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### ProblemCh1-72

Course: MATH 110, Fall 2009
School: UAB
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Word Count: 190

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Plurality-with-runoff (b) violates monotonicity criterion. Nbr of voters 1st 2nd 3rd 7 A B C 8 B C A 10 C A B 4 A C B Runoff is between A and C. Striking B from table gives result for first place votes: A --- 11, C --- 18. C wins. 1 Now suppose in a re-election, the voters in the last column change the order of A and C. New preference schedule. Nbr of voters 1st 2nd 3rd 7 A B C 8 B C A 14 C A B Now...

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Plurality-with-runoff (b) violates monotonicity criterion. Nbr of voters 1st 2nd 3rd 7 A B C 8 B C A 10 C A B 4 A C B Runoff is between A and C. Striking B from table gives result for first place votes: A --- 11, C --- 18. C wins. 1 Now suppose in a re-election, the voters in the last column change the order of A and C. New preference schedule. Nbr of voters 1st 2nd 3rd 7 A B C 8 B C A 14 C A B Now runoff is between B and C. Results: B --- 15, C --- 14. B wins. This violates the monotonicity C criterion: had the election won. In the re-election all changes were in favor ONLY of C. But C did NOT win the re-election. 2 (c) Plurality-with-runoff violates the Condorcet criterion. Nbr of voters 1st 2nd 3rd 10 A B C 8 C B A 6 B C A The runoff is between A and C, and C wins 14 to 10. Is there a Condorcet candidate? Compare: B to A: 14 to 10. B wins that comparison. B to C: 16 to 8. B wins that comparison. Hence, B is a Condorcet candidate. Thus, this election violates the Condorcet criterion. 3
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