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Voting Problems and Computer Science Applications

Fred Roberts, Rutgers University

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What do Mathematics and Computer Science have to do with Voting?

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Have you used Google lately?

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Have you used Google lately?

Did you know that Google has something to do with voting?

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Have you tried buying a book on online lately?

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Have you tried buying a book on online lately?

Did you get a message saying: If you are interested in this book, you might want to look at the following books as well?

Did you know that has something to do with voting?

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Have you ever heard of v-sis?

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Have you ever heard of v-sis?

•It’s a cancer-causing gene.

•Computer scientists helped discover how it works?

•How did they do it?

•The answer also has something to do with voting.

Cancer cell

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•Some connections between Computer Science and Voting are clearly visible.

•Some people are working on plans to allow us to vote from home – over the Internet.

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Electronic Voting

Security Risks in Electronic Voting

• Could someone put on a “denial of service attack?”

• That is, could someone flood your computer and those of other likely voters with so much spam that you couldn’t succeed in voting?

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Electronic VotingSecurity Risks in Electronic Voting

• How can we prevent random loss of connectivity that would prevent you from voting?

• How can your vote be kept private?

• How can you be sure your vote is counted?

• What will prevent you from selling your vote to someone else?

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Electronic VotingSecurity Risks in Electronic Voting

• These are all issues in modern computer science research.

• However, they are not what I want to talk about.

• I want to talk about how ideas about voting systems can solve problems of computer science.

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How do Elections Work?

• Typically, everyone votes for their first choice candidate.

• The votes are counted.• The person with the most

votes wins.• Or, sometimes, if no one

has more than half the votes, there is a runoff.

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But do we necessarily get the best candidate that way?

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Sometimes Having More Information about Voters’

Preferences is Very Helpful

•Sometimes it is helpful to have voters rank order all the candidates•From their top choice to their bottom choice.

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Rankings

Dennis Kucinich

Bill Richardson

John Edwards

Barack Obama Hillary Clinton

Ties are allowed

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Rankings • What if we have four voters and they give us the

following rankings? Who should win?

Voter 1 Voter 2 Voter 3 Voter 4Clinton Clinton Obama ObamaRichardson Kucinich Edwards RichardsonEdwards Edwards Richardson KucinichKucinich Richardson Kucinich EdwardsObama Obama Clinton Clinton

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Rankings • What if we have four voters and they give us the

following rankings? • There is one added candidate. • Who should win?

Voter 1 Voter 2 Voter 3 Voter 4Clinton Clinton Obama ObamaGore Gore Gore GoreRichardson Kucinich Edwards RichardsonEdwards Edwards Richardson KucinichKucinich Richardson Kucinich EdwardsObama Obama Clinton Clinton

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Rankings Voter 1 Voter 2 Voter 3 Voter 4Clinton Clinton Obama ObamaGore Gore Gore GoreRichardson Kucinich Edwards RichardsonEdwards Edwards Richardson KucinichKucinich Richardson Kucinich EdwardsObama Obama Clinton Clinton

Maybe someone who is everyone’s second choice is the best choice for winner.Point: We can learn something from ranking candidates.

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Consensus Rankings •How should we reach a decision in an election if every voter ranks the candidates?

•What decision do we want? − A winner− A ranking of all the candidates that is in some

sense a consensus ranking

•This would be useful in some applications• Job candidates are ranked by each interviewer• Consensus ranking of candidates• Make offers in order of ranking

•How do we find a consensus ranking?

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Consensus Rankings

These two rankings are very close:

Clinton ObamaObama ClintonEdwards EdwardsKucinich KucinichRichardson Richardson

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Consensus Rankings

These two rankings are very far apart:

Clinton ObamaRichardson KucinichEdwards EdwardsKucinich RichardsonObama Clinton

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Consensus Rankings

•This suggests we may be able to make precise how far apart two rankings are.

•How do we measure the distance between two rankings?

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Consensus Rankings• Kemeny-Snell distance between rankings: twice the

number of pairs of candidates i and j for which i is ranked above j in one ranking and below j in the other + the number of pairs that are ranked in one ranking and tied in another.

a bx y-zy xzOn {x,y}: +2On {x,z}: +2On {y,z}: +1d(a,b) = 5.

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Consensus Rankings

• One well-known consensus method: “Kemeny-Snell medians”: Given set of rankings, find ranking minimizing sum of distances to other rankings.

• Kemeny-Snell medians are having surprising new applications in CS.

John Kemeny,pioneer in time sharingin CS

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Consensus Rankings

• Kemeny-Snell median: Given rankings a1, a2, …, ap, find a ranking x so that

d(a1,x) + d(a2,x) + … + d(ap,x) is as small as possible.

• x can be a ranking other than a1, a2, …, ap.

• Sometimes just called Kemeny median.

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Consensus Rankings a1 a2 a3

Fish Fish ChickenChicken Chicken FishBeef Beef Beef

• Median = a1.

• If x = a1:d(a1,x) + d(a2,x) + d(a3,x) = 0 + 0 + 2 = 2

is minimized.• If x = a3, the sum is 4.• For any other x, the sum is at least 1 + 1 + 1 = 3.

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Consensus Rankings a1 a2 a3

Fish Chicken BeefChicken Beef FishBeef Fish Chicken

• Three medians = a1, a2, a3.

• This is the “voter’s paradox” situation.

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Consensus Rankings a1 a2 a3

Fish Chicken BeefChicken Beef FishBeef Fish Chicken

• Note that sometimes we wish to minimize

d(a1,x)2 + d(a2,x)2 + … + d(ap,x)2

• A ranking x that minimizes this is called a Kemeny-Snell mean.

• In this example, there is one mean: the ranking declaring all three alternatives tied.

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Consensus Rankings a1 a2 a3

Fish Chicken BeefChicken Beef FishBeef Fish Chicken

• If x is the ranking declaring Fish, Chicken and Beef tied, then

d(a1,x)2 + d(a2,x)2 + … + d(ap,x)2 = 32 + 32 + 32 = 27.

• Not hard to show this is minimum.

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Consensus Rankings

Theorem (Bartholdi, Tovey, and Trick, 1989; Wakabayashi, 1986): Computing the Kemeny-Snell median of a set of rankings is an NP-complete problem.

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Consensus Rankings

Okay, so what does this have to do with practical computer science questions?

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Consensus Rankings

I mean really practical computer science questions.

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Google Example

• Google is a “search engine”• It searches through web pages and rank orders

them. • That is, it gives us a ranking of web pages from

most relevant to our query to least relevant.

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Meta-search • There are other search engines besides Google.• Wouldn’t it be helpful to use several of them and

combine the results?• This is meta-search.• It is a voting problem• Combine page rankings from several search engines to

produce one consensus ranking• Dwork, Kumar, Naor, Sivakumar (2000): Kemeny-

Snell medians good in spam resistance in meta-search (spam by a page if it causes meta-search to rank it too highly)

• Approximation methods make this computationally tractable

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Collaborative Filtering • Recommending books or movies• Combine book or movie ratings by various

people• This too is voting• Produce a consensus ordered list of books or

movies to recommend• Freund, Iyer, Schapire, Singer (2003):

“Boosting” algorithm for combining rankings.• Related topic: Recommender Systems

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Meta-search and Collaborative Filtering

A major difference from the election situation

• In elections, the number of voters is large, number of candidates is small.

• In CS applications, number of voters (search engines) is small, number of candidates (pages) is large.

• This makes for major new complications and research challenges.

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Have you ever heard of v-sis?

•It’s a cancer-causing gene.

•Computer scientists helped discover how it works?

•How did they do it?

•The answer also has something to do with voting.

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Large Databases and Inference

• Decision makers consult massive data sets.• The study of large databases and gathering of

information from them is a major topic in modern computer science.

• We will give an example from the field of Bioinformatics.

• This lies at the interface between Computer Science and Molecular Biology

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Large Databases and Inference • Real biological data often in form of sequences.• GenBank has over 7 million sequences

comprising 8.6 billion “bases.” • The search for similarity or patterns has

extended from pairs of sequences to finding patterns that appear in common in a large number of sequences or throughout the database: “consensus sequences”

• Emerging field of “Bioconsensus”: applies consensus methods to biological databases.

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Large Databases and Inference

Why look for such patterns?

Similarities between sequences or parts of sequences lead to the discovery of shared phenomena.

For example, it was discovered that the sequence for platelet derived factor, which causes growth in the body, is 87% identical to the sequence for v-sis, that cancer-causing gene. This led to the discovery that v-sis works by stimulating growth.

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Large Databases and Inference DNA Sequences

A DNA sequence is a sequence of “bases”:

A = Adenine, G = Guanine, C = Cytosine, T = Thymine

Example:

ACTCCCTATAATGCGCCA

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Large Databases and Inference Example

Bacterial Promoter Sequences studied by Waterman (1989):

RRNABP1: ACTCCCTATAATGCGCCATNAA: GAGTGTAATAATGTAGCCUVRBP2: TTATCCAGTATAATTTGTSFC: AAGCGGTGTTATAATGCC

Notice that if we are looking for patterns of length 4, each sequence has the pattern TAAT.

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Large Databases and Inference Example

Bacterial Promoter Sequences studied by Waterman (1989):

RRNABP1: ACTCCCTATAATGCGCCATNAA: GAGTGTAATAATGTAGCCUVRBP2: TTATCCAGTATAATTTGTSFC: AAGCGGTGTTATAATGCC

Notice that if we are looking for patterns of length 4, each sequence has the pattern TAAT.

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Large Databases and Inference Example

However, suppose that we add another sequence:

M1 RNA: AACCCTCTATACTGCGCG

The pattern TAAT does not appear here.However, it almost appears, since the pattern

TACT appears, and this has only one mismatch from the pattern TAAT.

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Large Databases and Inference Example

However, suppose that we add another sequence:

M1 RNA: AACCCTCTATACTGCGCG

The pattern TAAT does not appear here.However, it almost appears, since the pattern

TACT appears, and this has only one mismatch from the pattern TAAT.

So, in some sense, the pattern TAAT is a good consensus pattern.

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Large Databases and Inference Example

We make this precise using best mismatch distance.

Consider two sequences a and b with b longer than a.

Then d(a,b) is the smallest number of mismatches in all possible alignments of a as a consecutive subsequence of b.

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Large Databases and Inference Example

a = 0011, b = 111010

Possible Alignments:111010111010 1110100011 0011 0011

The best-mismatch distance is 2, which is achieved in the third alignment.

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Large Databases and Inference Smith-Waterman Method from Bioinformatics

•Let S be a finite alphabet of size at least 2 and be a finite collection of sequences of length L with entries from S. •Let F() be the set of sequences of length k 2 that are our consensus patterns. (Assume L k.)•Let = {a1, a2, …, an}. •One way to define F() is as follows. •Let d(a,b) be the best-mismatch distance. •Then let F() consist of all those sequences x for which the sum of the distances to elements of is as small as possible. •That is, find x so that

d(a1,x) + d(a2,x) + … + d(an,x)is as small as possible.

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Large Databases and Inference•We call such an F a Smith-Waterman consensus.•(This is a special case of a more general Smith-Waterman consensus method.)•Notice that this consensus is the same as the consensus we used in voting.

Example:

•An alphabet used frequently is the purine/pyrimidine alphabet {R,Y}, where R = A (adenine) or G (guanine) and Y = C (cytosine) or T (thymine). •For simplicity, it is easier to use the digits 0,1 rather than the letters R,Y.

•Thus, let S = {0,1}, let k = 2. Then the possible pattern sequences are 00, 01, 10, 11.

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Large Databases and Inference•Suppose a1 = 111010, a2 = 111111. How do we find F(a1,a2)?•We have:

d(a1,00) = 1, d(a2,00) = 2d(a1,01) = 0, d(a2,01) = 1d(a1,10) = 0, d(a2,10) = 1d(a1,11) = 0, d(a2,11) = 0

•It follows that 11 is the consensus pattern, according to Smith-Waterman’s consensus.

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Example:•Let S ={0,1}, k = 3, and consider F(a1,a2,a3) where a1 = 000000, a2 = 100000, a3 = 111110. Possible pattern sequences are: 000, 001, 010, 011, 100, 101, 110, 111.

d(a1,000) = 0, d(a2,000) = 0, d(a3,000) = 2,d(a1,001) = 1, d(a2,001) = 1, d(a3,001) = 2,

d(a1,100) = 1, d(a2,100) = 0, d(a3,100) = 1, etc.

•The sum of distances from 000 is smaller than the sum of distances from 001 and the same as the sum of distances from 100. So, 001 is not a consensus. •It is easy to check that 000 and 100 minimize the sum of distances. •Thus, these are the two “Smith-Waterman” consensus sequences.

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Large Databases and Inference Other Topics in “Bioconsensus”

• Alternative phylogenies (evolutionary trees) are produced using different methods and we need to choose a consensus tree.

• Alternative taxonomies (classifications) are produced using different models and we need to choose a consensus taxonomy.

• Alternative molecular sequences are produced using different criteria or different algorithms and we need to choose a consensus sequence.

• Alternative sequence alignments are produced and we need to choose a consensus alignment.

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Large Databases and Inference Other Topics in “Bioconsensus”

• Several recent books on bioconsensus.• Day and McMorris [2003]• Janowitz, et al. [2003]

• Bibliography compiled by Bill Day: In molecular biology alone, hundreds of papers using consensus methods in biology.

• Large database problems in CS are being approached using methods of “bioconsensus” having their origin in the theory of voting and elections.

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Software & Hardware Measurement • A statement involving scales of measurement is considered meaningful if its truth or falsity is unchanged under acceptable

transformations of all scales involved.• Example: It is meaningful to say that I weigh more

than my daughter. • That is because if it is true in kilograms, then it is also

true in pounds, in grams, etc.• Even meaningful to say I weigh twice as much as my

daughter.• Not meaningful to say the temperature today is

twice as much as it was yesterday. • Could be true in Fahrenheit, false in Centigrade.

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Software & Hardware Measurement • Measurement theory has studied what statements you

can make after averaging scores.• Think of averaging as a consensus method.• One general principle: To say that the average score of

one set of tests is greater than the average score of another set of tests is not meaningful (it is meaningless) under certain conditions.

• This is often the case if the averaging procedure is to take the arithmetic mean: If s(xi) is score of xi, i = 1, 2, …, n, then arithmetic mean is

is(xi)/n = [s(x1) + s(x2) + … + s(xn)]/n• Long literature on what averaging methods lead to

meaningful conclusions.

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Software & Hardware Measurement A widely used method in hardware measurement:

Score a computer system on different benchmarks.

Normalize score relative to performance of one base system

Average normalized scoresPick system with highest average.Fleming and Wallace (1986): Outcome can

depend on choice of base system. Meaningless in sense of measurement theoryLeads to theory of merging normalized scores

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Software & Hardware Measurement Hardware Measurement

417 83 66 39,449 772

244 70 153 33,527 368

134 70 135 66,000 369

BENCHMARK

R

M

Z

PROCESSOR

E F G H I

Data from Heath, Comput. Archit. News (1984)

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Software & Hardware Measurement Normalize Relative to Processor R

417

1.00

83

1.00

66

1.00

39,449

1.00

772

1.00

244

.59

70

.84

153

2.32

33,527

.85

368

.48

134

.32

70

.85

135

2.05

66,000

1.67

369

.45

BENCHMARK

R

M

Z

PROCESSOR

E F G H I

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Software & Hardware Measurement Take Arithmetic Mean of Normalized Scores

417

1.00

83

1.00

66

1.00

39,449

1.00

772

1.00

244

.59

70

.84

153

2.32

33,527

.85

368

.48

134

.32

70

.85

135

2.05

66,000

1.67

369

.45

BENCHMARK

R

M

Z

PROCESSOR

E F G H I

ArithmeticMean

1.00

1.01

1.07

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Software & Hardware Measurement Take Arithmetic Mean of Normalized Scores

417

1.00

83

1.00

66

1.00

39,449

1.00

772

1.00

244

.59

70

.84

153

2.32

33,527

.85

368

.48

134

.32

70

.85

135

2.05

66,000

1.67

369

.45

BENCHMARK

R

M

Z

PROCESSOR

E F G H I

ArithmeticMean

1.00

1.01

1.07

Conclude that processor Z is best

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Software & Hardware Measurement Now Normalize Relative to Processor M

417

1.71

83

1.19

66

.43

39,449

1.18

772

2.10

244

1.00

70

1.00

153

1.00

33,527

1.00

368

1.00

134

.55

70

1.00

135

.88

66,000

1.97

369

1.00

BENCHMARK

R

M

Z

PROCESSOR

E F G H I

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Software & Hardware Measurement Take Arithmetic Mean of Normalized Scores

417

1.71

83

1.19

66

.43

39,449

1.18

772

2.10

244

1.00

70

1.00

153

1.00

33,527

1.00

368

1.00

134

.55

70

1.00

135

.88

66,000

1.97

369

1.00

BENCHMARK

R

M

Z

PROCESSOR

E F G H I

ArithmeticMean

1.32

1.00

1.08

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Software & Hardware Measurement Take Arithmetic Mean of Normalized Scores

417

1.71

83

1.19

66

.43

39,449

1.18

772

2.10

244

1.00

70

1.00

153

1.00

33,527

1.00

368

1.00

134

.55

70

1.00

135

.88

66,000

1.97

369

1.00

BENCHMARK

R

M

Z

PROCESSOR

E F G H I

ArithmeticMean

1.32

1.00

1.08

Conclude that processor R is best

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Software and Hardware Measurement • So, the conclusion that a given machine is best

by taking arithmetic mean of normalized scores is meaningless in this case.

• Above example from Fleming and Wallace (1986), data from Heath (1984)

• Sometimes, geometric mean is helpful.• Geometric mean is

is(xi)n

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Software & Hardware Measurement Normalize Relative to Processor R

417

1.00

83

1.00

66

1.00

39,449

1.00

772

1.00

244

.59

70

.84

153

2.32

33,527

.85

368

.48

134

.32

70

.85

135

2.05

66,000

1.67

369

.45

BENCHMARK

R

M

Z

PROCESSOR

E F G H I

GeometricMean

1.00

.86

.84

Conclude that processor R is best

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Software & Hardware Measurement Now Normalize Relative to Processor M

417

1.71

83

1.19

66

.43

39,449

1.18

772

2.10

244

1.00

70

1.00

153

1.00

33,527

1.00

368

1.00

134

.55

70

1.00

135

.88

66,000

1.97

369

1.00

BENCHMARK

R

M

Z

PROCESSOR

E F G H IGeometricMean

1.17

1.00

.99

Still conclude that processor R is best

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Software and Hardware Measurement• In this situation, it is easy to show that the conclusion

that a given machine has highest geometric mean normalized score is a meaningful conclusion.

• Even meaningful: A given machine has geometric mean normalized score 20% higher than another machine.

• Fleming and Wallace give general conditions under which comparing geometric means of normalized scores is meaningful.

• Research area: what averaging procedures make sense in what situations? Large literature.

• Note: There are situations where comparing arithmetic means is meaningful but comparing geometric means is not.

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Software and Hardware Measurement

• Message from measurement theory to computer science (and DM):

Do not perform arithmetic operations on data without paying attention to whether the conclusions you get are meaningful.

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Concluding Comment

• In recent years, interplay between computer science/mathematics and biology has transformed major parts of biology into an information science.• Led to major scientific breakthroughs in

biology such as sequencing of human genome.• Led to significant new developments in CS,

such as database search.• The interplay between CS and methods of the

social sciences such as the theory of voting and elections is not nearly as far along.

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Concluding Comment

• However, the interplay between computer science/mathematics and the social sciences has already developed a unique momentum of its own.

• One can expect many more exciting outcomes as partnerships between computer scientists/ mathematicians and social scientists expand and mature.

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