Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 11.1, Slide 1 11 Voting Using Mathematics to Make Choices.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 11.1, Slide 2 Voting Methods 11.1 Use the plurality method to determine the winner of an election. Understand the Borda count voting method.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 11.1, Slide 3 Voting Methods 11.1 Use the plurality-with-elimination method to determine the winner of an election. Determine the winner of an election using the pairwise comparison method.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 4 The Plurality Method

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 5 The Plurality Method Example: A group of 33 students had an election to choose the president of their club. The results of this election are shown. Using the plurality method, who is the winner of this election? Solution: Because Carim has the most votes, he is declared the winner.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 6 The Borda Count Method In the Borda count method, voters rank candidates on their ballots. Such a ballot is called a preference ballot. Votes are tallied and identical ballots are grouped in a table called a preference table.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 7 The Borda Count Method

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 8 Example: Suppose the 33 students had used the Borda count method and obtained the preference table shown. Who is the winner? (continued on next slide) The Borda Count Method

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 9 Solution: We tally each student’s votes. For example, for Ann we get Ann wins The Borda Count Method

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 10 The Borda Count Method

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 11 The Plurality-With-Elimination Method

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 12 The Plurality-With-Elimination Method Example: Suppose the 33 students had used the plurality-with-elimination method with the preference table shown. Who is the winner? (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 13 The Plurality-With-Elimination Method Solution: After eliminating Doreen we get a new preference table. (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 14 The Plurality-With-Elimination Method Combining identical ballots we get a new preference table. (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 15 The Plurality-With-Elimination Method After eliminating Ben we get a new preference table. We see that Ann has 22 first-place votes and Carim has 11, so Ann wins the election.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 16 The Pairwise Comparison Method

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 17 Example: Customers were asked to rank their preferences for (T)acos, (N)achos, and (B)urritos at a restaurant (see table). Using the pairwise comparison method, decide which item is preferred. The Pairwise Comparison Method (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 18 Solution: We first compare T with N. We award 1 point to T. The Pairwise Comparison Method (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 19 We next compare T with B. T and B each receive one-half point. The Pairwise Comparison Method (continued on next slide)

Copyright © 2014, 2010, 2007 Pearson Education, Inc.Section 11.1, Slide 20 Finally we compare N with B. We have that 2, , ,461 = 4,725 customers prefer N over B. Also, , ,080 = 3,531 customers prefer B over N. We award N 1 point. T has 1.5 points, N has 1 point, and B has 0.5 points. T is preferred. The Pairwise Comparison Method