MATH AND VOTING
October 22, 2009
Maura Bardos
OUTLINE

Two Candidates
 Majority

Three Candidates or More






Rule
Plurality
Borda
Condorcet
Sequential Pairwise
Instant Runoff
Arrow’s Theorem
Approval voting
 A better method?

3 PROPERTIES OF FAIR ELECTIONS

Sincere Ballot: A ballot that represents a voter’s true
preferences

3 Properties
1)
2)
3)
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?
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


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

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
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/Ralph_Nader
http://en.wikipedia.org/wiki/Ross_Perot
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)
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
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 COUNT
1)
2)
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
3)
Candidate with largest number of points is
declared the winner. (Hodge and Klima)
EXAMPLE
Rank
3
2
1
A
C
2
B
B
3
C
A
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
EXAMPLE
Rank
3
2
1
A
B
2
B
C
3
C
A
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
Fails the Independence of Irrelevant Alternatives
(IIA)
 IIA- a voting system satisfies this criteria if it is
impossible for a candidate to move from nonwinner 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:
Rank
6
5
4
1
Salad
Popcorn
Chips
2
Chips
Chips
Popcorn
3
Popcorn
Salad
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…
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.

BORDA COUNT MVP
2006 AL MVP Award
Voting results ¬
Player, Club
1st
Justin
Morneau,
MIN
Derek Jeter,
NYY
2nd
3rd
4th
15
8
3
2
12
14
5th
6th
7th
8th
9th
10th Points
320
1
1
306
David Ortiz,
BOS
1
11
5
7
3
1
193
Frank
Thomas, OAK
3
4
7
7
4
1
174
Jermaine
Dye, CWS
1
2
6
5
7
4
2
1
3
6
1
2
5
3
2
1
116
5
1
3
3
3
1
1
3
114
Joe Mauer,
MIN
Johan
Santana, MIN
1
156
Voting results ¬
Player, Club
Justin
Morneau,
MIN
Derek Jeter,
NYY
1st
2nd
3rd
4th
15
8
3
2
12
14
1
5th
6th
7th
8th
9th
10th Points
320
1
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
306
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-tohead 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-tohead contest (with the winner decided by
majority rule). (pg. 40)


Only one Condorcet loser and one Condorcet winner
per election
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
Photo source: http://en.wikipedia.org/wiki/Minnesota_gubernatorial_election,_1998
Jesse Ventura
(Reform Party)
St. Paul Mayor
Norm Coleman (R)
http://www.youtube.com/watch?v=TjU948
M0ARw
Attorney General Skip
Humphrey (D)
EXAMPLE: MINNESOTA GUBERNATORIAL
RACE
Rank
35
28
20
17
1
N
S
J
J
2
S
N
N
S
3
J
J
S
N
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
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

Rank
35
28
20
17
1
N
S
N
S
2
S
N
S
N
Now try Problem 2
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?

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

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
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).

EXAMPLE
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
Rank
1
1
1
1
a
c
b
2
b
a
d
3
d
b
c
4
c
d
a
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
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)
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 1st 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).

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
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

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
1st
6
G
3
G
4
C
3
A
1
H
2
I
3
B
2
D
1
D
5
F
2nd
3rd
4th
5th
C
E
F
I
A
E
C
H
I
E
A
F
B
E
D
C
B
E
I
D
B
E
H
G
A
E
I
D
A
E
B
G
F
E
C
A
D
E
C
H
Round 1: Does any candidate meet the Droop Quota?
Yes- G
9-6=3 excess votes are distributed to C and A
2
1
4
C
3
A
1
H
2
I
3
B
2
D
1
D
5
F
C
E
F
I
A
E
C
H
I
E
A
F
B
E
D
C
B
E
I
D
B
E
H
A
E
I
D
A
E
B
F
E
C
A
D
E
C
H
1st
2nd
3rd
4th
5th
Rounds 2 and 3- C reaches minimum number, E is eliminated
1
1st
2nd
A
3
1
2
3
2
1
5
A
H
I
B
D
D
F
B
B
B
A
A
F
D
D
I
H
I
B
A
H
3rd
4th
5th
H
D
D
Rounds 4 and 5- Eliminate H. Transfer one vote to B.
Eliminate I
1
1st
2n
d
3
1
2
A
A
B
B
B
3
2
1
5
B
D
D
F
A
A
F
D
3rd
4th
5th
D
B
D
D
Rounds 6 and 7- B is selected. D is eliminated.
A
FINAL SELECTIONS
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
SUMMARY: EVALUATING VOTING SYSTEMS
Anon.
Neutral
Monotone
MC
CWC
Plurality
Y
Y
Y
Y
N
Borda
Count
Y
Y
Y
N
N
Sequential
Pairs
Y
N
Y
Y
Y
Instant
Runoff
Y
Y
N
Y
N
Each fails to satisfy one desirable property
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:

Conditions:
Universality
2) Monotonicity
3) Independence of Irrelevant Alternatives
4) Citizen Sovereignty
5) Nondictatorship
(Hodge and Klima)
1)
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).

Example (not a proof)
Rank
7
6
5
1
A
B
C
2
B
C
A
3
C
A
B
In head to head:
A>B
B>C
C>A
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:
1)
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

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
12th Amendment- requires 270 votes in the
electoral college to win a presidential election.
Is 269 – 269 tie possible?
2008 PRESIDENTIAL ELECTION


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.
http://www.opinionjournal.com/ecc/calculator.htm

WHAT DOES A TIE LOOK LIKE?
CONCLUSION



“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.
HOMEWORK
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.
SOURCES
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
 http://web.wm.edu/sacs/accdoc/3/7/5/documents/B
ylawsoftheFacultyoftheSchoolofBusinessAdminis
tration.pdf?svr=www
 http://sa.wm.edu/other/aia/constitution.php




Voting and Social Choice, Princeton University.
http://www.math.princeton.edu/math_alive/6/index.sh
tml
“Voting and Elections: An Introduction.” American
Mathematical Society.
http://www.ams.org/featurecolumn/archive/votingintroduction.html
Delvin, Kevin. “The perplexing mathematics of
presidential elections.” Mathematics Assocation of
America. November 2000.
http://www.maa.org/devlin/devlin_11_00.html
Mackenzie, Dana. “Making Sense out of consensus.”
October 21, 2000. Society for Industrial and Applied
Mathematics.
http://www.siam.org/news/news.php?id=674
SOURCES









http://dev.whydomath.org/node/voting/voting_vectors_mvp.
html
http://blogs.wsj.com/numbersguy/voting-math-doesntalways-add-up-564/
http://blogs.wsj.com/numbersguy/numbers-guy-interviewsteven-brams-340/
http://dev.whydomath.org/node/voting/academy_awards.ht
ml
http://www.oscars.org/press/pressreleases/2009/20090831a.
html
http://online.wsj.com/article/SB123388752673155403.html
http://blogs.wsj.com/numbersguy/some-theorists-withholdbest-voting-system-award-794/
http://www.fivethirtyeight.com/2009/03/colorado-becomesfront-line-in-battle.html
http://www.fivethirtyeight.com/search/label/12th%20amend
ment