and C weren’t
present, B would have 6 1 ^{st} place votes to 3 for a, so B gets 3^{rd},
and A gets 4^{th} (last) place.*

D-C-B-A

4. a. In ** this** example, did the

*No. *

*There was no majority
candidate, as no candidate got over ˝ the first place votes, so the majority
criterion couldn’t possibly be violated no matter who won.*