14.2 Flaws of the Voting Methods 917 By looking at Example 3, we can conclude that the only voting method that does not have the potential to violate the head-to-head criterion is the pairwise comparison method. Czechanski defeats Davis by a count of 297 to 129 (you should verify these results). Therefore, Czechanski is elected using the plurality with elimination method. Therefore, this result violates the head-to-head criterion. This example demonstrates that the plurality with elimination method has the potential to violate the head-to-head criterion. e) Using the pairwise comparison method, we see that Alvarez is favored over Buchannon, Czechanski, and Davis, thereby giving Alvarez 3 points. Buchannon is favored using head-to-head comparison over Czechanski and Davis, thereby giving Buchannon 2 points. Czechanski is favored over Davis, thereby giving Czechanski 1 point. Davis is not favored over any of the other candidates, thereby giving Davis 0 points. (You should verify these results.) Therefore, Alvarez is elected using the pairwise comparison method. It should be clear that if a certain candidate were favored to all other candidates, then this candidate would be elected using the pairwise comparison method. Therefore, the pairwise comparison method never violates the head-to-head criterion. 7 Now try Exercise 33 At this point in our discussion, we should note the differences between the head-tohead fairness criterion and the pairwise comparison voting method. The head-to-head criterion is one of the fairness criteria used to determine the fairness of a voting method and therefore is not used to determine the winner of an election. The pairwise comparison method is a voting method and is used to determine the winner of an election. When applying the head-to-head criterion, we are trying to determine if there is one candidate who is favored when compared head-to-head with every other candidate. If there is such a candidate, this candidate, according to the head-to-head criterion, should win an election, regardless of which voting method is used to determine the winner. As we have seen in Example 3, it is possible for a candidate to be favored when compared head-tohead with every other candidate yet lose an election. When a candidate is favored when compared with all other candidates and does not win the election, as in Example 3, we say that the voting method used violated the head-to-head criterion. Example 3 highlighted how the head-to-head criterion uncovered flaws in the plurality method, Borda count method, and the plurality with elimination method. Now, we will introduce a third criterion for fair elections, the monotonicity criterion, and use it to uncover an additional flaw in the plurality with elimination method. Monotonicity Criterion The third criterion for fair elections is only relevant when an election is repeated. A repeated election refers to the same number of voters choosing among the same candidates. One example of a repeated election is a straw vote followed by another vote. Frequently, preceding an election, voters discuss the candidates’ strengths and weaknesses. Prior to holding an official election, voters may take a straw vote to gain a preliminary measure of the candidates’ strength. After discussions, some voters may decide to change their preferences. If a candidate gains votes at the expense of the other candidates, this candidate’s chances of winning should increase. If that candidate was already leading, this candidate should still win with additional votes in their favor. The repeated election leads us to our next criterion, the monotonicity criterion. The monotonicity criterion uncovers a strange flaw in the plurality with elimination voting method. As we soon shall see, it is possible for the winner of the first election to gain additional support before the second election and then lose the second election!
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