920 CHAPTER 14 Voting and Apportionment Solution a) Using the pairwise comparison method, since we have five candidates, n 5 = and the number of pairwise comparisons needed is = − = = c n n( 1) 2 5(4) 2 10 We will do the comparisons in the following order: Anderson versus Bacari, Anderson versus Mendolson, Anderson versus Sanchez, Anderson versus Tompkins, Bacari versus Mendolson, Bacari versus Sanchez, Bacari versus Tompkins, Mendolson versus Sanchez, Mendolson versus Tompkins, and Sanchez versus Tompkins. These are the 10 possible comparisons. The following results can be verified from Table 14.20. 1. Bacari defeats Anderson (56 to 32), so Bacari gets 1 point. 2. Anderson defeats Mendolson (48 to 40), so Anderson gets 1 point. 3. Sanchez defeats Anderson (72 to 16), so Sanchez gets 1 point. 4. Tompkins defeats Anderson (72 to 16), so Tompkins gets 1 point. 5. Mendolson defeats Bacari (48 to 40), so Mendolson gets 1 point. 6. Bacari defeats Sanchez (60 to 28), so Bacari gets 1 point. 7. Bacari ties Tompkins (44 to 44), so Bacari gets 1 2 point and Tompkins gets 1 2 point. 8. Sanchez defeats Mendolson (64 to 24), so Sanchez gets 1 point. 9. Mendolson defeats Tompkins (48 to 40), so Mendolson gets 1 point. 10. Sanchez defeats Tompkins (52 to 36), so Sanchez gets 1 point. We have made all possible one-to-one comparisons. Sanchez has 3 points, Bacari has 21 2 points, Mendolson has 2 points, Tompkins has 1 1 2 points, and Anderson has 1 point. Therefore, Sanchez is the winner when Mendolson is included. b) Once Mendolson is eliminated from consideration, there is a new election and a new preference table. In the new election we will assume that the voters voted in the same order as in the first election. The new preference table is obtained by eliminating Mendolson from Table 14.20 and keeping the remaining candidates in the same order. See Table 14.21. Since we now have four candidates, n 4 = and the number of comparisons needed is c n n( 1) 2 4(3) 2 6 = − = = Table 14.21 Savannah Bird Watching Club President Preference Table, After Mendolson Drops Out Number of Votes 24 16 16 16 8 4 4 First B B T A S B T Second S S S T T S S Third T T A B B T B Fourth A A B S A A A The pairwise comparison method, using Table 14.21, produces the following 6 results: 1. Bacari defeats Anderson (56 to 32), so Bacari gets 1 point. 2. Sanchez defeats Anderson (72 to 16), so Sanchez gets 1 point. 3. Tompkins defeats Anderson (72 to 16), so Tompkins gets 1 point. 4. Bacari defeats Sanchez (60 to 28), so Bacari gets 1 point.
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