14.2 Flaws of the Voting Methods 921 In Example 5, we showed that the pairwise comparison method has the potential to violate the irrelevant alternatives criterion. Note that each voting method we have discussed—the plurality method, the Borda count method, the plurality with elimination method, and the pairwise comparison method—has the potential to violate the irrelevant alternatives criterion. 5. Bacari ties Tompkins (44 to 44), so Bacari gets 1 2 point and Tompkins gets 1 2 point. 6. Sanchez defeats Tompkins (52 to 36), so Sanchez gets 1 point. We have made all 6 possible one-to-one comparisons. Now Bacari has 21 2 points, Sanchez has 2 points, Tompkins has 11 2 points, and Anderson has 0 points. Thus, Bacari is selected. c) The first election produced Sanchez as the winner. Then, even though Mendolson was not the winning candidate, Mendolson’s removal from the election produced Bacari as the winner. Therefore, this result violates the irrelevant alternatives criterion. This example demonstrates that the pairwise comparison method has the potential for violating the irrelevant alternatives criterion. 7 Now try Exercise 25 Example 6 Satisfying the Four Fairness Criteria Suppose that the plurality method is used to determine the winner of an election. The results of the election are given in Table 14.22. Does this method satisfy the four fairness criteria, that is, the majority criterion, the head-to-head criterion, the monotonicity criterion, and the irrelevant alternatives criterion? Now try Exercise 35 Number of Votes 14 6 4 First A B C Second B C B Third C A A Table 14.22 Solution By the plurality method, A is the winner with 14 votes. Since a total of 14 votes is also a majority of the 24 votes cast (58%), the majority criterion is satisfied. Examining pairwise comparison matchups, we see that A is favored to B (14 to 10) and that A is also favored to C (14 to 10). Therefore, the head-to-head criterion is satisfied. Since A holds a majority of first-place votes, the monotonicity criterion is satisfied when a second election is held in which A picks up additional votes; A still wins by plurality. Last, if B or C drops out, A still wins by plurality and the irrelevant alternatives criterion is satisfied. Therefore, this election satisfies all four fairness criteria. 7 We saw in Example 6 that in a particular election, it is possible that all of the fairness criteria are satisfied. However, in a different election, with a different preference table, it is quite possible for the same voting method to violate one or more of the fairness criteria. Table 14.23 summarizes the four fairness criteria, and Table 14.24 summarizes which voting methods always satisfy the criteria. Table 14.23 Summary of Fairness Criteria Majority criterion If a candidate receives a majority of first-place votes, that candidate should be declared the winner. Head-to-head criterion If a candidate is favored when compared individually with every other candidate, that candidate should be declared the winner. Monotonicity criterion A candidate who wins a first election and then gains additional support without losing any of the original support should also win a second election. Irrelevant alternatives criterion If a candidate is the winner of an election and in a second election one or more of the other candidates are removed, the previous winner should still be the winner. Instructor Resources for Section 14.2 in MyLab Math • Objective-Level Videos 14.2 • PowerPoint Lecture Slides 14.2 • MyLab Exercises and Assignments 14.2
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