Math 115

Quiz 1

September 4, 2013                                           Name:                          Key                                                     

 

3

4

2

a

d

b

c

c

a

b

b

c

d

a

d

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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 1st place votes.  Then eliminate b, who has 2 1st 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 1st place votes, with 4, and so wins.

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

B has the 3rd most 1st place votes, 2, and gets 3rd place.

C has the fewest 1st place votes, 0, and gets 4th place.

D-A-B-C

 

 

   b. Again rank the candidates, this time using the recursive Plurality method.

 

D has the most 1st place votes, and so wins.

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

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

D-C-B-A

 

 

 

4.  a.  In this example, did the Borda count violate the Majority criterion?  Carefully justify your answer in 1 or 2 complete sentences.

 

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.