908 CHAPTER 14 Voting and Apportionment of Order, the president of any group votes only when there is a tie or to create a tie. There are other ways of breaking a tie that are less arbitrary than flipping a coin. If a tie results from using the Borda count method, it could be broken by choosing the candidate with the most first-place votes. If a tie results from the pairwise comparison method, it could be broken by choosing the winner of the one-to-one comparison between the candidates involved in the tie. Different tie-breaking methods could produce different winners. Therefore, to remain fair, the tie-breaking method should be decided upon in advance. Suppose the University of Georgia Student Government Association (SGA) is trying to decide whom to invite as the keynote speaker for commencement exercises. The choices are David Muir (M), John Krasinski (K), and Tiffany Haddish (H). To make this difficult decision, the SGA decides to hold an election among graduating seniors. It also decides that the plurality method will be used to determine the winner and that the Borda count method will decide the winner in the event that the plurality method leads to a tie. The results of the election are represented in Table 14.13. Who will be asked to speak at the commencement ceremony? Table 14.13 University of Georgia Commencement Speaker Election Number of Votes 400 250 150 First H M M Second K K H Third M H K David Muir and Tiffany Haddish each receive 400 first-place votes; therefore, the plurality method leads to a tie. Thus, the Borda count method is used to break the tie between Muir and Haddish. From Table 14.13, we award the following points: + + = Tiffany Haddish: 400(3) 150(2) 250(1) 1750 + + = DavidMuir: 400(3) 0(2) 400(1) 1600 Since the Borda count method breaks the tie, Tiffany Haddish is asked to present the commencement address. Table 14.14 summarizes the voting methods we discussed in this section. m John Krasinski Table 14.14 Summary of Voting Methods Voting Method Description Plurality method Each voter votes for one candidate. The candidate receiving the most votes is declared the winner. Borda count method Voters rank candidates from the most favorable to the least favorable. Each last-place vote is awarded 1 point, each next-to-last-place vote is awarded 2 points, each third-from-last-place vote is awarded 3 points, and so forth. The candidate receiving the most points is the winner. Plurality with elimination method Each voter votes for one candidate. If a candidate receives a majority of first-place votes, that candidate is declared the winner. If no candidate receives a majority of first-place votes, eliminate the candidate with the fewest number of first-place votes and hold another election. (If there is a tie for the fewest number of first-place votes, eliminate all candidates tied for the fewest number of first-place votes.) Repeat this process until a candidate receives a majority of first-place votes. Pairwise comparison method Voters rank the candidates. A series of comparisons in which each candidate is compared with each of the other candidates follows. If candidate A is preferred to candidate B, A receives 1 point. If candidate B is preferred to candidate A, B receives 1 point. If the candidates tie, each receives 1 2 point. The candidate receiving the most points is declared the winner. Instructor Resources for Section 14.1 in MyLab Math • Objective-Level Videos 14.1 • Animation: Borda Count Method • PowerPoint Lecture Slides 14.1 • MyLab Exercises and Assignments 14.1 S_bukley/Shutterstock
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