914 CHAPTER 14 Voting and Apportionment In Section 14.1, we discussed four voting methods: the plurality method, the Borda count method, the plurality with elimination method, and the pairwise comparison method. We discovered that the voting method chosen can be as important to the outcome as the preferences in the voting. Sometimes all four methods will result in the same winner. At other times, the four methods may produce four different winners. Although each of these methods seems to provide a reasonable means for determining a winner, we soon shall see that there are certain flaws with each method. In this section, we will examine four criteria, known as the fairness criteria, that mathematicians and political scientists have agreed that a voting method should meet to be considered fair. The fairness criteria include: 1. Majority criterion 2. Head-to-head criterion 3. Monotonicity criterion 4. Irrelevant alternatives criterion. The first fairness criterion we will discuss is the majority criterion. Majority Criterion If a single candidate is the first choice of a majority (more than 50%) of voters, most voters would agree that that candidate should be declared the winner. If that does not happen with a voting method, that voting method violates the majority criterion. Majority Criterion If a candidate receives a majority (more than 50%) of first-place votes, that candidate should be declared the winner. Example 1 Park Maintenance Company Selection The members of the Fayetteville Town Board are holding an election to select a company to maintain the property at the town park. The choices are Green World Landscaping (G), Lawn and Garden Haven (L), and Jubilant Gardens (J). The 13 board members rank the three choices and then use the Borda count method to make their selection. The preference table is shown in Table 14.15. a) Which company is chosen using the Borda count method? b) Does the winner from part (a) have a majority of first-place votes? Solution a) Using the Borda count method, Lawn and Garden Haven is chosen. Verify this outcome yourself. b) Lawn and Garden Haven has 6 out of 13 first-place votes, which is less than 50% So the winner of the election, Lawn and Garden Haven does not have a majority of first-place votes. In fact, a majority of town board members, 7 out of 13, chose Green World Landscaping. 7 Now try Exercise 9 Table 14.15 Property Maintenance Company Preference Table Number of Votes 7 4 2 First G L L Second L J G Third J G J In Example 1, although a majority of town board members preferred one choice, Green World Landscaping, the Borda count method yielded a different choice, Lawn and Garden Haven. This example demonstrates that the Borda count method has the potential to violate the majority criterion. Why This Is Important Understanding flaws of different voting methods can help determine which method to use for an election.
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