October 22 2009 Maura Bardos Outline Two Candidates Majority Rule Three Candidates or More Plurality Borda Condorcet Sequential Pairwise Instant Runoff Arrows Theorem Approval voting ID: 300660
Download Presentation The PPT/PDF document "Math and Voting" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Math and Voting
October 22, 2009
Maura
BardosSlide2
Outline
Two Candidates
Majority Rule
Three Candidates or More
Plurality
Borda
Condorcet
Sequential
Pairwise
Instant Runoff
Arrow’s Theorem
Approval voting
A better method?Slide3
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?Slide4
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
)Slide5
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/2Slide6
Example
2008 Presidential Election
Obama
:
1,959,532
votes
53%
McCain
:
1,725,005
votes
47%
Total Votes cast:
3,864,537
Quota: 1,842,528.5Slide7
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_PerotSlide8
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?Slide9
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!Slide10
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. Slide11
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 changeSlide12
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 basedSlide13
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
)Slide14
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 winnerSlide15
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. Slide16
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 criterionSlide17
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…Slide18
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.Slide19
Borda Count MVP
2006 AL MVP Award
Voting results ¬
Player, Club
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
Points
Justin
Morneau
, MIN
15
8
3
2
320
Derek Jeter, NYY
12
14
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
156
Joe Mauer, MIN
3
6
1
2
5
3
2
1
116
Johan Santana, MIN
1
5
1
3
3
3
1
1
3
114Slide20
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, Club
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
Points
Justin Morneau, MIN
15
8
3
2
320
Derek Jeter, NYY
12
14
1
1
306Slide21
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 electionSlide22
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.Slide23
Example: Minnesota Gubernatorial Race
Jesse Ventura (Reform Party)
Attorney General Skip Humphrey (D)
St. Paul Mayor Norm Coleman (R)
Photo source: http://en.wikipedia.org/wiki/Minnesota_gubernatorial_election,_1998
http://www.youtube.com/watch?v=TjU948M0ARwSlide24Slide25
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 systemsSlide26
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 2Slide27
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?Slide28
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]Slide29
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.” Slide30
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). Slide31
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 winnerSlide32
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: ABDCa vs. b: a a
vs. d
:
a
a
vs. d
:
c
Rank
1
1
1
1
a
c
b
2
b
a
d
3
d
b
c
4
c
d
aSlide33
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’ Slide34
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 1
st
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
).Slide35Slide36
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.”Slide37
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: PluralitySlide38
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 & MeSlide39
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 = 6Slide40
6
3
4
3
1
2
3
2
1
5
1st
G
G
C
A
H
I
B
D
D
F
2nd
C
A
I
B
B
B
A
A
F
D
3rd
E
E
E
E
E
E
E
E
E
E
4th
F
C
A
D
I
H
I
B
C
C
5th
I
H
F
C
D
G
D
G
A
H
Round 1: Does any candidate meet the Droop Quota?
Yes- G
9-6=3 excess votes are distributed to C and ASlide41
2
1
4
3
1
2
3
2
1
5
1st
C
A
H
I
B
D
D
F
2nd
C
A
I
B
B
B
A
A
F
D
3rd
E
E
E
E
E
E
E
E
E
E
4th
F
C
A
D
I
H
I
B
C
C
5th
I
H
F
C
D
D
A
H
Rounds 2 and 3- C reaches minimum number, E is eliminatedSlide42
1
3
1
2
3
2
1
5
1st
A
H
I
B
D
D
F
2nd
A
B
B
B
A
A
F
D
3rd
4th
D
I
H
I
B
5th
H
D
D
A
H
Rounds 4 and 5- Eliminate H. Transfer one vote to B. Eliminate ISlide43
1
3
1
2
3
2
1
5
1st
A
B
D
D
F
2nd
A
B
B
B
A
A
F
D
3rd
4th
D
B
5th
D
D
A
Rounds 6 and 7- B is selected. D is eliminated. Slide44
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. Slide45
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?Slide46
…stay tuned for February 2, 2010Slide47
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 propertySlide48
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:Slide49
Conditions:
Universality
Monotonicity
Independence of Irrelevant Alternatives
Citizen Sovereignty
Nondictatorship
(Hodge and
Klima
)Slide50
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).Slide51
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>ASlide52
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 VotingSlide53
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. Slide54
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 momentSlide55
What If: Electoral college Tie
12
th
Amendment- requires 270 votes in the electoral college to win a presidential election.
Is 269 – 269 tie possible?Slide56
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.htmSlide57
What does a Tie Look Like?Slide58
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.Slide59
Homework
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.Slide60
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/BylawsoftheFacultyoftheSchoolofBusinessAdministration.pdf?svr=www
http://sa.wm.edu/other/aia/constitution.phpSlide61
Voting and Social Choice, Princeton University. http://www.math.princeton.edu/math_alive/6/index.shtml
“Voting and Elections: An Introduction.” American Mathematical Society. http://www.ams.org/featurecolumn/archive/voting-introduction.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=674Slide62
Sources
http://dev.whydomath.org/node/voting/voting_vectors_mvp.html
http://blogs.wsj.com/numbersguy/voting-math-doesnt-always-add-up-564/
http://blogs.wsj.com/numbersguy/numbers-guy-interview-steven-brams-340/
http://dev.whydomath.org/node/voting/academy_awards.html
http://www.oscars.org/press/pressreleases/2009/20090831a.html
http://online.wsj.com/article/SB123388752673155403.html
http://blogs.wsj.com/numbersguy/some-theorists-withhold-best-voting-system-award-794/
http://www.fivethirtyeight.com/2009/03/colorado-becomes-front-line-in-battle.html
http://www.fivethirtyeight.com/search/label/12th%20amendment