1. Section 14.1
Voting Methods

2. What You Will Learn
Upon completion of this section, you will be able to:
Use the plurality method to determine the winner of an election.
Use the Borda count method to determine the winner of an election.
Use the plurality with elimination method to determine the winner of an election.
Use the pairwise comparison method to determine the winner of an election.
Understand tie-breaking procedures for an election.

3. Example 2: Voting for the Honor Society President
Four students are running for president of the Honor Society: Antoine (A), Betty (B), Camille (C), and Don (D). The club members were asked to rank all candidates. The resulting preference table for this election is given in the table on the next slide.

4. Example 2: Voting for the Honor Society President

5. Example 2: Voting for the Honor Society President
a) How many students voted in the election?
b) How many students selected the candidates in this order: C, A, D, B?
c) How many students selected C as their first choice?

6. Example 2: Voting for the Honor Society President
Solution
a) Add numbers in row labeled Number of Votes
= 54

7. Example 2: Voting for the Honor Society President
Solution
b) From the 2nd column, 15 voted in the given order.

8. Example 2: Voting for the Honor Society President
Solution
c) Read across row that says First, find C, and read number above it. Add those: = 17.

9. Plurality Method
This is the most commonly used method, and it is the easiest method to use when there are more than two candidates.
Each voter votes for one candidate.
The candidate receiving the most votes is declared the winner.

10. Example 4: Electing the Honor Society President by the Plurality Method
Consider the Honor Society election given in Example 2. Who is elected president using the plurality method?

11. Example 4: Electing the Honor Society President by the Plurality Method
Solution
Antoine: 7; Betty: 19; Don: 11;
Camille: = 17
Betty is elected president.
She received 19/54 = 35% of the 1st place votes; not a majority.

12. Borda Count Method
Voters rank candidates from the most favorable to the least favorable.
Each last-place vote is awarded one point, each next-to-last-place vote is awarded two points, each third-from-last-place vote is awarded three points, and so forth.
The candidate receiving the most points is the winner of the election.

13. Example 6: Electing the Honor Society President Using the Borda Count Method
Use the Borda count method to determine the winner of the election for president of the Honor Society discussed in Example 2. Recall that the candidates are Antoine (A), Betty (B), Camille (C), and Don (D). For convenience, the preference table is reproduced on the next slide.

14. Example 6: Electing the Honor Society President Using the Borda Count Method
1st place worth 4, 2nd place worth 3, 3rd place worth 2, 4th place worth 1

15. Example 6: Electing the Honor Society President Using the Borda Count Method
Solution
Antoine: 7 - 1st; 7 × 4 = 28
34 - 2nd; 34 × 3 = 102
13 - 3rd;
13 × 2 = 26
Total 156

16. Example 6: Electing the Honor Society President Using the Borda Count Method
Solution
Betty: st; 19 × 4 = 76
35 - 4th; 35 × 1 = 35
Total 111

17. Example 6: Electing the Honor Society President Using the Borda Count Method
Solution
Camille: st; 17 × 4 = 68
11 - 2nd; 11 × 3 = 33
26 - 3rd;
26 × 2 = 52
Total 153

18. Example 6: Electing the Honor Society President Using the Borda Count Method
Solution
Don: st; 11 × 4 = 44
9 - 2nd; 9 × 3 = 27
15 - 3rd;
15 × 2 = 30
19 – 4th;
19 × 1 = 19
Total 120

19. Example 6: Electing the Honor Society President Using the Borda Count Method
Solution
Antoine, with 156 points, receives the most points using the Borda count method and is declared the winner.

20. Plurality with Elimination
Each voter votes for one candidate.
If a candidate receives a majority of votes, that candidate is declared the winner.

21. Plurality with Elimination
If no candidate receives a majority, eliminate the candidate with the fewest votes and hold another election. (If there is a tie for the fewest votes, eliminate all candidates tied for the fewest votes.)
Repeat this process until a candidate receives a majority.

22. Example 8: Electing the Honor Society President Using the Plurality with Elimination Method
Use the plurality with elimination method to determine the winner of the election for president of the Honor Society from Example 2. The preference table is shown on the next slide. Recall that A represents Antoine, B represents Betty, C represents Camille, and D represents Don.

23. Example 8: Electing the Honor Society President Using the Plurality with Elimination Method

24. Example 8: Electing the Honor Society President Using the Plurality with Elimination Method
Solution
1st place votes: Antoine: 7, Betty:19, Camille: 17, Don: 11

25. Example 8: Electing the Honor Society President Using the Plurality with Elimination Method
Solution
Total of 54 votes. No one has majority.
Antoine is eliminated. We assume all voters rank their preferences the same.
Column 1, 19 voters ranked the four as B, A, C, D; now it is B, C, D.
Column 2 was C, A, D, B; now it is C, D, B.

26. Example 8: Electing the Honor Society President Using the Plurality with Elimination Method
Solution
Column 3 was D, C, A, B; now it is D, C, B.
Column 4 was A, D, C, B; now it is D, C, B.
Column 5 was C, D, A, B; now it is C, D, B.

27. Example 8: Electing the Honor Society President Using the Plurality with Elimination Method
Solution
The new preference table is:

28. Example 8: Electing the Honor Society President Using the Plurality with Elimination Method
Solution
1st place votes:
Betty: 19, Camille: 17, Don: 18

29. Example 8: Electing the Honor Society President Using the Plurality with Elimination Method
Solution
No one has majority. Camille is eliminated. New preference table is:

30. Example 8: Electing the Honor Society President Using the Plurality with Elimination Method
Solution
1st place votes:
Betty: 19, Don: 35

31. Example 8: Electing the Honor Society President Using the Plurality with Elimination Method
Solution
Don has a majority of first-place votes and is declared the winner using the plurality with elimination method.

32. Pairwise Comparison Method
Voters rank the candidates.
A series of comparisons in which each candidate is compared with each of the other candidates follows.
If candidate A is preferred to candidate B, A receives one point. If candidate B is preferred to candidate A, B receives 1 point. If the candidates tie, each receives ½ point.

33. Pairwise Comparison Method
After making all comparisons among the candidates, the candidate receiving the most points is declared the winner.

34. Number of Comparison
The number of comparisons, c, needed when using the pairwise comparison method when there are n candidates is

35. Example 10: Electing the Honor Society President Using the Pairwise Comparison Method
Use the pairwise comparison method to determine the winner of the election for president of the Honor Society that was originally discussed in Example 2.

36. Example 10: Electing the Honor Society President Using the Pairwise Comparison Method
Solution
Antoine, Betty, Camille, Don

37. Example 10: Electing the Honor Society President Using the Pairwise Comparison Method
Solution
4 candidates, n = 4, the number of comparisons needed is
The 6 comparisons are A versus B, A versus C, A versus D, B versus C, B versus D, and C versus D.

38. Example 10: Electing the Honor Society President Using the Pairwise Comparison Method
Solution
1. The pairwise comparison of Antoine versus Betty is
Antoine: = 35 votes
Betty: 19 votes
Antoine wins this comparison and is awarded 1 point.

39. Example 10: Electing the Honor Society President Using the Pairwise Comparison Method
Solution
2. The pairwise comparison of Antoine versus Camille is
Antoine: = 26 votes
Camille: = 28 votes
Camille wins this comparison and is awarded 1 point.

40. Example 10: Electing the Honor Society President Using the Pairwise Comparison Method
Solution
3. The pairwise comparison of Antoine versus Don is
Antoine: = 41 votes
Don: = 13 votes
Antoine wins this comparison and is awarded a second point.

41. Example 10: Electing the Honor Society President Using the Pairwise Comparison Method
Solution
4. The pairwise comparison of Betty versus Camille is
Betty: 19 votes
Camille: = 35 votes
Camille wins this comparison and is awarded a second point.

42. Example 10: Electing the Honor Society President Using the Pairwise Comparison Method
Solution
5. The pairwise comparison of Betty versus Don is
Betty: 19 votes
Don: = 35 votes
Don wins this comparison and is awarded 1 point.

43. Example 10: Electing the Honor Society President Using the Pairwise Comparison Method
Solution
6. The pairwise comparison of Camille versus Don is
Camille: = 36 votes
Don: = 18 votes
Camille wins this comparison and is awarded a third point.

44. Example 10: Electing the Honor Society President Using the Pairwise Comparison Method
Solution
Antoine received 2 points, Betty received 0 points, Camille received 3 points, and Don received 1 point. Since Camille received 3 points, the most points from the pairwise comparison method, Camille wins the election.

45. Tie Breaking
Breaking a tie can be achieved by either making an arbitrary choice, such as flipping a coin, or by bringing in an additional voter.
Robert’s Rule of Order: president of group votes only to break a tie or create a tie.
Borda method: could choose person with most 1st place votes.

46. Tie Breaking
Pairwise comparison method: could choose the winner of a one-to-one comparison between the two candidates involved in the tie.
Different tie-breaking methods could produce different winners.
To remain fair, the method should be chosen in advance.