912 CHAPTER 14 Voting and Apportionment Sprint Distance Relay Medley First C R R T Second W T W R Third T W T W Fourth C W W C Choosing a Computer In Exercises 51–54, the Wizards Computer Club is ordering five new computers. To obtain a discount price, the club has decided that all five should be the same brand. The choices they are considering are Apple (A), Hewlett-Packard (H), Dell (D), and Lenovo (L). The 142 members’ choices are reflected in the following preference table. 62. A Lost Preference Table A ski club is having an election for president. There are three candidates and 15 voters. Each voter casts a ballot listing their preferences. Just before the winner is to be announced, the previous president announces that the preference table has been lost. The only information he can remember from the preference table is that there were only two columns in the table. a) Explain why there must be a majority winner. * b) Explain why there must always be a majority winner, regardless of the number of candidates, when all voters vote, when there are an odd number of voters and only two columns in the preference table. * 63. Cat Competition Friskies Cat Food is looking for a new cat for the cover of their 2025 calendar. Four cats are finalists: A, B, C, and D. They are ranked first, second, third, and fourth in each of the following four categories: looks, eye color, size, and attitude. The preference table is shown below. Number of Votes 43 30 29 26 14 First L A H D D Second A D D H H Third H H A L A Fourth D L L A L 51. Determine which brand of computer is selected using the pairwise comparison method. Apple 52. Determine which brand of computer is selected using the plurality with elimination method. Dell 53. Determine which brand of computer is selected using the Borda count method. Hewlett-Packard 54. Determine which brand of computer is selected using the plurality method. Lenovo 55. What does this example demonstrate about the need for choosing a voting method prior to the election? You must choose the voting method to use before the election takes place. Concept/Writing Exercises 56. Describe a benefit of the Borda count method over the plurality method. Voters are able to provide more information, such as ranking choices, with the Borda count method. 57. Describe one way other than flipping a coin to settle a tied election. One way to settle a tie in an election is to bring in an additional voter. Other answers are possible. Challenge Problems/Group Activities 58. Another way to determine the winner if the plurality with elimination method is used is to eliminate the candidate with the most last-place votes at each step. Using the preference table given for Exercises 31–34, determine the winner if the plurality with elimination method is used and the candidate with the most last-place votes is eliminated at each step. Choi 59. Determine the winner using the plurality with elimination method, with the preference table for Exercises 35–38, if the candidate with the most last-place votes is eliminated at each step. Erie Road 60. Determine the winner using the plurality with elimination method, with the preference table for Exercises 43– 46, if the candidate with the most last-place votes is eliminated at each step. Friends 61. Determine the winner using the plurality with elimination method, with the preference table for Exercises 47–50, if the candidate with the most last-place votes is eliminated at each step. Petunias Categories Cat Looks Eyes Size Attitude A 4 3 2 3 B 2 4 1 2 C 1 1 4 4 D 3 2 3 1 a) Which cat wins if a Borda count method is used that awards 4 points, 3 points, 2 points, and 1 point for first, second, third, and fourth place, respectively, in each category? Tie between B and D b) Which cat wins if a modified Borda count method is used where 5, 3, 1, and 0 points are awarded for first, second, third, and fourth place, respectively, in each category? B 64. Ranking the Swim Teams The results of a swim meet between the Comets (C), Rams (R), Wildcats (W), and Tigers (T) are shown in the following preference table. Notice, for example, that in the sprint competition, the Comets (C) took both first and fourth place. *See Instructor Answer Appendix Chad McDermott/123RF
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