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Final+exam+practice+problems

# Final+exam+practice+problems - Review for Final Exam for...

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Review for Final Exam for Math 103, Section 11 5/11/11 1. Problem 1 refers to the election given by the following preference schedule: Number of voters 10 7 5 5 4 1 st choice A D B C B 2 nd choice C B C D C 3 rd choice B A A A D 4 th choice D C D B A a. How many people voted in this election? 10+7+5+5+4=31 b. Is there a majority candidate in this election? No Explain. Need 31/2=15.5 Need 16 votes c. Which candidate wins using the plurality method? A has the most 1 st choice votes. 10 votes. d. Which candidate wins using the Borda count method? Show your work! Number of voters 10 7 5 5 4 1 st choice A 40 D 28 B 20 C 20 B 16 2 nd choice C 30 B 21 C 15 D 15 C 12 3 rd choice B 20 A 14 A 10 A 10 D 8 4 th choice D 10 C 7 D 5 B 5 A 4 A: 40+14+10+10+4=78 B: 20+21+20+5+16=82 C: 30+7+15+20+12=84 D: 10+28+5+15+8=72 C wins with 84 votes! e. Which candidate wins using plurality with elimination? Show your work! A B C D 10 9 5 7 Out 10 9 out 7+5 10 out out 12 10+5 out out 12+4 15 out out 16 D has a majority of 1 st place votes. D wins. f. How many pairwise comparisons are there? Answer: There are 4 candidates. 3+2+1=6 or using (n-1)*n/2=(3*4)/2=6 Which candidate wins using the method of pairwise comparisons? A versus B: 10+5=15 prefer A. 16 prefer B B wins. B gets 1 point. A versus C: 10+7=17 prefer A. 14 prefer B. A wins. A gets 1 point. A versus D: 10+5=15 prefer A. 16 preferD. D wins. D gets 1 point. B versus C: 7+5+4=16 prefer B. 15 prefer C. B wins. B gets 1 point. B versus D: 10+5+4=19 prefer B. 12 prefer D. B wins. B gets 1 point. C versus D: 10+5+5+4=24 prefer C. 7 prefer D. C wins. C gets 1 point.

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In summary, A gets 1 point. B gets 3 points. C gets 1 point. D gets 1 point. B has the most points. B wins the method of pairwise comparisons. g. Is there a Condorcet candidate? Yes Explain. B beats out each of the other candidates head to head. So B is a Condorcet candidate. h. Does this election, under the plurality method, violate the Condorcet criterion? Explain. Yes, By plurality, A wins. But by the Condorcet criterion, B should win. i. Another election is held because candidate D is disqualified. Now which candidate wins using the plurality method? Which fairness criterion is violated by this second election B wins. Independence of irrelevant alternatives j. Another election is held using the plurality method. The monotonicity criterion is violated. Explain what must be true for the monotonicity criterion to be violated. 2.  An election is held in which there are 10 candidates. (Candidates A,B,C,D,E,F,G,H,I,J)  There are 100 voters, and they use the Borda count method. a.   How many Borda count points are given out by one ballot?   1+2+3+…+8+9+10=(10*11)/2=55 b. How many total Borda count points are given out?  One ballot has 55 Borda count points. There are  100 voters, making 100 ballots. So there are 100*55=5500 Borda count points.
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Final+exam+practice+problems - Review for Final Exam for...

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