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Math 103, Section 11 Exam #1-Sample February 27, 2011 Name____________________ Please show your work. There are 11 problems. Formulas: 1+2+3+…+L= L (L+1) 2 Formulas: 1+2+3+…+N-1= (N-1)N 2 Part I: Use the preference schedule below to answer the questions that follow. Number of  voters 27 19 9 15 2 1 st  choice B A D C A 2 nd  choice A D C A C 3 rd  choice D C A D D 4 th  choice C B B B B 1. 20 points a. How many people voted in this election? 27 + 19 + 9 + 15 + 2 = 72 voters b. Rank the candidates using the extended Borda Count method. A: 3*27+4*19+2*9+3*15+4*2= = 228 points 1 st place = A B: 4*27+1*19+1*9+1*15+1*2== 153 points 2 nd place = D C: 1*27+2*19+3*9+4*15+3*2= 158 points 3 rd place = C D: 2*27+3*19+4*9+2*15+2*2= 181 points 4 th place = B c. Use the method of pairwise comparisons to determine the winner of the election. A Wins with 3 points. (B had 0 points, C had 1 point, and D had 2 points.) Compare A:B 19+9+15+2=45 prefer A. 27 prefer B. A gets 1 point. Compare A:C 27+19+2=48 prefer A. 24 prefer C. A gets 1 point. Compare A:D 27+19+15+2=63 prefer A. 9 prefer D. A gets 1 point. Compare B:C 27 prefer B. 19+9+15+2=45 prefer C. C gets 1 point. Compare B:D 27 prefer B. 45 prefer D. D gets 1 point.
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Compare C:D15+2=17 prefer C. 27+19+9=55 prefer D. D gets 1 point. d. Is there a Condorcet candidate? If Yes, which candidate is a Condorcet candidate? Explain your answer. Yes, A is the Condorcet candidate as A beats each of the other candidates in a head-to-head match! e. Use the extended plurality-with-elimination method to rank the candidates. A majority of votes is 72/2+1=36+1=37 votes. First place: C Second place: B Third place: A Fourth place: D A B C D 21 27 15 9 ROUND 1 21 27 15+9=24 OUT ROUND2 OUT 27 24+19+2=45 OUT C wins with a majority of votes. f. Does the plurality-with-elimination method violate the Condorcet criterion? Yes Explain your answer. A should win according to the Condorcet criterion, but C wins using the plurality-with-elimination method. g.  After the election, it is decided for some reason that it is necessary to hold a revote.  What would have to happen to make us conclude that the plurality method violates the  independence of irrelevant alternatives (IIA) criterion? Be as specific as possible. That is, say  which candidate won, etc  Then what happened. B wins by plurality (with 27 votes). A, a non-winning candidate, withdraws. Then D wins 
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This note was uploaded on 01/20/2012 for the course MATH 103 taught by Professor Berkowitz during the Spring '07 term at Rutgers.

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