M ATH AND V OTING October 22, 2009 Maura Bardos

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

3 P ROPERTIES OF F AIR E LECTIONS Sincere Ballot: A ballot that represents a voter’s true preferences 3 Properties 1) Anonymous. All voters are treated equally 2) Neutral. Both candidates are treated equally 3) 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?

M AY ’ S T HEOREM 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)

M AJORITY R ULE 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

E XAMPLE 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

E NTER : T HIRD C ANDIDATE 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…

T HIRD C ANDIDATE ( 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?

S IMPLE E XAMPLE (S AARI ) 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!

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.

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

B ORDA C OUNT Developed by Jean Charles de Borda in 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

B ORDA C OUNT 1) Each voter ranks candidates based on preferences 2) 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 3) Candidate with largest number of points is declared the winner. (Hodge and Klima)

E XAMPLE Rank32 1AC 2BB 3CA 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

E XAMPLE Rank32 1AB 2BC 3CA 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.

P ARADOX WITH B ORDA S CHEME 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

L ETS R ETURN TO THE P ARTY E XAMPLE : Rank654 1SaladPopcornChips 2 Popcorn 3 Salad 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…

B ORDA C OUNT IN P RACTICE 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.

B ORDA C OUNT MVP 2006 AL MVP Award Voting results ¬ Player, Club1st2nd3rd4th5th6th7th8th9th10thPoints Justin Morneau, MIN Derek Jeter, NYY David Ortiz, BOS Frank Thomas, OAK Jermaine Dye, CWS Joe Mauer, MIN Johan Santana, MIN

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 Voting results ¬ Player, Club1st2nd3rd4th5th6th7th8th9th10thPoints Justin Morneau, MIN Derek Jeter, NYY

C ONDORCET M ETHOD 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

C ONDORCET 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.

E XAMPLE : M INNESOTA G UBERNATORIAL R ACE Jesse Ventura (Reform Party) Attorney General Skip Humphrey (D) St. Paul Mayor Norm Coleman (R) Photo source: M0ARw

E XAMPLE : M INNESOTA G UBERNATORIAL R ACE Rank NSJJ 2SNNS 3JJSN 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

E XAMPLE : M INNESOTA G UBERNATORIAL R ACE 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 Rank NSNS 2SNSN Now try Problem 2

E XAMPLE : M INNESOTA G UBERNATORIAL R ACE 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?

R ELATIONSHIP BETWEEN B ORDA AND C ONDORCET 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]

B ORDA C OUNT AND C ONDORCET ’ S M ETHOD AT W ILLIAM AND M ARY 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.”

S EQUENTIAL P AIRWISE V OTING 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).

E XAMPLE Steps: 1) Determine an Agenda (ordering candidates for future comparison) 2) Compare the first two candidates, use majority rule to decide the winner. 3) Next choose between the winner of step one and third candidate in agenda. 4) Continue sets of majority rules head to head contests to find the overall winner

Agenda: ABCD a vs. b : a a vs. c : c c vs. d : d Agenda: 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 : b Agenda: ABDC a vs. b : a a vs. d : a a vs. d : c Rank111 1 acb 2 bad 3 dbc 4 cda

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’

I NSTANT R UNOFF ( OR S INGLE T RANSFERABLE V OTE ) 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: 1) Each voter submits preferences in order 2) Candidate with least number of 1 st place votes is eliminated from each voter’s preference order, and the remaining candidates are moved up and “wasted votes” are redistributed 3) Repeat step 2 until only a single candidate, the winner, remains. (Hodge and Klima).

I N P RACTICE 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.”

E XAMPLE : A CADEMY A WARDS Original Procedure (for awards ) 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

E XAMPLE B EST P ICTURE A: Milk B: Slumdog Millionaire C: Curious Case of Benjamin Button D: The Reader E: Quantum of Solace F: Transporter 3 G: Frost/Nixon H: Twilight I: Marley & Me

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

stGGCAHIBDDF 2ndCAIBBBAAFD 3rdEEEEEEEEEE 4thFCADIHIBCC 5thIHFCDGDGAH Round 1: Does any candidate meet the Droop Quota? Yes- G 9-6=3 excess votes are distributed to C and A

st CAHIBDDF 2ndCAIBBBAAFD 3rdEEEEEEEEEE 4thFCADIHIBCC 5thIHFCD D AH Rounds 2 and 3- C reaches minimum number, E is eliminated

st AHIBDDF 2nd A BBBAAFD 3rd 4th DIHIB 5th H D D AH Rounds 4 and 5- Eliminate H. Transfer one vote to B. Eliminate I

st A BDDF 2n d A BBBAAFD 3rd 4th D B 5th D D A Rounds 6 and 7- B is selected. D is eliminated.

F INAL S ELECTIONS Films G, C, B, A and D: A: Milk B: Slumdog Millionaire C: Curious Case of Benjamin Button D: The Reader G: Frost/Nixon Note that E, Quantum of Solace, was the Condorcet winner.

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?

…stay tuned for February 2, 2010

S UMMARY : E VALUATING V OTING S YSTEMS Anon.NeutralMonotoneMCCWC PluralityYYYYN Borda Count YYYNN Sequential Pairs YNYYY Instant Runoff YYNYN Each fails to satisfy one desirable property

A RROW ’ S T HEOREM “ 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:

Conditions: 1) Universality 2) Monotonicity 3) Independence of Irrelevant Alternatives 4) Citizen Sovereignty 5) Nondictatorship (Hodge and Klima)

E XAMPLE 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).

Example (not a proof) Rank765 1ABC 2BCA 3CAB In head to head: A > B B > C C>A

I S THERE A B ETTER W AY ? For 2 Candidates- no problems For 3 or more Candidates- no system that satisfies all properties Possibilities supported by scholars: 1) Approval Voting

A PPROVAL V OTING A better way? Approval Voting- Each voter is allowed to give one vote to as many candidates that are acceptable. Voters show disapproval by not voting for them. The winner is determined by the largest number of approval votes. (COMAP) Uses: Baseball Hall of Fame, Selection of UN Secretary General Supported by Academics In general, favors consensus. Scholars, such as Steven Brams, have argued that AV selects the strongest nominee and avoids extremists. He advocates for this method especially during the primaries.

S O W HAT Is there any evidence to suggest that our political system,especially method for electing president, will change based on these mathematical findings? No substantive evidence of incentive at the moment

W HAT I F : E LECTORAL COLLEGE T IE 12 th Amendment- requires 270 votes in the electoral college to win a presidential election. Is 269 – 269 tie possible?

2008 P RESIDENTIAL E LECTION Analysis and modeling by Nate Silver of fivethirtyeight.com As of October 2008, a tie in the electoral college occurred 3.2% of the time. There were various combinations that produced this result, but 92% of the ties were the following: Obama- wins the Kerry states plus Iowa, New Mexico and Colorado, but loses New Hampshire.

W HAT DOES A T IE L OOK L IKE ?

C ONCLUSION “A society made up of rational people can vote irrationally.” (SIAM) We have seen that when three (or more) candidates are enter a race, strange things begin to happen. While there is no ‘perfect’ method to arrive at a decision, it is important to understand the relative strengths and weaknesses of each.

H OMEWORK 1) Class Election Rank the following: Paul’s Green Leaf Aroma’s 2) Research a ranking/decision making method (such as sports, Olympic games, election method in a foreign country). What method is used? Pick a particular occurrence and describe a surprising outcome.

S OURCES COMAP text Hodge, Jonathan and Richard Klima. The Mathematics of Voting and Elections: A Hands on Approach. Providence: American Mathematical Society, William and Mary Links ylawsoftheFacultyoftheSchoolofBusinessAdminis tration.pdf?svr=www

Voting and Social Choice, Princeton University. tml “Voting and Elections: An Introduction.” American Mathematical Society. introduction.html Delvin, Kevin. “The perplexing mathematics of presidential elections.” Mathematics Assocation of America. November Mackenzie, Dana. “Making Sense out of consensus.” October 21, Society for Industrial and Applied Mathematics.

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