Excursions in Modern Mathematics Sixth Edition

1. Excursions in Modern MathematicsSixth Edition Peter Tannenbaum

2. Chapter 1The Mathematics of Voting The Paradoxes of Democracy

3. The Mathematics of VotingOutline/learning Objectives • Construct and interpret a preference schedule for an election involving preference ballots. • Implement the plurality, Borda count, plurality-with-elimination, and pairwise comparisons vote counting methods. • Rank candidates using recursive and extended methods. • Identify fairness criteria as they pertain to voting methods. • Understand the significance of Arrows’ impossibility theorem.

4. The Mathematics of Voting 1.1 Preference Ballots and Preference Schedules

5. Voting Theory Why Vote? • Think about all the elections you can vote in: • Presidential • Local elections • School board • Homecoming queen • American Idol • Clearly, not all voting is equal

6. Voting Theory • Casting a vote is only part of the story. What matters more is how the votes are counted to determine a winner. • To analyze the various voting methods we need: • Candidates – the choices • Voters – the ones who are voting • Ballots – the way the votes are collected

7. Voting Theory • In 1940, Kenneth Arrow discovered an incredible fact: for elections involving three or more candidates, there is no consistently fair democratic method for choosing a winner. • In fact, a method for determining election results that is democratic and always fair is a mathematical impossibility. • This is known as Arrow’s Impossibility Theorem.

8. Preference Ballots and Schedules • Preference ballots A ballot in which the voters are asked to rank the candidates in order of preference. • Linear ballot A ballot in which ties are not allowed.

9. Preference Ballots and Schedules • An example of a preference ballot is each person’s vote in our math class party election. • Preference ballots allow voters to express an opinion on all candidates instead of just choosing their choice.

10. Preference Ballots and Schedules 18 voters

11. Preference Ballots and Schedules • There are a limited number of ways the candidates can be ranked, so some ballots may repeat. If we collect all the repeats and organize them in a table, we get a preference schedule.

12. Preference Ballots and Schedules

13. Preference Ballots and Schedules

14. Preference Ballots and Schedules • A preference schedule:

15. Preference Ballots and Schedules Things to keep in mind: • Voter preferences are transitive • If I like Sprite better than Dr. Pepper and I like Dr. Pepper more than Mt. Dew, then I like Sprite better than Mt. Dew. • If we need to know which candidate a voter would vote for if it all came down to candidate X and candidate Y, we just look at where X and Y are on that person’s ballot.

16. Preference Ballots and Schedules Things to keep in mind: • Relative preferences are not affected by elimination of a candidate. • If a candidate drops out of a race, then the votes shift up accordingly. • Example: If I remove Starburst from the candy options (because the wrappers will end up everywhere), to recalculate a winner, I just move everyone’s votes up accordingly.

17. Preference Ballots and Schedules Relative Preferences by elimination of one or more candidates

18. Preference Ballots and Schedules • How many people voted in this election? • If D gets sick and can’t run, what is the new schedule?

19. Preference Ballots and Schedules • The Mathematics Society is holding an election for the president. The three candidates are A, B, and C. Forty-five percent of voters like A the most and B the least. Thirty percent of voters like B the most and C the least. Twenty-five percent of voters like C the most and A the least. Write out the preference schedule for this election.

20. Preference Ballots and Schedules • In an election involving 6 candidates, what is the maximum number of columns possible in the preference schedule? N!

21. The Mathematics of Voting 1.2 The Plurality Method

22. The Plurality Method • Plurality method In the plurality method, all we care about is first-place votes. The candidate with the most first-place votes wins. • Plurality candidate The Candidate with the most 1st place votes

23. The Plurality Method • The vast majority of our elections are decided using the plurality method. Since the only votes that count are first-place votes, we don’t bother to rank the other candidates.

24. The Plurality Method • Find the winner of this election using the plurality method:

25. The Plurality Method • Majority rule The candidate with a more than half the votes should be the winner. • Majority candidate The candidate with the majority of 1st place votes .

26. The Plurality Method • The plurality method is appealing because it is simple and it is a natural extension of the principle of majority rule. • In a democratic election between 2 candidates, the candidate with a majority of the votes should be the winner. The candidate with a majority of first-place votes is called the majority candidate. • However, with 3 or more candidates, there is no guarantee that one candidate will win a majority of votes.

27. The Plurality Method The Majority Criterion (a fairness criterion) If candidate X has a majority of the 1st place votes, then candidate X should be the winner of the election. Good News: The plurality method satisfies the majority criterion! Bad News: The plurality method fails a different fairness criterion.

28. The Plurality Method • Under the plurality method, a majority candidate is guaranteed to be the winner. • Why?

29. The Plurality Method • There are widely used voting methods that can produce violations of the majority criterion. Specifically, a violation of the majority criterion occurs in an election in which there is a majority candidate but that candidate does not win the election. If this can happen under some voting method, then the voting method itself violates the majority criterion. • Note: violations may occur, not that they always will occur.

30. The Plurality Method • Other than the majority criterion, the plurality method has little appeal and is undesirable when choosing between more than two candidates.

31. The Plurality Method • The principal weakness of the plurality method is that it fails to take into consideration a voter’s other preferences beyond first choice and in doing so can lead to some very bad election results.

32. The Plurality Method • Tasmania State University has a superb marching band. They are so good that they have invitations to perform at five different bowl games: the Rose Bowl (R), the Hula Bowl (H), the Fiesta Bowl (F), the Orange Bowl (O), and the Sugar Bowl (S). An election is held among the 100 members of the band to decide in which bowl game they will perform.

33. The Plurality Method • Tasmania State University has a superb marching band. They are so good that they have invitations to perform at five different bowl games: the Rose Bowl (R), the Hula Bowl (H), the Fiesta Bowl (F), the Orange Bowl (O), and the Sugar Bowl (S). An election is held among the 100 members of the band to decide in which bowl game they will perform.

34. The Plurality Method • Find the winner using the plurality method. • Is this a good outcome? Why or why not?

35. The Plurality Method • By contrast, the Hula Bowl has 48 first-place votes and 52 second-place votes. Common sense tells us that the Hula Bowl is a far better choice to represent the wishes of the entire band. • In fact, if we compare the Hula Bowl with any other bowl on a head-to-head basis, the Hula Bowl is always the preferred choice.

36. The Plurality Method • For example, compare the Hula Bowl to the Rose Bowl.

37. The Plurality Method The Condorcet Criterion If candidate X is preferred by the voters over each of the other candidates in a head-to-head comparison, then candidate X should be the winner of the election. Not every election has a Condorcet candidate but if there is one, it is a good sign that this candidate represents the voice of the voters better than any other candidate.

38. The Plurality Method • Consider once again the marching band bowl game example. R v H H v F F v S R v F H v O O v S R v O H v S R v S F v O

39. The Plurality Method • The Hula Bowl is a Condorcet candidate but under the plurality method, the Hula bowl is not the winner. Therefore, the plurality method violates the Condorcet criterion.

40. The Plurality Method Insincere Voting (or Strategic Voting) If we know that the candidate we really want doesn’t have a chance of winning, then rather than “wasting our vote” on our favorite candidate we can cast it for a lesser choice that has a better chance of winning the election.

41. The Plurality Method • Of the voting methods, the plurality method is most susceptible to insincere voting. In the US, presidential races are often decided by insincere voters.

42. The Plurality Method Jill Stein Barack Obama • This candidate was my first choice, but I knew that she had zero chance of winning the national election. • On the other hand, this was my second choice and had a much better chance of winning, so I voted for him instead.

43. The Plurality Method • Back to the marching band example. If the three people who voted for the Fiesta Bowl as their first choice realized that it had little chance of winning but that the Hula bowl, their second choice, had a good chance of winning, they could change their votes and therefore, change the outcome.

44. The Plurality Method

45. The Plurality Method • The plurality method of voting reinforces an entrenched two-party system that often leaves voters with no real choice. This is called Duverger’s Law.

46. The Plurality Method • Who is the plurality method winner? • Is there a majority candidate? • Is there a Condorcet candidate?

47. The Plurality Method • Consider an election with 456 voters and seven candidates. What is the smallest number of votes that a plurality candidate could have?

48. The Mathematics of Voting 1.3 The Borda Count Method

49. The Borda Count Method • In the Borda Count Methodeach place on a ballot is assigned points. In an election with N candidates we give 1 point for last place, 2 points for second from last place, and so on. • The points for each candidate are tallied and the candidate with the highest total is the winner. This candidate is called the Borda winner.

50. The Borda Count Method • This method can sometimes result in a tie. In this unit, assume that a tie can stand and will not be decided using some other method.