1. Which units are you most interested in covering? Unit A –Management Science Unit B – Growth Unit C – Shape and Form Unit D – Statistics

2. Here’s how you voted (remember?) Student (Voter) # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

3. Here’s how you voted If we group identical ballots together, it becomes a little more manageable:

4. Here’s how you voted If we group identical ballots together, it becomes a little more manageable: 6 4 4 4 1 2

5. Summarizing: How many

6. Let’s simplify the question a little: Of the 4 choices, which is the most popular? How many

7. Plurality method: Which has the most first place votes?

8. Since A has 10 first place votes, and none of the others have more than 4, A is the easy winner by this method. The Plurality winner is Unit A – Management Science

9. Borda Count We give 4 points for each 1st place vote, 3 for a 2nd place vote, 2 for a 3rd, and 1 for a 4th. Method #2:

10. Borda Count A has 10 first place and 11 4th place votes, so its point total is 10*4 +11*1 = 51.

11. Borda Count A has 10 first place and 11 4th place votes, so its point total is 10*4 +11*1 = 51. Similarly, B gets 4*4 + 7*3 + 10*2 = 57 C gets 4*4 + 2*3 + 11*2 + 4*1 = 48 D gets 3*4 + 12*3 + 6*1 = 54

12. Borda Count A has 10 first place and 11 4th place votes, so its point total is 10*4 +11*1 = 51. Similarly, B gets 4*4 + 7*3 + 10*2 = 57 C gets 4*4 + 2*3 + 11*2 + 4*1 = 48 D gets 3*4 + 12*3 + 6*1 = 54 The Borda winner is Unit B – Growth

13. Plurality with elimination method: Get a majority, “survivor-style” Method #3:

14. Plurality with elimination method: Get a majority, “survivor-style” Keep eliminating losing options and re-casting ballots until a candidate has a majority of the 1 st place votes.

15. Plurality with elimination method: Get a majority, “survivor-style” Since D has the fewest 1st place votes (3), it is “voted off the island”. All ballots are recast, keeping A, B and C in their original order, but crossing D off.

16. Plurality with elimination method: Get a majority, “survivor-style”

18. Note that some columns can now be combined (the first 2 are now the same, as are the 3rd and 5th, and the 4th and 6th)

19. Plurality with elimination method: Get a majority, “survivor-style” Note that some columns can now be combined (the first 2 are now the same, as are the 3rd and 5th, and the 4th and 6th)

20. Plurality with elimination method: Get a majority, “survivor-style”

21. Next to be voted off the island is B:

22. Plurality with elimination method: Get a majority, “survivor-style” Now, we can combine columns 2 and 3:

23. Plurality with elimination method: Get a majority, “survivor-style” And we now have a “survivor” - a majority winner, C. The Plurality with Elimination winner is Unit C –Shape and Form

24. Head-to-head comparisons Given a choice of only A or D, which would most people prefer? Method #4:

25. Head-to-head comparisons Given a choice of only A or D, which would most people prefer? Method #4:

26. Head-to-head comparisons Given a choice of only B or D, which would most people prefer?

27. Head-to-head comparisons Given a choice of only B or D, which would most people prefer?

28. Head-to-head comparisons Given a choice of only C or D, which would most people prefer?

29. Head-to-head comparisons Given a choice of only C or D, which would most people prefer?

30. Head-to-head comparisons Normally, we would continue comparing all pairs of choices, (A-B, A-C, and B-C) but in this example that is not really necessary to find the most popular option (why not?)

31. Head-to-head comparisons In fact, if we did compare all pairs of options, we would find that: A wins 0 comparisons; B wins 2 comparisons (B beats A, 11-10, and B beats C, 15-6); C wins 1 comparison (C beats A, 11-10); and D wins 3 comparisons, as we’ve already seen

32. Head-to-head comparisons The winner by the method of Head-to-Head Comparisons is Unit D, Statistics

33. Ok, back to the question… Which is the most popular choice? A, which got the most 1 st place votes? B, which got the most points in the 4-3-2-1 point counting method? C, which won the “reality show” elimination method? Or D, which the majority of you preferred head- to-head against each and every other choice?

34. Which method is best? To answer that, we need to decide which of the voting methods is the most fair. But what do we mean by “fair” anyway? Weren’t all of the voting methods used so far fair?

35. Which method is best? The book presents 4 criteria that most people would agree any fair (democratic?) voting method should satisfy:

36. The 4 fairness criteria: 1) Majority: If one candidate is the favorite of more than half (a majority) of the voters, then that candidate should win. What is fair: Majority rules! What is fair: Majority rules! What isn’t fair: 15 out of 21 like A the best, but B beats A in the election Note: If no candidate receives a majority of the 1 st place votes, then this criterion does not apply – any result would be considered fair

37. The 4 fairness criteria: 2) Condorcet: If one candidate is preferred in head-to-head comparisons with every other candidate, then that candidate should win. Fair: If one candidate is preferred to each and every other candidate, it is should win. Not fair: More voters would choose candidate A over each possible alternative, and yet A loses Note: There may not be a candidate that wins every head- to-head comparison. In that case, this criterion would not apply, and any result could be fair.

38. The 4 fairness criteria: 3) Monotonicity: If candidate X would win, and a voter then changes his ballot in a way that favors X, then X should still win. Not fair: “I snuck a peak at the votes, and Ryan is about to win the election” “Oh –I wasn’t going to vote for him, but I might as well just change my vote to support him.” With this ballot changed to support Ryan … Ryan loses?!?

39. The 4 fairness criteria: 4) Independence-of-Irrelevant-Alternatives: Adding (or removing) losing candidates to the list of options should not change the winner. Not fair (and rather weird!): “Would you like blueberry or cherry pie?” “Cherry, please.” “Oh – I just noticed, we also have apple pie.” “Oh great! In that case I’ll have blueberry”

40. The 4 fairness criteria: 4) Independence-of-Irrelevant-Alternatives: Adding (or removing) losing candidates to the list of options should not change the winner. Not fair (alternative version): “Would you like blueberry, cherry, or apple pie?” “Cherry, please.” “Oh – I just noticed, we are out of apple today.” “In that case give me blueberry.”

41. The 4 fairness criteria: Majority: If one candidate is the favorite of more than half (a majority) of the voters, then that candidate should win. Condorcet: If one candidate is preferred in head-to-head comparisons with every other candidate, then that candidate should win. Monotonicity: If candidate X would win, and a voter then changes his ballot in a way that favors X, then X should still win. Independence-of-Irrelevant-Alternatives: If X would win, and one or more of the losing candidates drops out of the election, then X should still win.

42. So, back to the question (again)… Which voting method is the “fair” one? And, most importantly, which candidate won our class vote? What will we be studying this semester?

43. ?