1. Copyright © 2009 Pearson Education, Inc. Chapter 15 Section 1 - Slide 1 5-1 Election Theory Voting Methods

2. Chapter 15 Section 1 - Slide 2 Copyright © 2009 Pearson Education, Inc. WHAT YOU WILL LEARN Preference tables Voting methods

3. Chapter 15 Section 1 - Slide 3 Copyright © 2009 Pearson Education, Inc. Example: Voting Voting for Math Club President: Four students are running for president of the Math Club: Jerry, Thomas, Annette and Becky. The club members were asked to rank all candidates. The resulting preference table for this election is shown on the next slide. a) How many students voted in the election? b) How many students selected the candidates in this order: A, J, B, T? c) How many students selected A as their first choice?

4. Chapter 15 Section 1 - Slide 4 Copyright © 2009 Pearson Education, Inc. Example: Voting (continued) a) How many students voted in the election? Add the row labeled Number of Votes 14 + 12 + 9 + 4 + 1 = 40 Therefore, 40 students voted in the election. TTTTBFourth A B J 4 JBJAThird BJAJSecond AABTFirst 191214 # of Votes

5. Chapter 15 Section 1 - Slide 5 Copyright © 2009 Pearson Education, Inc. Example: Voting (continued) b) How many students selected the candidates in this order: A, J, B, T? 3 rd column of numbers, 9 people TTTTBFourth A B J 4 JBJAThird BJAJSecond AABTFirst 191214 # of Votes

6. Chapter 15 Section 1 - Slide 6 Copyright © 2009 Pearson Education, Inc. Example: Voting (continued) c) How many students selected A as their first choice? 9 + 1 = 10 TTTTBFourth A B J 4 JBJAThird BJAJSecond AABTFirst 191214 # of Votes

7. Chapter 15 Section 1 - Slide 7 Copyright © 2009 Pearson Education, Inc. 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.

8. Chapter 15 Section 1 - Slide 8 Copyright © 2009 Pearson Education, Inc. Example: Plurality Method Who is elected math club president using the plurality method? We will assume that each member would vote for the person he or she listed in first place. TTTTBFourth A B J 4 JBJAThird BJAJSecond AABTFirst 191214 # of Votes

9. Chapter 15 Section 1 - Slide 9 Copyright © 2009 Pearson Education, Inc. Example: Plurality Method (continued) Thomas would be elected since he received the most votes. Note that Thomas received 14/40, or 35%, of the first-place votes, which is less than a majority. Thomas received 14 votes Becky received 12 votes Annette received 10 votes Jerry received 4 votes

10. Chapter 15 Section 1 - Slide 10 Copyright © 2009 Pearson Education, Inc. 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.

11. Chapter 15 Section 1 - Slide 11 Copyright © 2009 Pearson Education, Inc. Example: Borda Count Use the Borda count method to determine the winner of the election for math club president. Since there are four candidates, a first-place vote is worth 4 points, a second-place vote is worth 3 points, a third-place vote is worth 2 points, and a fourth-place vote is worth 1 point.

12. Chapter 15 Section 1 - Slide 12 Copyright © 2009 Pearson Education, Inc. Example: Borda Count (continued) Thomas 14 first place votes 0 second place 0 third place 26 fourth place 14(4) + 0 + 0 + 26(1) = 82 Annette 10 first place votes 12 second place 18 third place 0 fourth place 10(4) + 12(3) + 18(2) + 0 = 112 TTTTBFourth A B J 4 JBJAThird BJAJSecond AABTFirst 191214 # of Votes

13. Chapter 15 Section 1 - Slide 13 Copyright © 2009 Pearson Education, Inc. Example: Borda Count (continued) Betty 12 first place votes 5 second place 9 third place 14 fourth place 12(4) + 5(3) + 9(2) + 14 = 95 Jerry 4 first place votes 23 second place 13 third place 0 fourth place 4(4) + 23(3) + 13(2) + 0 = 111 TTTTBFourth A B J 4 JBJAThird BJAJSecond AABTFirst 191214 # of Votes

14. Chapter 15 Section 1 - Slide 14 Copyright © 2009 Pearson Education, Inc. Example: Borda Count (continued) Thomas - 82 Annette - 112 Betty - 95 Jerry - 111 Annette, with 112 points, receives the most points and is declared the winner.

15. Chapter 15 Section 1 - Slide 15 Copyright © 2009 Pearson Education, Inc. Plurality with Elimination Each voter votes for one candidate. If a candidate receives a majority of votes, that candidate is declared the winner. 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.

16. Chapter 15 Section 1 - Slide 16 Copyright © 2009 Pearson Education, Inc. Example: Plurality with Elimination Use the plurality with elimination method to determine the winner of the election for president of the math club. Count the number of first place votes Annette 10 Betty 12 Thomas 14 Jerry 4 TTTTBFourth A B J 4 JBJAThird BJAJSecond AABTFirst 191214 # of Votes

17. Chapter 15 Section 1 - Slide 17 Copyright © 2009 Pearson Education, Inc. Example: Plurality with Elimination (continued) Since 40 votes were cast, a candidate must have 20 first place votes to receive a majority. Jerry had the fewest number of first place votes, so he is eliminated. Redo the table. Thomas 14 Annette 10 Betty 16 T A B 4 TTTBThird BBAASecond AABTFirst 191214# of Votes

18. Chapter 15 Section 1 - Slide 18 Copyright © 2009 Pearson Education, Inc. Example: Plurality with Elimination (continued) Still, no candidate received a majority. Annette has the fewest number of first-place votes, so she is eliminated. New preference table Betty 26 Thomas 14 Betty is the winner. T B 4 TTTBSecond BBBTFirst 191214# of Votes

19. Chapter 15 Section 1 - Slide 19 Copyright © 2009 Pearson Education, Inc. 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. After making all comparisons among the candidates, the candidate receiving the most points is declared the winner.

20. Chapter 15 Section 1 - Slide 20 Copyright © 2009 Pearson Education, Inc. Example: Pairwise Comparison Use the pairwise comparison method to determine the winner of the election for math club president. Number of comparisons needed:

21. Chapter 15 Section 1 - Slide 21 Copyright © 2009 Pearson Education, Inc. Example: Pairwise Comparison (continued) Thomas versus Jerry  T = 14J = 12 + 9 + 4 + 1 = 26 Jerry = 1 Thomas versus Annette  T = 14A = 12 + 9 + 4 + 1 = 26Annette = 1 Thomas versus Betty  T = 14B = 12 + 9 + 4 + 1 = 26Betty = 1 TTTTBFourth A B J 4 JBJAThird BJAJSecond AABTFirst 191214 # of Votes

22. Chapter 15 Section 1 - Slide 22 Copyright © 2009 Pearson Education, Inc. Example: Pairwise Comparison (continued) Betty versus Annette  B = 12 + 4 = 16A = 14 + 9 + 1 = 24Annette = 1 Betty versus Jerry  B = 12 + 1 = 13J = 14 + 9 + 4 = 27Jerry = 1 Annette versus Jerry  A = 12 + 9 + 1 = 22J = 14 + 4 = 18Annette = 1 TTTTBFourth A B J 4 JBJAThird BJAJSecond AABTFirst 191214 # of Votes Annette would win with 3 total points, the most from the pairwise comparison method.

23. Slide 15 - 23 Copyright © 2009 Pearson Education, Inc. The employees at a local law firm are voting on the entrée for their annual holiday party. Their choices are chicken (C), salmon (S), and steak (T). The preference table follows. Number of Votes30181223 FirstTCTS SecondSTCC ThirdCSST

24. Slide 15 - 24 Copyright © 2009 Pearson Education, Inc. Here’s the preference table, again. How many employees voted? a. 48b. 58c. 60d. 83 Number of Votes30181223 FirstTCTS SecondSTCC ThirdCSST

25. Slide 15 - 25 Copyright © 2009 Pearson Education, Inc. Here’s the preference table, again. How many employees voted? a. 48b. 58c. 60d. 83 Number of Votes30181223 FirstTCTS SecondSTCC ThirdCSST

26. Slide 15 - 26 Copyright © 2009 Pearson Education, Inc. Here’s the preference table, again. Does any choice have a majority of votes? a. Yes.b. No.c. Can’t determine. Number of Votes30181223 FirstTCTS SecondSTCC ThirdCSST

27. Slide 15 - 27 Copyright © 2009 Pearson Education, Inc. Here’s the preference table, again. Does any choice have a majority of votes? a. Yes.b. No.c. Can’t determine. Number of Votes30181223 FirstTCTS SecondSTCC ThirdCSST

28. Slide 15 - 28 Copyright © 2009 Pearson Education, Inc. Here’s the preference table, again. Determine the winner using the plurality method. a. Chickenb. Salmon c. Steakd. Can’t determine Number of Votes30181223 FirstTCTS SecondSTCC ThirdCSST

29. Slide 15 - 29 Copyright © 2009 Pearson Education, Inc. Here’s the preference table, again. Determine the winner using the plurality method. a. Chickenb. Salmon c. Steakd. Can’t determine Number of Votes30181223 FirstTCTS SecondSTCC ThirdCSST

30. Slide 15 - 30 Copyright © 2009 Pearson Education, Inc. Here’s the preference table, again. Determine the winner using the Borda count method. a. Chickenb. Salmon c. Steakd. Can’t determine Number of Votes30181223 FirstTCTS SecondSTCC ThirdCSST

31. Slide 15 - 31 Copyright © 2009 Pearson Education, Inc. Here’s the preference table, again. Determine the winner using the Borda count method. a. Chickenb. Salmon c. Steakd. Can’t determine Number of Votes30181223 FirstTCTS SecondSTCC ThirdCSST

32. Slide 15 - 32 Copyright © 2009 Pearson Education, Inc. Here’s the preference table, again. Determine the winner using the plurality method with elimination method. a. Chickenb. Salmon c. Steakd. Can’t determine Number of Votes30181223 FirstTCTS SecondSTCC ThirdCSST

33. Slide 15 - 33 Copyright © 2009 Pearson Education, Inc. Here’s the preference table, again. Determine the winner using the plurality method with elimination method. a. Chickenb. Salmon c. Steakd. Can’t determine Number of Votes30181223 FirstTCTS SecondSTCC ThirdCSST

34. Slide 15 - 34 Copyright © 2009 Pearson Education, Inc. Here’s the preference table, again. Determine the winner using the pairwise comparison method. a. Chickenb. Salmon c. Steakd. Can’t determine Number of Votes30181223 FirstTCTS SecondSTCC ThirdCSST

35. Slide 15 - 35 Copyright © 2009 Pearson Education, Inc. Here’s the preference table, again. Determine the winner using the pairwise comparison method. a. Chickenb. Salmon c. Steakd. Can’t determine Number of Votes30181223 FirstTCTS SecondSTCC ThirdCSST

36. Slide 15 - 36 Copyright © 2009 Pearson Education, Inc. The employees also must pick the dessert for their holiday party. The choices are cheese cake (C), apple tart (A), chocolate mousse cake (M), or vanilla ice cream (V). Here’s the preference table. Number of Votes1916122214 FirstACMVA SecondMAVMM ThirdCVACC FourthVMCAV

37. Slide 15 - 37 Copyright © 2009 Pearson Education, Inc. Which dessert wins this election if the plurality method is used? a. Cheese cakeb. Apple tart c. Chocolate Moussed. Vanilla ice cream Number of Votes1916122214 FirstACMVA SecondMAVMM ThirdCVACC FourthVMCAV

38. Slide 15 - 38 Copyright © 2009 Pearson Education, Inc. Which dessert wins this election if the plurality method is used? a. Cheese cakeb. Apple tart c. Chocolate Moussed. Vanilla ice cream Number of Votes1916122214 FirstACMVA SecondMAVMM ThirdCVACC FourthVMCAV

39. Slide 15 - 39 Copyright © 2009 Pearson Education, Inc. Which dessert wins this election if the Borda count method is used? a. Cheese cakeb. Apple tart c. Chocolate Moussed. Vanilla ice cream Number of Votes1916122214 FirstACMVA SecondMAVMM ThirdCVACC FourthVMCAV

40. Slide 15 - 40 Copyright © 2009 Pearson Education, Inc. Which dessert wins this election if the Borda count method is used? a. Cheese cakeb. Apple tart c. Chocolate Moussed. Vanilla ice cream Number of Votes1916122214 FirstACMVA SecondMAVMM ThirdCVACC FourthVMCAV

41. Slide 15 - 41 Copyright © 2009 Pearson Education, Inc. Which dessert wins this election if the plurality with elimination method is used? a. Cheese cakeb. Apple tart c. Chocolate Moussed. Vanilla ice cream Number of Votes1916122214 FirstACMVA SecondMAVMM ThirdCVACC FourthVMCAV

42. Slide 15 - 42 Copyright © 2009 Pearson Education, Inc. Which dessert wins this election if the plurality with elimination method is used? a. Cheese cakeb. Apple tart c. Chocolate Moussed. Vanilla ice cream Number of Votes1916122214 FirstACMVA SecondMAVMM ThirdCVACC FourthVMCAV

43. Slide 15 - 43 Copyright © 2009 Pearson Education, Inc. Which dessert wins this election if the pairwise comparison method is used? a. Cheese cakeb. Apple tart c. Chocolate Moussed. Vanilla ice cream Number of Votes1916122214 FirstACMVA SecondMAVMM ThirdCVACC FourthVMCAV

44. Slide 15 - 44 Copyright © 2009 Pearson Education, Inc. Which dessert wins this election if the pairwise comparison method is used? a. Cheese cakeb. Apple tart c. Chocolate Moussed. Vanilla ice cream Number of Votes1916122214 FirstACMVA SecondMAVMM ThirdCVACC FourthVMCAV

45. Slide 15 - 45 Copyright © 2009 Pearson Education, Inc. Practice Problems

46. Slide 15 - 46 Copyright © 2009 Pearson Education, Inc.

47. Slide 15 - 47 Copyright © 2009 Pearson Education, Inc.

48. Slide 15 - 48 Copyright © 2009 Pearson Education, Inc.

49. Slide 15 - 49 Copyright © 2009 Pearson Education, Inc.

50. Slide 15 - 50 Copyright © 2009 Pearson Education, Inc.