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Unformatted text preview: Math 103, Section 11 Exam #1 March 4, 2011 Name____________________ Please show me all youve learned in the course so far! I know youll do very well! Please show your work. There are 9 problems and 2 extra credit problems. Formulas: 1+2+3++L= L (L+1) 2 Formulas: 1+2+3++N-1= (N-1)N 2 20 points 1. Use the preference schedule below to answer the questions that follow. Number of voters 7 5 4 3 2 1 st choice A 28 B 20 C 16 D 12 D 8 2 nd choice B 21 D 15 D 12 B 9 A 6 3 rd choice C 14 C 10 B 8 A 6 B 4 4 th choice D 7 A 5 A 4 C 3 C 2 1. a. How many people voted in this election? 21 b. Rank the candidates using the extended Borda Count method. First place: B Second place: D Third place: A Fourth place: C A:28+5+4+6+6=49 B:21+20+8+9+4=62 C:14+10+16+3+2=45 D:7+15+12+12+8=54 c. Use the method of pairwise comparisons to determine the winner of the election. A:B 7+2=9 prefer A. 12 prefer B. B gets 1 point. A:C 7+3+2=12 prefer A. 9 prefer C. A gets 1 point. A:D 7 prefer A. 14 prefer D. D gets 1 point. B:C 7+5+3+2=17 prefer B. 4 prefer C. B gets 1 point. B:D 7+5=12 prefer B. 9 prefer D. B gets 1 point. C:D 7+4=11 prefer C. 10 prefer D. C gets 1 points. A has 1 point. B has 3 points. C has 1 point. D has 1 point. B is the winner with 3 points. In fact, B is a Condorcet candidate since B beat each of the other candidates, head to head. d. Is there a Condorcet candidate? B If Yes, which candidate is a Condorcet candidate? Explain your answer. B is a Condorcet candidate since B beat each of the other candidates, head to head. See part c. e. Use the extended plurality-with-elimination method to rank the candidates. First place: D Second place: A Third place: B Fourth place:C A B C D 7 5 4 5 Round 1 7 5 Out 5+4=9 Round 2 7 Out Out 9+5=1 4 D has 14, a majority. So D wins by plurality with elimination. f. In this election, which, if any, of the 4 voting methods violate the Condorcet criterion? Explain. Any of the 4 voting methods that have a winner other than candidate B. The plurality winner is A. So the plurality method violates the Condorcet criterion. The Borda count method winner is B. So the Borda count method does not violate the Condorcet criterion. The pairwise comparisons method winner is B. So the pairwise comparisons method does not violate the Condorcet criterion. The plurality with elimination method winner is D. So the plurality with elimination method does violate the Condorcet criterion. In summary the plurality and plurality with elimination methods violate the Condorcet criterion. g. In this election, which, if any, of the 4 voting methods violate the majority criterion? Explain. There is no majority candidate. So none of the 4 voting methods violate the majority criterion....
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