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The Mathematics of Social Choice:Is Democracy Mathematically Unsound?

Jose Maria P. Balmaceda

ProfessorInstitute of Mathematics

University of the Philippines Dilimanjoey@math.upd.edu.ph

Science, Technology and Society

Prelude

Order and Mathematics

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Order in Mathematics

Concept of order appears everywherein mathematics

Order theory studies various binaryrelations that capture the intuitivenotion of mathematical ordering

Examples: Usual order, ≤, on natural numbers Lexicographic ordering of words

Pictorial Representation ofOrder (Lattices)

Example: Subsets ofa set {a,b}

{a,b}

{a} {b}

∅

Example:Subsets of a 4-element

set

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Ordering the divisors of an integer:

“x ≤ y” , if x is a divisor of y

6 is the LCM of2 and 3

5 is the GCD of 20 and 15

Example: divisors of 60

3 Diagrams of the Lattice ofSubsets of a 4-Element Set

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Lattices are studied in many areasof math and computer science

Crystallography Number theory Cryptography Coding theory Sphere packing

Order and Society

• Social Choice and Voting• Paradoxes of Democracy• Ideal Voting Systems and Arrow’s

Theorem

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Social Choice Theory andVoting Systems

Social Choice Theory:deals with process bywhich varied andconflicting choices areconsolidated into asingle choice of thegroup (or society)

Voting: vehicle bywhich decisions aremade in a democraticsociety

Underlyingprinciple is that ofordering orranking(of preferencesor choices)

Social C

hoice

and

Individual

Values

By

Kenneth

Arrow

1952

Voting Systems

Voting system: a way for a group to select one(winner) from among several candidates

If there are only two alternatives, choosing is easy:the one preferred by the majority wins

If there is only one person doing the choosing, thingsare again easy (but this option is probablyundesirable)

When several people choose from among three ormore alternatives, the process is trickier

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Some assumptions on our votingsystem:

Individual preferences are assumed to betransitive: if a voter prefers X to Y and Y to Z,it is reasonable to assume that the voterprefers X to Z

Relative preferences are not altered by theelimination of one or more candidates

These are so-called “fairness” assumptions.Later, we shall impose other fairnessprinciples on our voting system

Non-transitivity can cause problems

Situation : Choosing a Suitor

• 3 suitors: ALEX , BUDDY , CALOY

• Girl ranks the 3 men according to : intelligence, physical

attractiveness, income

Intelligence Physical Income

Rank 1 A B C

Rank 2 B C A

Rank 3 C A B

! Taken in pairs, she prefers: A to B

B to C

C to A

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Situation : Choosing a Candidate

• 3 candidates: A, B, C, ranked by all voter s

1/3 of

voters

1/3 of

voters

1/3 of

voters

Rank 1

A B C

Rank 2

B C A

Rank 3

C A B

! Taken in pairs: 2/3 prefer A to B

2/3 prefer B to C

2/3 prefer C to A

Choosing a fastfood place

Choose first between McDo and Wendy’s McDo vs Wendy’s → Wendy’s Then, Wendy’s vs Jollibee → Jollibee (winner)

But Vic is unhappy.

McDoJollibeeWendy’s3rd Choice

Wendy’sMcDoJollibee2nd Choice

JollibeeWendy’sMcDo1st Choice

JoeyVicTito

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Insincere or Strategic Voting

Suppose Vic is insincere. He votes for McDo(instead of his real 1st choice, Wendy’s).

McDoJollibeeWendy’s3rd Choice

Wendy’sMcDoJollibee2nd Choice

JollibeeWendy’sMcDo1st Choice

JoeyVicTito

Insincere or Strategic Voting

Suppose Vic is insincere. He votes for McDo(instead of his real 1st choice, Wendy’s). McDo vs Wendy’s → McDo McDo vs Jollibee → McDo

Vic is satisfied (he gets his 2nd choice)

McDoJollibeeWendy’s3rd Choice

Wendy’sWendy’sJollibee2nd Choice

JollibeeMcDoMcDo1st Choice

JoeyVicTito

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Voting Methods

The most popular method of voting isplurality voting.

A candidate with the most number ofvotes, or most 1st-place votes wins.

This seems like a very reasonablemethod, right? Yes, but…

Example: Plurality isn’t always the best method

• 100 residents elect their barangay leader.

• The candidates are R, H, C, O, and S.

• The results (given by a preference schedule):

No. of voters 49 48 3

1st choice R H C

2nd choice H S H

3rd choice C O S

4th choice O C O

5th choice S R R

• Using plurality, the choice is R, despite being the last

choice of a majority (51).

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Consider H :

• H is the first choice of 48 (only 1

less than R) and also has 52 second

place votes.

• Under any reasonable interpretation,

H is more representat ive of the

town’s choice than R, but plurality

method fails to choose H.

Number of vot ers:

49 48 3

1st R H C

2nd

H S H

3rd C O S

4th O C O

5th S R R

Consider H :

• H is the first choice of 48 (only 1

less than R) and also has 52 second

place votes.

• Under any reasonable interpretation,

H is more representat ive of the

town’s choice than R, but plurality

method fails to choose H.

Number of vot ers:

49 48 3

1st R H C

2nd

H S H

3rd C O S

4th O C O

5th S R R

In a one-to-one comparison H would always get a majority of

the votes.

• Compare H and R: H would get 51 votes (48 from the

second column and 3 from the third) versus 49 for R.

• Comparing H and C would result in 97 votes for H and only

3 for C.

• Finally H is preferred to both O and S by all 100 voters.

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Fairness Criteria

In the language of voting theory, we saythat the plurality method fails to satisfya basic principle of fairness called theCondorcet criterion.

Condorcet’s Criterion:Marquis de Condorcet, 1743-1794

If there is acandidate who winsin a one-to-onecomparison withany otheralternative, thenthat candidateshould be thewinner of theelection.

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We shall examine several votingmethods and discuss other fairnesscriteria.

Common Voting Methods (preferential voting)

The STS Club Election

There are four candidate s for the position of President:

Alice (A) Ben (B) Cris (C) Dina (D)

37 members of the club each submit a ballot indicating his or h er

1st, 2nd, 3rd, 4 th choices.

No. of voters 14 10 8 4 1

1st choice A C D B C

2nd

choice B B C D D

3rd choice C D B C B

4th choice D A A A A

We shall use several voting methods to pick the winner.

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Plurality Method: candidate with most 1st place votes wins

A : 14 first-place votes C : 11 first-place votesB : 4 first-place votes D : 8 first-place votes

A (Alice) wins using plurality.

STSC election results:

No. of voters 14 10 8 4 1 1st A C D B C 2nd B B C D D 3rd C D B C B 4th D A A A A

Borda Count Method : weighted voting method Jean-Charles de Borda, 1733 -1799

No. of voters 14 10 8 4 1

1st choice A C D B C

2nd choice B B C D D

3rd choice C D B C B

4th choice D A A A A

• Each place on a ballot is assi gned points . In there are

N candidates, give N points for first place, N -1 points for

second, and so on, until the last place, to be given 1

point.

• The points are tallied for each candidate, and the

candidate with the highest total wins.

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Borda count for the STS Club election:

Rank \ # Vote 14 10 8 4 1

1st : 4 pts A: 56 C: 40 D: 32 B: 16 C: 4

2nd

: 3 pts B: 42 B: 30 C: 24 D: 12 D: 3

3rd

: 2 pts C: 28 D:20 B: 16 C: 8 B: 2 4th : 1 pt D: 14 A: 10 A: 8 A: 4 A: 1

A gets 56 + 10 + 8 + 4 + 1 = 79 points

B gets 42 + 30 + 16 + 16 + 2 = 106 points

C gets 28 + 40 + 24 + 8 + 4 = 104 points

D gets 14 + 20 + 32 + 12 + 3 = 81 points

The winner is Ben (B) !

Method of Pairwise Comparisons : Head-to-head match -ups

• Every candidate is matched on a one -to-one basis with every other candidate.

• Each of these one -to-one pairings is called a pairwise

comparison .

• When pairing two candidates (say X or Y) one on one , each vote is assigned to either X or Y by the order of preference indicated by the voter. (X gets the votes of all voters ranking X higher than Y.)

• Who will win the STS Club election using this method?

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No. of voters 14 10 8 4 1

1st choice A C D B C

2nd

choice B B C D D

3rd

choice C D B C B 4

th choice D A A A A

1. Compare A versus B

• A is preferred by 14 over B

• B is preferred by 23 ove r A ! B gets 1 point.

2. Compare all other p airs

A vs C (14 to 23) ! C gets 1 pt

A vs D (14 to 23) ! D gets 1 pt

B vs C (18 to 19) ! C gets 1 pt

B vs D (28 to 9) ! B gets 1 pt

C vs D (25 to 12) ! C gets 1 pt

! C has the most (3 pts), so Cris is the winner!

Plurality-with-Elimination

Method:

• a sophisticated version of plurality met hod and is carried out

in round s

• eliminate candidates with fewest number of 1st place votes

one at a time, until a candidate with a majority of 1 stplace

votes emerges.

Example: Math Lovers Club election

# of voters 14 10 8 4 1

1st choice A C D B C A has 14 first places

2nd

choice B B C D D B has 4 first places

3rd

choice C D B C B C has 11 first places

4th

choice D A A A A D has 8 first places

Round 1: Eliminate B

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Plurality with elimination, round 2:

In round 1, B got fewest 1 st place votes and was eliminated.

# 14 10 8 4 1 Round 2

1st A C D B C # 14 10 8 4 1

2nd

B B C D D ! 1st A C D D C

3rd

C D B C B 2nd

C D C C D

4th D A A A A 3

rd D A A A A

A: 14 first places

C: 11 first places

D: 12 first places

Therefore, eliminate C

Round 3: 14 23 A: 14 first places

1st A D D: 23 first places

2nd

D A

Therefore Dave (D) wins, using plurality with

elimination!

Round 2

# 14 11 12

1st A C D

2nd

C D C

3rd D A A

Summary: STS Club Election

Voting Method Winner

Plurality AliceBorda count BenPairwise comparison CrisPlurality with elimination Dave

4 methods, 4 different winners! Which method is best?

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Problems with the Different VotingMethods

Earlier, we saw that the pluralitymethod violates the Condorcetcriterion

Are the other voting methods better?

Problems with the Borda method:4 candidates, 11 voters

B gets 32 ptsC gets 30 ptsD gets 19 pts

B wins under the Borda method, even if A has themajority of first place votes (6 of 11).

This violates the Majority Criterion.

AAD4th: 1 pt

BDC3rd: 2 pts

DCB2nd: 3 pts

CBA1st: 4 pts

326# votersA gets 29 pts(4x6 + 1x2 + 1x3)

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Majority Criterion:

If there is a candidate that is the firstchoice of the majority of the voters,then that candidate should be thewinner.

Problems with Plurality with Elimination method:

Example: 3 candidates : A, B, C

# of votes 7 8 10 4 1

st choice A B C A

2nd

choice B C A C B has fewest 1st places.

3rd

choice C A B B Therefore, eliminate B.

# of votes 11 18

1st choice A C C has majority of 1

st place

2nd

choice C A Therefore, C wins.

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Problems with Plurality with Elimination method:

Example: 3 candidates : A, B, C

# of votes 7 8 10 4

1st choice A B C A

2nd

choice B C A C B has fewest 1st places.

3rd

choice C A B B Therefore, eliminate B.

# of votes 11 18 1st choice A C C has majority of 1 st place

2nd

choice C A Therefore, C wins.

Suppose the election was declared null and void (due to

irregularities).

A second election is held. The 4 voters (in the last column) change

their vote and switch their 1st and 2

nd choice s (between A and C)

Since C won the first election, and the new votes only

increased C’s votes, we expect C to win again.

New election:

# of votes 7 8 10 4 Voters in last column 1

st choice A B C C switch A and C.

2nd choice B C A A

3rd

choice C A B B

Since A has fewest first places (7) eliminate A.

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New election:

# of votes 7 8 10 4 Voters in last column 1

st choice A B C C switch A and C.

2nd choice B C A A

3rd

choice C A B B

Since A has fewest first places (7) , eliminate A.

# of votes 15 14 1

st choice B C

2nd choice C B B wins this time!

This violates another fairness principle called the

Monotonicity Criterion .

Monotonicity Criterion

If a candidate X is the winner of anelection, and in a re-election, allvoters who change their preferencesdo so in a way that is favorable onlyto X, then X should still be thewinner.

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Problems with Method of PairwiseComparison

DEABEE5th

CAEACD4th

ABBDBC3rd

ECCEDB2nd

BDDCAA1st

423535# A vs B: 13 to 9, A winsA vs C: 12 to 10, A winsA vs D: 12 to 10, A winsA vs E: 10 to 12, E wins

Since A has 3 pts, B has 2 pts, C and D have 2 pts, E has 1 pt,the winner using this method is A.

B vs C: B winsB vs D: D winsB vs E: B winsC vs D: C winsC vs E: C winsD vs E: D wins

Now, suppose, that for some reason, the votes have to berecounted. But before they are, candidates B, C, D becomediscouraged and drop out.

EAEE5th

AEA4th

A3rd

EE2nd

AA1st

423535# Eliminating B,C,Dgives:

The winner is now E ! Originally, the winner was A, butwhen some candidates dropped out, and no re-vote wasmade, the winner became E. This violates still anothercriterion.

AE2nd

EA1st

1210#

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Independence of IrrelevantAlternatives Criterion:

If a candidate X is the winner of anelection, and one or more candidatesare removed and votes are recounted,then X should still be the winner.

Other fairness criteria:

Unanimity : if every individual prefersa certain option to another, then somust the resulting societal choice

Non-dictatorship : the social choicefunction should not simply follow thepreference order of a single individualwhile ignoring all others

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Summary: STS Club Election

Voting Method Winner

Plurality AliceBorda count BenPairwise comparison CrisPlurality with elimination Dave

4 methods, 4 different winners!

Each method fails to satisfy some fairnesscriterion!

More precisely, it is possible that in aparticular election, a particular outcome ofvotes will violate some fairness criterion.

Which is the best voting method?

There is no ideal voting method!

CHAOTIC ELECTIONS!A Mathematician Looks at

Voting,by Donald Saari, 2001American Math. Soc.

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Arrow’s Impossibility Theorem

There is no consistent method by which ademocratic society can make a choice thatis always fair when that choice must bemade from among several (three or more)alternatives.

Kenneth Arrow, in 1952 essay “A Difficulty in the Concept ofSocial Welfare” and PhD dissertation “Social Choice andIndividual Values”

Kenneth J. Arrow, b. 19211972 Nobel Prize, Economics

Arrow proved that it isimpossible to design a set ofrules for social decision thatwould simultaneously obeyall of the fairness criteriabelow:• Transitivity• Monotonicity• Independence of irrelevant

alternatives• Unanimity• Non-dictatorship

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An Implication of Arrow’s Theorem:

The only social choice functionthat respects, transitivity,unanimity, monotonicity, andindependence of irrelevantalternatives is a dictatorship!

Arrow’s mathematical proof usesconcepts of order theory.

Remarks

Arrow’s theorem applies only toranked or preferential voting systems

It doesn’t prescribe a “best” method;certainly doesn’t say dictatorship isbetter

It does prove that no voting methodcan satisfy at the same time allreasonable fairness criteria (for allpossible outcomes of votes)

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Are there alternative methods?

• Non-preferential methods - voters are not asked to rank

candidates according to p reference

• Best known method is the Approval Method – given a set of

candidates, voters can give their approval to as many (or as few)

of the choices. No ranking is made.

Features of Approval Voting (according to advocates ):

1. Easy to understand and simple to implement

2. Gives voters flexible options and increases voter turnout

3. Helps elect the strongest candidates

4. Unaffected by the number of candidates

5. Will reduce ne gative campaigning

6. Will give minority candidates their proper due

Conclusion The concept of order is ubiquitous and important in

mathematics

Lattice theory and order theory formalize and systematize thestudy of order; lattices are important structures for both theoryand application

Order is also critical in society, especially in decision-making

Mathematics allows us to analyze voting systems and othersocial issues, but mathematics does not provide all answers

Arrow’s Impossibility Theorem shows that there can be noperfect voting system; and that fairness and democracy areinherently incompatible

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End NotesThe Math Lovers Election example is from P.

Tannembaum and R. Arnold’s book, Excursionsin Modern Mathematics, Prenctice Hall, 1995.

There are various websites on the mathematicsof voting theory and Arrow’s Theorem (easy tosearch via Google©) as well as sites devoted tolattice and order theory.

Many problems in society and government,such as fair division and apportionment, themeasurement of power, can be helpedanalyzed using mathematics.

Hello…

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Hello… hello…

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