14.2 Flaws of the Voting Methods 919 In Example 4, we showed that the plurality with elimination method has the potential to violate the monotonicity criterion. Note that the Borda count method also has the potential to violate the monotonicity criterion. We have thus far shown how the majority criterion, the head-to-head criterion, and the monotonicity criterion have all uncovered flaws in our voting methods. Our next criterion, the irrelevant alternatives criterion, demonstrates a potential flaw with the pairwise comparison method. Irrelevant Alternatives Criterion The last criterion we will discuss, the irrelevant alternatives criterion, will address the result of removing a candidate from an election who has no chance of winning. Suppose that four candidates, A, B, C, and D, are involved in an election. The voting takes place and it is determined that candidate B is the winner. However, before the results of the election are announced, it is discovered that candidate C had dropped out of the election. We might conclude that candidate C’s action should not have any effect on the outcome of the election. We shall soon see that such action can negatively affect the winner. We now have our fourth fairness criterion. 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. Example 5 Selecting a Bird Watching Club President The members of the Savannah Bird Watching Club are holding an election to select a club president. The choices are Anderson (A), Bacari (B), Mendolson (M), Sanchez (S), and Tompkins (T). The election results are shown in Table 14.20. Table 14.20 Savannah Bird Watching Club President Preference Table Number of Votes 24 16 16 16 8 4 4 First B B T A S M M Second S S S M T B T Third M T A T M S S Fourth T A M B B T B Fifth A M B S A A A Prior to the announcement of the election results, it is discovered that Mendolson had dropped out of the election. The election officials then decide to eliminate Mendolson’s name from the election results. a) Using the pairwise comparison method, which candidate wins this election if Mendolson is included? b) Using the pairwise comparison method, which candidate wins the election if Mendolson is eliminated from the preference table? c) Does this result violate the irrelevant alternatives criterion? Doug Lemke/Shutterstock
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