Math and Voting

1. Math and Voting October 22, 2009 Maura Bardos

2. Outline • Two Candidates • Majority Rule • Three Candidates or More • Plurality • Borda • Condorcet • Sequential Pairwise • Instant Runoff • Arrow’s Theorem • Approval voting • A better method?

3. 3 Properties of Fair Elections • Sincere Ballot: A ballot that represents a voter’s true preferences • 3 Properties • Anonymous. All voters are treated equally • Neutral. Both candidates are treated equally • Monotone Can you think of an examples where these criteria fail? Dictatorship Imposed Rule Minority Rule Can you think of an example where all three properties are satisfied for a two candidate election?

4. May’s Theorem • In a two candidate election with an odd number of voters, majority rule is the only system that is anonymous, neutral, and monotone, and that avoids the possibilities of ties. (Hodge and Klima)

5. Majority Rule • Each voter indicates a preference for one of the candidates. The candidate with the most votes wins. In a two candidate election, the candidate that is preferred by more than half of the voters is the winner. • What is the quota for majority rule in a two candidate election with n voters? • If n is even: (n/2) + 1 • If n is odd: n/2

6. Example • 2008 Presidential Election Obama: 1,959,532 votes 53% McCain: 1,725,005 votes 47% Total Votes cast: 3,864,537 Quota: 1,842,528.5

7. Enter: Third Candidate • If there are only two candidates, it is easy to determine the winner • The candidate that is preferred by the majority wins • With more than two candidates, things change… http://en.wikipedia.org/wiki/Ross_Perot http://en.wikipedia.org/wiki/Ralph_Nader

8. Third Candidate (or more) • Plurality method- voting system that elects the candidate who receives the largest number of votes even if that number is less than half of the total number of votes cast. • Questions to consider • Do we really elect the winner? • Do our voting systems reflect what the voters really want?

9. Simple Example (Saari) • Let’s pretend Math 490 is having a party during our next Tuesday class at 2pm. • We need to choose a snack to serve. The party planner asks all students to rank their preferences: 6 Students: Salad > Chips > Popcorn 5 Students: Popcorn > Chips > Salad 4 Students: Chips > Popcorn > Salad Observations: Plurality: Salad Wins!

10. 6 Students (40%): Salad > Chips > Popcorn 5 Students (33%): Popcorn > Chips > Salad 4 Students (27%): Chips > Popcorn > Salad We get to the store…we see that Bloom is sold out of Popcorn. What difference does it make? Lets Revisit our preferences 6 Students (40%): Salad > Chips 5 Students (33%): Chips > Salad 4 Students (27%): Chips > Salad 60% prefer chips to Salad.

11. 6 Students (40%): Salad > Popcorn 5 Students (33%): Popcorn > Salad 4 Students (27%): Popcorn > Salad Either way- voters prefer anything to Salad. With majority rule- we select a “winner” that the voters don’t really want. Note that voter preferences did not change

12. Borda Count • Developed by Jean Charles de Borda in 1770. • Definition: A voting system for elections with several candidates in which points are assigned to voters’ preferences and theses points are summed for each candidate to determine a winner. • Uses rank by preference order • Violates majority criterion • Possible for a candidate to be viewed as the most desirable by the majority but still not win • Consensus based

13. Borda Count • Each voter ranks candidates based on preferences • For each ballot, points are allocated: First Place is worth n-1 points Second Place is worth n-2 points …Last Place is worth n-n=0 points • Candidate with largest number of points is declared the winner. (Hodge and Klima)

14. Example How many points to award? Top Rank = n-1 points, where n is the number of candidates ….Last Ranked = 0 points Borda Score for : A = 3 (2 points) + 2 (0 points) = 6 B = 3 ( 1 point) + 2 (1 point) = 5 C = 3 (0 points) + 2 (2 points) = 4 Candidate A is the winner

15. Example Lets switch the rank of B and C. Now recalculate the Borda Score A = 6 (same as last time) B = 3 (1 point) + 2( 2 points) = 7 C = 3 (0 points) + 2(1 point) = 2 Candidate B is the winner.

16. Paradox with Borda Scheme • Fails the Independence of Irrelevant Alternatives (IIA) • IIA- a voting system satisfies this criteria if it is impossible for a candidate to move from non-winner to winner unless at least one voter reverses the order in which the candidate was ranked. • So in our example, A changed from winner to non-winner, even though no one changed their mind on A compared to B preference • Other issue: Borda Count is capable of violating the majority criterion

17. Lets Return to the Party Example: Presentation packet Problem #1: Salad: 6 (2 points) + 5 ( 0 points) + 4 ( 0 points) = 12 Chips: 6 (1 points) + 5 ( 1 points) + 4 ( 4 points) = 27 Popcorn: 6 (0 points) + 5 ( 2 points) + 4 ( 1 points) = 14 Chips Win Salad loses…

18. Borda Count in Practice • Grade Point Average: A=4 points, B = 3 points… • Think if majority system was used instead • National Assembly of Slovenia • Kiribati and Nauru (Pacific Island Countries) • Sports: • MVP in MLB • Heisman Trophy • Borda count is used to break ties for member elections of the faculty personnel committee of the School of Business Administration at the College of William and Mary.

19. Borda Count MVP

20. The following method is used to calculate the winner: Morneau: (15 x 14) + (8 x 9) + (3 x 8) + (2 x 7) = 320 Jeter: (12 x 14) + (14 x 9) + (1 x 7) + (1 x 5) = 306

21. Condorcet Method • Developed in 1785 by Marquis de Condorcet • Contemporary of Borda • Condorcet winner: A candidate in an election who would defeat ever other candidate in a head-to-head contest (with the winner decided by majority rule). • Condorcet loser: A candidate in an election who would lose to ever other candidate in a head-to-head contest (with the winner decided by majority rule). (pg. 40) • Only one Condorcet loser and one Condorcet winner per election

22. Condorcet continued • Other important properties • If a candidate in an election receives a majority of the first place votes cast, then that candidate will be a Condorcet winner. • If a voting system satisfies the Condorcet winner criterion, then it will also satisfy the majority criterion • If a voting system violates the majority criterion, then it will also violate the Condorcet winner criterion.

23. Example: Minnesota Gubernatorial Race Photo source: http://en.wikipedia.org/wiki/Minnesota_gubernatorial_election,_1998 Jesse Ventura (Reform Party) St. Paul Mayor Norm Coleman (R) Attorney General Skip Humphrey (D) http://www.youtube.com/watch?v=TjU948M0ARw

25. Example: Minnesota Gubernatorial Race 1998 Minnesota Governors race with Jesse Ventura (Reform Party), Attorney General Skip Humphrey (D), and St. Paul Mayor Norm Coleman (R). Lets examine who wins the election under a variety of systems

26. Example: Minnesota Gubernatorial Race • In a head-to-head race between just Skip and Norm, who would win? • Norm is ranked first by 55% of the voters • Skip is ranked first by 45% of the voters • Norm would defeat Skip in a head-to-head race Now try Problem 2

27. Example: Minnesota Gubernatorial Race • Condorcet winner: Norm Coleman • Condorcet loser: Jesse Ventura What about other voting Systems: Majority: Plurality: Borda: • In actuality: Ventura is proclaimed the winner. Ventura is similar to salad in the party example • Ventura- “extreme candidate.” Coincidence he only held one term?

28. Relationship between Borda and Condorcet • Theorem: If there is a Condorcet winner, this candidate is NEVER ranked last by the Borda count. • Note that this theorem is only applicable when the weights are [ (n-1), (n-2)….., 2, 1, 0]

29. Borda Count and Condorcet’s Method at William and Mary • Article 5, Section 3 of the by-laws of the faculty of School of Business Administration • Voting systems at use for the selection of a Faculty Personnel Committee “The Condorcet Criterion shall be used to determine the results, and if there is a tie, the Adjusted Borda Count, direct paired comparisons, the Borda Count, and a deciding vote by the Dean, are to be used sequentially, until the tie is broken.”

30. Sequential Pairwise Voting • Uses concept of head-to-head elections for elections with more than two candidates • Definition: Pits the first candidate against the second in a one-on-one contest. The winner then moves on to confront the third candidate in the list. Losers are deleted. Process continues until there is one candidate remaining (COMAP).

31. Example Steps: • Determine an Agenda (ordering candidates for future comparison) • Compare the first two candidates, use majority rule to decide the winner. • Next choose between the winner of step one and third candidate in agenda. • Continue sets of majority rules head to head contests to find the overall winner

32. Agenda: ABCD a vs. b: a a vs. c: c c vs. d: dAgenda: BCAD b vs. c: b a vs. b: a a vs. d: a Agenda: ACBD a vs. c: c b vs. c: b b vs. d: bAgenda: ABDC a vs. b: a a vs. d: a a vs. d: c

33. This method satisfies the Condorcet voter criteria. But a Condorcet winner doesn’t always exist. In these situations, the result is contingent in the agenda. In general, the later an alternative is introduced, the better its chances of winner. Obviously not applicable for elections Used in single elimination tournaments, such as tournaments where teams are ‘seeded’

34. Instant Runoff (or Single Transferable Vote) • Definition: Arrive at a winner by repeatedly deleting candidates that are “least preferred” in the sense of being at the top of the fewest ballots (COMAP). • A version of this is known as the Hare system • General Steps: • Each voter submits preferences in order • Candidate with least number of 1st place votes is eliminated from each voter’s preference order, and the remaining candidates are moved up and “wasted votes” are redistributed • Repeat step 2 until only a single candidate, the winner, remains. (Hodge and Klima).

36. In Practice • Fails monotonicity • Elections of public officials in Australia, Malta, Ireland • Academy Awards (nominating stage) • William and Mary Student Assembly Elections • Article 5, Section 3 of the Constitution of the Student Assembly • “III. Undergraduate Senatorial Elections shall be by plurality, with each Class' candidates being chosen together on the same ballot. Undergraduate Class Officers shall be elected by the Instant Runoff System.”

37. Example: Academy Awards Original Procedure (for awards 1936-2008) • Nominating: STV. All voters are allowed to nominate for best picture. 5 nominees are selected for best picture • Final Ballot for determining the winner: Plurality

38. Example- 2008 Best Picture • A: MilkB: Slumdog MillionaireC: Curious Case of Benjamin ButtonD: The ReaderE: Quantum of SolaceF: Transporter 3G: Frost/NixonH: TwilightI: Marley & Me

39. We need to nominate 5 films for the Awards show. Droop Quota: Minimum number of votes a candidate must receive to be the winner For our example, lets assume that there are n=30 voters (total valid poll) and k=5 films to nominate (seats) Quota = 6

40. Round 1: Does any candidate meet the Droop Quota? Yes- G 9-6=3 excess votes are distributed to C and A

41. Rounds 2 and 3- C reaches minimum number, E is eliminated

42. Rounds 4 and 5- Eliminate H. Transfer one vote to B. Eliminate I

43. Rounds 6 and 7- B is selected. D is eliminated.

44. Final Selections Films G, C, B, A and D: A: MilkB: Slumdog MillionaireC: Curious Case of Benjamin ButtonD: The ReaderG: Frost/Nixon Note that E, Quantum of Solace, was the Condorcet winner.

45. In previous Oscars- the nomination processes narrowed down the film to five nominees • As of Aug 31, 2009, there will be 10 nominees for best picture. Voters will rank these 10 nominees to determine the winner. The same method we just went through will be conducted for the 10 films, requiring a 50% threshold for the winner. • The Academy- “Though no voting system is perfect, for the Academy’s purposes, it is difficult to point to a better system than the preferential system.” • Do Scholars like this system any better?

46. …stay tuned for February 2, 2010

47. Summary: Evaluating Voting Systems Each fails to satisfy one desirable property

48. Arrow’s Theorem “The only voting method that isn't flawed is a dictatorship“ With three of more candidates an any number of voters, there does not exist a voting system that always produces a winner that satisfies the following criteria:

49. Conditions: • Universality • Monotonicity • Independence of Irrelevant Alternatives • Citizen Sovereignty • Nondictatorship (Hodge and Klima)

50. Example • Lets look at an example of the weaker version of the theorem: Theorem: With three or more candidates and an odd number of votes, there does not exist- and there will never exist a voting system that satisfies both the Condorcet winner criterion and the independence of irrelevant alternatives and that always produces at least one winner in every election (COMAP).