14.2 Flaws of the Voting Methods 927 32. Electing a Communications Director The City of Phoenix is electing a communications director. The choices for the position are Allen (A), Brady (B), Conrad (C), Davis (D), and Evers (E). The election is conducted, and the preference table is shown below. Number of Votes 4 3 1 1 First A B B A Second B C D B Third C D C D Fourth D A A C Number of Votes 60 40 40 40 20 10 10 First B B E A D C C Second D D D C E B E Third C E A E C D D Fourth E A C B B E B Fifth A C B D A A A Before calculating the results of this election, it is learned that Conrad has dropped out of the race for personal reasons. a) Using the pairwise comparison method, which candidate is chosen if Conrad is included? Davis b) Using the pairwise comparison method, which candidate is chosen if Conrad is eliminated from the preference table? Brady c) Does this second result violate the irrelevant alternatives criterion? Yes 33. Preference for Coffee Twenty- five people are asked to taste and rank four different brands of coffee. The following preference table shows the rankings of the 25 voters. Number of Votes 10 2 5 8 First A B D C Second B C B B Third D A C A Fourth C D A D a) Is there one brand that is favored over all others using a head-to-head comparison? B b) Which brand wins this election if the plurality method is used? A c) Which brand wins this election if the Borda count method is used? B d) Which brand wins this election if the plurality with elimination method is used? C e) Which brand wins this election if the pairwise comparison method is used? B f) For this preference table, which voting method(s), if any, violate the head-to-head criterion? Plurality and Plurality with Elimination 34. Selecting a Club Secretary The 9 members of the Chittenango Sewing Club are electing a club secretary. The candidates for the position are Abdullah (A), Baker (B), Cabrera (C), and DiLaura (D). The following preference table shows the rankings of the 9 voters. a) Is there one candidate who is favored over all others using a head-to-head comparison? A b) Who wins this election if the plurality method is used? A c) Who wins this election if the Borda count method is used? B d) Who wins this election if the plurality with elimination method is used? A e) Who wins this election if the pairwise comparison method is used? A f) For this preference table, which voting method(s), if any, violate the head-to-head criterion? Borda count 35. Plurality Method Suppose the plurality method is used on the following preference table. Does this method satisfy the four fairness criteria? Yes. Number of Votes 12 5 3 First A B C Second B C B Third C A A 36. Borda count Method Suppose the Borda count method is used on the following preference table. Does this method satisfy the four fairness criteria? Yes Number of Votes 10 5 4 First R S T Second S R S Third T T R Challenge Problems/Group Activities 37. Explain why the plurality method always satisfies the monotonicity criterion. A candidate who holds a plurality only gains strength and holds an even larger lead if more favorable votes are added. 38. Construct a preference table with four candidates and four rankings of the candidates, similar to the preference table in Exercise 10, such that the plurality method violates the head-to-head criterion. Have the total number of votes be 25. Answers will vary.
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