906 CHAPTER 14 Voting and Apportionment The tiger wins the pairwise comparison between the tiger and the kangaroo and is awarded 1 point. The pairwise comparison of the tiger versus the giraffe is + + = + + = Tiger: 2 3 3 8 votes Giraffe: 5 1 1 7 votes The tiger wins the pairwise comparison between the tiger and the giraffe and is awarded another point. The pairwise comparison of the kangaroo versus the giraffe is + + = + + = Kangaroo: 2 1 3 6 votes Giraffe: 3 5 1 9 votes The giraffe wins the pairwise comparison between the kangaroo and the giraffe and is awarded 1 point. Since the tiger received 2 points, the giraffe received 1 point, and the kangaroo received 0 points, using the pairwise comparison method the tiger is chosen for the zoo logo. 7 Now try Exercise 25 Let’s look at how we arrived at the comparisons. Each candidate was compared one-to-one with all the other candidates. In Example 9, we started with the tiger. Since the tiger had to be compared with each of the other candidates, we compared the tiger with the kangaroo and then we compared the tiger with the giraffe. Next, we chose the kangaroo. Since the kangaroo had already been compared with the tiger, we only needed to compare the kangaroo with the giraffe. The last candidate was the giraffe. Since the giraffe was already compared with each of the other candidates, we have made all the necessary comparisons. This process is used regardless of the number of candidates. In Example 9, there were 3 candidates and we made 3 comparisons. The formula to determine the number of comparisons that are needed when n candidates are being considered follows. Number of Comparisons The number of comparisons, c, needed when using the pairwise comparison method when there are n candidates is c n n( 1) 2 = − For example, when there are 5 candidates, n 5 = and the number of comparisons is c n n( 1) 2 5(4) 2 20 2 10 = − = = = Example 10 Electing the Veterans Club President Using the Pairwise Comparison Method Use the pairwise comparison method to determine the winner of the election for president of the Veterans Club that was originally discussed in Example 2. Solution The preference table is shown again in Table 14.12. Recall that A represents Antoine, B represents Betty, C represents Carlos, and D represents Don. Table 14.12 Veterans Club Preference Table Number of Votes 19 15 11 7 2 First B C D A C Second A A C D D Third C D A C A Fourth D B B B B
RkJQdWJsaXNoZXIy NjM5ODQ=