Voting

1. Voting Using Mathematics to Make Choices.

2. Voting Methods 11.1 • Use the plurality method to determine the winner of an election. • Understand the Borda count voting method.

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.

4. The Plurality Method

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.

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.

7. The Borda Count Method

8. The Borda Count Method • 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)

9. The Borda Count Method • Solution: We tally each student’s votes. For example, for Ann we get Ann wins

10. The Borda Count Method

11. The Plurality-With-Elimination Method

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)

13. The Plurality-With-Elimination Method • Solution: After eliminating Doreen we get a new preference table. (continued on next slide)

14. The Plurality-With-Elimination Method Combining identical ballots we get a new preference table. (continued on next 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.

16. The Pairwise Comparison Method

17. The Pairwise Comparison Method • 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. (continued on next slide)

18. The Pairwise Comparison Method • Solution: We first compare T with N. • We award 1 point to T. (continued on next slide)

19. The Pairwise Comparison Method We next compare T with B. T and B each receive one-half point. (continued on next slide)

20. The Pairwise Comparison Method Finally we compare N with B. We have that 2,108 + 1,156 + 1,461 = 4,725 customers prefer N over B. Also, 864 + 1,587 + 1,080 = 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.