14.2 Flaws of the Voting Methods 923 7. A voting method that always satisfies the majority criterion but may not satisfy any other criterion is the ________ method. Plurality with elimination 8. A voting method that always satisfies the majority criterion and the monotonicity criterion, but may not satisfy any other criterion, is the ________ method. Plurality Practice the Skills/Problem Solving 9. Annual Meeting Members of the board of directors of the National Association of Realtors are voting to select a city to host the association’s annual meeting. The board is considering the following cities: Charleston (C), Seattle (S), and Detroit (D). The preference table for the 19 members of the board of directors is shown below. Show that the Borda count method violates the majority criterion. Detroit wins with the Borda count method, but Seattle has a majority of first-place votes. b) Suppose that the plurality method is used to determine the winner. Is the head-to-head criterion satisfied? Explain. No. C wins by the plurality method, but A is favored when compared to B and to C. Number of Votes 9 6 2 First S D D Second D C S Third C S C m Charleston, SC 10. Hiring a New Director Delaware County Community College is hiring a new director of counseling. The hiring committee ranks the four candidates, Alvarez (A), Brown (B), Singleton (S), and Yu (Y), according to the preference table below. Suppose that the Borda count method is used to determine the winner. Is the majority criterion satisfied? Explain. No. Brown is the Borda count winner, but Alvarez has a majority of first-place votes. Number of Votes 5 4 1 1 First A B B A Second B S Y B Third S Y S Y Fourth Y A A S 11. Restructuring a Company The board of directors at The Limited is considering three different administrative restructuring plans, A, B, and C. The 11 members of the board of directors rank the plans according to the following preference table. a) Which restructuring plan is preferred to all the others in a head-to-head comparison? Plan A is preferred. Number of Votes 2 4 2 3 First B A C C Second A B A B Third C C B A 12. Party Theme The children in Ms. Cohn’s seventh-grade class are voting on a theme for their class party. Their choices are beach (B), the world (W), superheroes (S), or rock and roll (R). The results are shown in the preference table below. a) Which theme is preferred to all the other themes in a head-to-head comparison? Beach is preferred. b) Suppose that the plurality method is used to determine the winner. Is the head-to-head criterion satisfied? Explain. Yes. Beach is the winner by the plurality method. Number of Votes 12 6 4 3 First B S R W Second W B S R Third S W B B Fourth R R W S 13. Residence Hall Improvements The administration at Erie Community College is considering three possible areas of improvement to the school’s residence halls: parking (P), security (S), and lounge areas (L). The nine members of the executive committee of the Student Senate rank the choices according to the preference table shown below. Suppose that the Borda count method is used to determine the winner. Is the head-to-head criterion satisfied? Explain. No. Lounge areas wins by the Borda count method, but parking is favored when compared to security and to lounge areas. Number of Votes 4 1 1 1 2 First P L S S L Second L P L P S Third S S P L P AVAVA/Shutterstock f11photo/Shutterstock
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