Voting Methods

1. Voting Methods Chapter 12 sec 1

2. Plurality Method • Def. • Each person votes for his or her favorite candidate. The candidate receiving the most votes is declared the winner.

3. Plurality method is the simplest way to determine the outcome of an election; • Many state and local elections use the plurality method because it is easy to determine the winner.

4. Example • The students have organize a union in order to improve their salaries and working conditions. A group of 33 students, have just had and election to choose their president. • Using the plurality method, who won the election?

5. Example

6. Mike won, for he had the most votes. • Mike won the election even though 22/33 = 66.7% of the group voted against him.

7. Borda Count Method • Permits the voter to “fine-tune” his or her vote in the sense that the voter can designate not only a first choice but also a second choice, a third choice, and so on.

8. Borda • In the previous election, we could have used the Borda Method by specifying that on the voter’s ballot the first choice would be 4 points, the second choice 3 points, third choice 2 points, and the fourth choice 1 point.

9. Def. • If there are k candidates in an election, each voter ranks all candidates on the ballot. Then the first choice is given k pts, the second choice is given k-1 pts, the third choice is given k-2 pts, and so on. The candidate who receives the most total pts wins the election.

10. Example • The top numbers are the number of people that voted in the particular preference ballot. The first column means that there were 6 people choose Mike first, Ann second, Ben third, and Zach fourth.

11. To determine the winner we calculate the ballots. Ann = 10x4+18x3+0x2+5x1 = 99 pts Ben = 9x4+0x3+21x2+3x1 = 81 pts Mike = 11x4+7x3+3x2+12x1 = 83 pts Zach = 3x4+8x3+9x2+13x1 = 67 pts

12. We notice that Ann wins the election. • Many polls use the Borda to rank sports teams.

13. Try to calculate the winner • A WINS WITH 78 POINTS

14. Plurality-with-Elimination Method • Def. • Each voter votes for on candidate. A candidate receiving a majority of votes is declared the winner. If no candidate receives a majority of votes, then the candidate(or candidates) with the fewest votes is dropped from the ballot and a new election is held. This process is continues until a candidate receives a majority of votes.

15. Viewing the first place, Zach has the fewest votes so he is eliminated. The remaining candidates move up.

16. By looking at the 3rd column and the last column they are the same, therefore we can combine them.

17. Viewing the first row Ben has the fewest votes, therefore he is eliminated.

18. Combine the same columns together. • Ann wins with 22 votes and Mike has 11 votes.

19. Try this • C wins

20. Pairwise Comparison Method • Def. • Voters first rank all candidates. If A and B are a pair of candidates, we count how many voters prefer A to B and vise versa. Whichever candidate is preferred the most receives 1 pt. If A and B are tied, then each receives ½ pt. Do this comparison, assigning pts., for each pair of candidates. Candidates with the most pts wins.

21. Example • You are a owner of a restaurant and want to add to its menu. You did a survey in which the customers were asked to rank their preferences for (B)urritos, (S)ushi, & (H)amburger.

22. We must compare : • a) S with H, b) S with B, and c) H with B

23. A) S with H • Comparing S with H, we will ignore all references to B • 4, 559 prefer S over H • 3, 697 prefer H over S • Therefore we award S, 1 pt.

24. Compare S with B • 4, 128 prefer S over B • 4, 128 prefer B over S • Therefore S and B are tied, so each receive ½ pt each.

25. Compare H and B • 4, 725 prefer H over B • 3, 531 prefer B over H • Therefore award 1 point to H

26. Who is the winner? • S has1½ pt • H has 1 pt • B has ½ pt • Therefore the winner is S.

27. Try this • Use the pairwise comparison to determine the winner. Also list how many pts each item gets. • H has 2 pts. B has 1 pt. S has 0 pts.