Voting Problems and Computer
Science Applications
Fred Roberts, Rutgers University
1
What do Mathematics and Computer
Science have to do with Voting?
2
Have you used Google lately?
3
4
Have you used Google lately?
Did you know that Google has something
to do with voting?
5
Have you tried buying a book on online
lately?
6
7
8
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?
9
Have you ever heard of v-sis?
10
Have you ever heard of v-sis?
•It’s a cancer-causing gene.
Cancer cell
•Computer scientists helped discover how it
works?
•How did they do it?
•The answer also has something to do with
voting.
11
•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.
12
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?
13
Electronic Voting
Security 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?
14
Electronic Voting
Security 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.
15
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.
16
But do we necessarily get the best
candidate that way?
17
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.
18
Rankings
Dennis Kucinich
Bill Richardson
John Edwards
Ties are allowed
Barack Obama Hillary Clinton
19
Rankings
• What if we have four voters and they give us the
following rankings? Who should win?
Voter 1
Clinton
Richardson
Edwards
Kucinich
Obama
Voter 2
Voter 3
Voter 4
Clinton
Obama
Obama
Kucinich Edwards Richardson
Edwards Richardson Kucinich
Richardson Kucinich Edwards
Obama
Clinton
Clinton
20
Rankings
• What if we have four voters and they give us the
following rankings?
• There is one added candidate.
• Who should win?
Voter 1
Clinton
Gore
Richardson
Edwards
Kucinich
Obama
Voter 2
Voter 3
Voter 4
Clinton
Obama
Obama
Gore
Gore
Gore
Kucinich Edwards Richardson
Edwards Richardson Kucinich
Richardson Kucinich Edwards
Obama
Clinton
Clinton
21
Rankings
Voter 1
Clinton
Gore
Richardson
Edwards
Kucinich
Obama
Voter 2
Voter 3
Voter 4
Clinton
Obama
Obama
Gore
Gore
Gore
Kucinich Edwards Richardson
Edwards Richardson Kucinich
Richardson Kucinich Edwards
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.
22
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?
23
Consensus Rankings
These two rankings are very close:
Clinton
Obama
Edwards
Kucinich
Richardson
Obama
Clinton
Edwards
Kucinich
Richardson
24
Consensus Rankings
These two rankings are very far apart:
Clinton
Richardson
Edwards
Kucinich
Obama
Obama
Kucinich
Edwards
Richardson
Clinton
25
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?
26
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
b
x
y-z
y
x
z
On {x,y}: +2
On {x,z}: +2
On {y,z}: +1
d(a,b) = 5.
27
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 sharing
in CS
28
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.
29
Consensus Rankings
a1
Fish
Chicken
Beef
a2
Fish
Chicken
Beef
a3
Chicken
Fish
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.
30
Consensus Rankings
a1
Fish
Chicken
Beef
a2
Chicken
Beef
Fish
a3
Beef
Fish
Chicken
• Three medians = a1, a2, a3.
• This is the “voter’s paradox” situation.
31
Consensus Rankings
a1
Fish
Chicken
Beef
a2
Chicken
Beef
Fish
a3
Beef
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.
32
Consensus Rankings
a1
Fish
Chicken
Beef
a2
Chicken
Beef
Fish
a3
Beef
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.
33
Consensus Rankings
Theorem (Bartholdi, Tovey, and Trick, 1989;
Wakabayashi, 1986): Computing the KemenySnell median of a set of rankings is an NPcomplete problem.
34
Consensus Rankings
Okay, so what does this have to do with practical
computer science questions?
35
Consensus Rankings
I mean really practical computer science
questions.
36
37
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.
38
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): KemenySnell 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
39
tractable
40
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
41
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.
42
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.
43
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
44
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.
45
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.
46
Large Databases and Inference
DNA Sequences
A DNA sequence is a sequence of “bases”:
A = Adenine, G = Guanine,
C = Cytosine, T = Thymine
Example:
ACTCCCTATAATGCGCCA
47
Large Databases and Inference
Example
Bacterial Promoter Sequences studied by
Waterman (1989):
RRNABP1:
TNAA:
UVRBP2:
SFC:
ACTCCCTATAATGCGCCA
GAGTGTAATAATGTAGCC
TTATCCAGTATAATTTGT
AAGCGGTGTTATAATGCC
Notice that if we are looking for patterns of length
4, each sequence has the pattern TAAT.
48
Large Databases and Inference
Example
Bacterial Promoter Sequences studied by
Waterman (1989):
RRNABP1:
TNAA:
UVRBP2:
SFC:
ACTCCCTATAATGCGCCA
GAGTGTAATAATGTAGCC
TTATCCAGTATAATTTGT
AAGCGGTGTTATAATGCC
Notice that if we are looking for patterns of length
4, each sequence has the pattern TAAT.
49
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.
50
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.
51
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.
52
Large Databases and Inference
Example
a = 0011, b = 111010
Possible Alignments:
111010
111010
111010
0011
0011
0011
The best-mismatch distance is 2, which is achieved
in the third alignment.
53
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.
54
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
55
sequences are 00, 01, 10, 11.
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) = 2
d(a1,01) = 0, d(a2,01) = 1
d(a1,10) = 0, d(a2,10) = 1
d(a1,11) = 0, d(a2,11) = 0
•It follows that 11 is the consensus pattern, according to
Smith-Waterman’s consensus.
56
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
57
sequences.
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. 58
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.
59
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.
60
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.
61
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 scores
Pick system with highest average.
Fleming and Wallace (1986): Outcome can
depend on choice of base system.
Meaningless in sense of measurement theory
Leads to theory of merging normalized scores
62
Software & Hardware Measurement
Hardware Measurement
P R
R
O
C
M
E
S
S
O Z
R
E
417
BENCHMARK
F
G
H
83
66 39,449
I
772
244
70
153
33,527
368
134
70
135
66,000
369
63
Data from Heath, Comput. Archit. News (1984)
Software & Hardware Measurement
Normalize Relative to Processor R
P R
R
O
C
M
E
S
S
O Z
R
E
417
1.00
BENCHMARK
F
G
H
I
83
66 39,449 772
1.00 1.00
1.00 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
64
Software & Hardware Measurement
Take Arithmetic Mean of Normalized Scores
P R
R
O
C
M
E
S
S
O Z
R
E
417
1.00
Arithmetic
BENCHMARK
Mean
F
G
H
I
83
66 39,449 772
1.00
1.00 1.00
1.00 1.00
244
.59
70
.84
153
2.32
33,527
.85
368
.48
1.01
134
.32
70
.85
135
2.05
66,000
1.67
369
.45
1.07
65
Software & Hardware Measurement
Take Arithmetic Mean of Normalized Scores
P R
R
O
C
M
E
S
S
O Z
R
E
417
1.00
Arithmetic
BENCHMARK
Mean
F
G
H
I
83
66 39,449 772
1.00
1.00 1.00
1.00 1.00
244
.59
70
.84
153
2.32
33,527
.85
368
.48
1.01
134
.32
70
.85
135
2.05
66,000
1.67
369
.45
1.07
Conclude that processor Z is best
66
Software & Hardware Measurement
Now Normalize Relative to Processor M
P R
R
O
C
M
E
S
S
O Z
R
E
417
1.71
BENCHMARK
F
G
H
I
83
66 39,449 772
1.19
.43
1.18 2.10
244
1.00
70
1.00
153
1.00
33,527 368
1.00 1.00
134
.55
70
1.00
135
.88
66,000 369
1.97 1.00
67
Software & Hardware Measurement
Take Arithmetic Mean of Normalized Scores
P R
R
O
C
M
E
S
S
O Z
R
E
417
1.71
Arithmetic
BENCHMARK
Mean
F
G
H
I
83
66 39,449 772
1.32
1.19
.43
1.18 2.10
244
1.00
70
1.00
153
1.00
33,527 368
1.00 1.00
1.00
134
.55
70
1.00
135
.88
66,000 369
1.97 1.00
1.08
68
Software & Hardware Measurement
Take Arithmetic Mean of Normalized Scores
P R
R
O
C
M
E
S
S
O Z
R
E
417
1.71
Arithmetic
BENCHMARK
Mean
F
G
H
I
83
66 39,449 772
1.32
1.19
.43
1.18 2.10
244
1.00
70
1.00
153
1.00
33,527 368
1.00 1.00
1.00
134
.55
70
1.00
135
.88
66,000 369
1.97 1.00
1.08
69
Conclude that processor R is best
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
n

 is(xi)
70
Software & Hardware Measurement
Normalize Relative to Processor R
P R
R
O
C
M
E
S
S
O Z
R
E
417
1.00
Geometric
BENCHMARK
Mean
F
G
H
I
83
66 39,449 772
1.00
1.00 1.00
1.00 1.00
244
.59
70
.84
153
2.32
33,527
.85
368
.48
.86
134
.32
70
.85
135
2.05
66,000
1.67
369
.45
.84
Conclude that processor R is best
71
Software & Hardware Measurement
Now Normalize Relative to Processor M
P R
R
O
C
M
E
S
S
O Z
R
E
417
1.71
Geometric
BENCHMARK
Mean
F
G
H
I
83
66 39,449 772
1.17
1.19
.43
1.18 2.10
244
1.00
70
1.00
153
1.00
33,527 368
1.00 1.00
1.00
134
.55
70
1.00
135
.88
66,000 369
1.97 1.00
.99
Still conclude that processor R is best
72
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
73
not.
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.
74
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.
75
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.
76
77