918 CHAPTER 14 Voting and Apportionment Our next example demonstrates this unusual occurrence. 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. Example 4 Choosing a New Service The marketing committee at Family Groceries is considering three new services to offer: delivery (D), self-serve checkout (S), and catering (C). To obtain an initial measure of the intentions of the committee, a straw vote is taken and the results are given in Table 14.18. Table 14.18 Marketing Committee Preference Table: First Election (straw vote) Number of Votes 8 9 5 11 First C D C S Second D S S C Third S C D D After several hours of debate and discussion, a second vote is taken. In the second election, the five people who previously voted C, S, D now vote S, C, D. This change has the effect of eliminating the C, S, D column of Table 14.18 and increasing the number voting for S, C, D from 11 to 16. The results of the second election are given in Table 14.19. Table 14.19 Marketing Committee Preference Table: Second Election Number of Votes 8 9 16 First C D S Second D S C Third S C D Now try Exercise 21 a) Using the plurality with elimination method, which service wins the first election? b) Using the plurality with elimination method, which service wins the second election? c) Does this result violate the monotonicity criterion? Solution a) Refer to Table 14.18. Using the plurality with elimination method, the first election (the straw vote) results in delivery being eliminated and in self-serve checkout gaining a majority over catering by a vote of 20 to 13. Thus, self-serve checkout would be chosen as the new service to offer. b) Refer to Table 14.19. Using the plurality with elimination method, the second election results in catering being eliminated and in delivery gaining a majority over self-serve checkout by a vote of 17 to 16. Thus, delivery would be chosen as the new service to offer. c) The five voters who changed their ballots did so in a way to add support to selfserve checkout. However, self-serve checkout’s victory was not repeated in the second election. Therefore, this result violates the monotonicity criterion. This occurrence shows that the plurality with elimination method has the potential to violate the monotonicity criterion. 7
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