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Math and Voting. October 22, 2009 Maura Bardos. 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 Properties of Fair Elections.

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math and voting

Math and Voting

October 22, 2009

Maura Bardos

    • Two Candidates
      • Majority Rule
  • Three Candidates or More
    • Plurality
    • Borda
    • Condorcet
    • Sequential Pairwise
    • Instant Runoff
  • Arrow’s Theorem
    • Approval voting
    • A better method?
3 properties of fair elections
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?


Imposed Rule

Minority Rule

Can you think of an example where all three properties are satisfied for a two candidate election?

may s theorem
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)
majority rule
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
  • 2008 Presidential Election

Obama: 1,959,532 votes


McCain: 1,725,005 votes


Total Votes cast:


Quota: 1,842,528.5

enter third candidate
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…

third candidate or more
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?
simple example saari
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


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

borda count
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
borda count1
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)

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


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.

paradox with borda scheme
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
lets return to the party example
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…

borda count in practice
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.

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

condorcet method
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
condorcet continued
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.
example minnesota gubernatorial race
Example: Minnesota Gubernatorial Race

Photo source:,_1998

Jesse Ventura (Reform Party)

St. Paul Mayor Norm Coleman (R)

Attorney General Skip Humphrey (D)

example minnesota gubernatorial race1
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

example minnesota gubernatorial race2
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

example minnesota gubernatorial race3
Example: Minnesota Gubernatorial Race
  • Condorcet winner: Norm Coleman
  • Condorcet loser: Jesse Ventura

What about other voting Systems:




  • 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?
relationship between borda and condorcet
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]
borda count and condorcet s method at william and mary
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.”

sequential pairwise voting
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).


  • 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

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


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’

instant runoff or single transferable vote
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).
in practice
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.”
example academy awards
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
example 2008 best picture
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

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


Round 1: Does any candidate meet the Droop Quota?

Yes- G

9-6=3 excess votes are distributed to C and A

final selections
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.


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?
summary evaluating voting systems
Summary: Evaluating Voting Systems

Each fails to satisfy one desirable property

arrow s theorem
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:



    • Universality
    • Monotonicity
    • Independence of Irrelevant Alternatives
    • Citizen Sovereignty
    • Nondictatorship

(Hodge and Klima)

  • 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)

In head to head:

A > B

B > C


is there a better way
Is there A Better Way?
  • For 2 Candidates- no problems
  • For 3 or more Candidates- no system that satisfies all properties
  • Possibilities supported by scholars:
    • Approval Voting
approval voting
Approval Voting
  • 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.
so what
So What
  • 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
what if electoral college tie
What If: Electoral college Tie

12th Amendment- requires 270 votes in the electoral college to win a presidential election.

Is 269 – 269 tie possible?

2008 presidential election
2008 Presidential Election
  • Analysis and modeling by Nate Silver of
  • 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.

  • “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.
  • Class Election

Rank the following:


Green Leaf


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.

  • COMAP text
  • Hodge, Jonathan and Richard Klima. The Mathematics of Voting and Elections: A Hands on Approach. Providence: American Mathematical Society, 2005.

William and Mary Links


Voting and Social Choice, Princeton University.

  • “Voting and Elections: An Introduction.” American Mathematical Society.
  • Delvin, Kevin. “The perplexing mathematics of presidential elections.” Mathematics Assocation of America. November 2000.
  • Mackenzie, Dana. “Making Sense out of consensus.” October 21, 2000. Society for Industrial and Applied Mathematics.