Social Choice and Computer
Science
Fred Roberts, Rutgers University
1
Social Choice
• How do societies (or groups) reach good
decisions?
• The theory of social choice deals
with this question.
• They argue.
• They find a dictator.
• They vote.
2
What do Mathematics and Computer
Science have to do with Voting?
3
Have you used Google lately?
4
5
Have you used Google lately?
Did you know that Google has something
to do with voting?
6
Have you tried buying a book on online
lately?
7
8
9
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?
10
Have you ever heard of v-sis?
11
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.
12
Computer Science and the Social
Sciences
•Many recent applications in CS involve
issues/problems of long interest to social scientists:
preference, utility
conflict and cooperation
allocation
incentives
measurement
social choice or consensus
•Methods developed in SS beginning to be used in
CS
13
CS and SS
•CS applications place great strain on SS methods
Sheer size of problems addressed
Computational power of agents an issue
Limitations on information possessed by
players
Sequential nature of repeated applications
•Thus: Need for new generation of SS methods
•Also: These new methods will provide powerful
tools to social scientists
14
Social Choice and CS: Outline
1.Consensus Rankings
2.Meta-search and Collaborative Filtering
3.Large Databases and Inference
4.Computational Intractability of Consensus
Functions
5.Electronic Voting
6.Software and Hardware Measurement
7.Power of a Voter
15
Social Choice and CS: Outline
1.Consensus Rankings
2.Meta-search and Collaborative Filtering
3.Large Databases and Inference
4.Computational Intractability of Consensus
Functions
5.Electronic Voting
6.Software and Hardware Measurement
7.Power of a Voter
16
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.
17
But do we necessarily get the best
candidate that way?
Let’s look back at the 2008 Democratic primaries.
18
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.
19
Rankings
Dennis Kucinich
Bill Richardson
John Edwards
Ties are allowed
Barack Obama Hillary Clinton
20
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
21
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
22
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.
23
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?
24
Consensus Rankings
Background: Arrow’s Impossibility Theorem:
• There is no “consensus method” that satisfies
certain reasonable axioms about how societies
should reach decisions.
• Input to Arrow’s Theorem: rankings
of alternatives (ties allowed).
• Output: consensus ranking.
Kenneth Arrow
Nobel prize winner
25
Consensus Rankings
• There are widely studied and widely used
consensus methods that violate one or
more of Arrow’s conditions.
• One well-known consensus method:
“Kemeny-Snell medians”: Given set
of rankings, find ranking minimizing
sum of distances to other rankings.
John Kemeny,
pioneer in time sharing
in CS
• Kemeny-Snell medians are having
surprising new applications in CS.
26
Consensus Rankings
These two rankings are very close:
Clinton
Obama
Edwards
Kucinich
Richardson
Obama
Clinton
Edwards
Kucinich
Richardson
27
Consensus Rankings
These two rankings are very far apart:
Clinton
Richardson
Edwards
Kucinich
Obama
Obama
Kucinich
Edwards
Richardson
Clinton
28
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?
29
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.
30
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 minimized.
• x can be a ranking other than a1, a2, …, ap.
• Sometimes just called Kemeny median.
31
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.
32
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.
33
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.
34
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.
35
Consensus Rankings
Theorem (Bartholdi, Tovey, and Trick, 1989;
Wakabayashi, 1986): Computing the KemenySnell median of a set of rankings is an NPcomplete problem.
36
Consensus Rankings
Okay, so what does this have to do with practical
computer science questions?
37
Consensus Rankings
I mean really practical computer science
questions.
38
Social Choice and CS: Outline
1.Consensus Rankings
2.Meta-search and Collaborative Filtering
3.Large Databases and Inference
4.Computational Intractability of Consensus
Functions
5.Electronic Voting
6.Software and Hardware Measurement
7.Power of a Voter
39
40
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.
41
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
42
tractable
43
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
44
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.
45
Social Choice and CS: Outline
1.Consensus Rankings
2.Meta-search and Collaborative Filtering
3.Large Databases and Inference
4.Computational Intractability of Consensus
Functions
5.Electronic Voting
6.Software and Hardware Measurement
7.Power of a Voter
46
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.
47
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
48
Large Databases and Inference
• Real data often in form of sequences
• Here, concentrate on bioinformatics
• 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 SS
consensus methods to biological databases.
49
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, a cancer-causing gene. This led to the
discovery that v-sis works by stimulating
growth.
50
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.
51
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.
52
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.
53
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.
54
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.
55
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.
56
Large Databases and Inference
Smith-Waterman
•Let  be a finite alphabet of size at least 2 and  be a finite
collection of words of length L on .
•Let F() be the set of words 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.
•Consider nonnegative parameters sd that are monotone
decreasing with d and let F(a1,a2, …, an) be all those
words w of length k that maximize
S(w) = isd(w,ai)
57
Large Databases and Inference
•We call such an F a Smith-Waterman consensus.
•In particular, Waterman and others use the parameters
sd = (k-d)/k.
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  = {0,1}, let k = 2. Then the possible pattern
words are 00, 01, 10, 11.
58
Large Databases and Inference
•Suppose a1 = 111010, a2 = 111111. How do we find
F(a1,a2)?
•We have:
d(00,a1) = 1, d(00,a2) = 2
d(01,a1) = 0, d(01,a2) = 1
d(10,a1) = 0, d(10,a2) = 1
d(11,a1) = 0, d(11,a2) = 0
S(00) =  sd(00,ai) = s1 + s2,
S(01) =  sd(01,ai) = s0 + s1
S(10) =  sd(10,ai) = s0 + s1
S(11) =  sd(11,ai) = s0 + s0
•As long as s0 > s1 > s2, it follows that 11 is the consensus
59
pattern, according to Smith-Waterman’s consensus.
Example:
•Let  ={0,1}, k = 3, and consider F(a1,a2,a3) where
a1 = 000000, a2 = 100000, a3 = 111110. The possible
pattern words are: 000, 001, 010, 011, 100, 101, 110, 111.
d(000,a1) = 0, d(000,a2) = 0, d(000,a3) = 2,
d(001,a1) = 1, d(001,a2) = 1, d(001,a3) = 2,
d(100,a1) = 1, d(100,a2) = 0, d(100,a3) = 1, etc.
S(000) = s2 + 2s0, S(001) = s2 + 2s1, S(100) = 2s1 + s0,
etc.
•Now, s0 > s1 > s2 implies that S(000) > S(001).
•Similarly, one shows that the score is maximized by
S(000) or S(100).
• Monotonicity doesn’t say which of these is highest.
60
Large Databases and Inference
The Special Case sd = (k-d)/k
•Then it is easy to show that the words w that maximize
s(w) are exactly the words w that minimize
id(w,ai).
•In other words: In this case, the Smith-Waterman
consensus is exactly the median.
Algorithms for computing consensus sequences
such as Smith-Waterman are important in modern
molecular biology.
61
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. 62
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 social sciences.
63
Social Choice and CS: Outline
1.Consensus Rankings
2.Meta-search and Collaborative Filtering
3.Large Databases and Inference
4.Computational Intractability of Consensus
Functions
5.Electronic Voting
6.Software and Hardware Measurement
7.Power of a Voter
64
Computational Intractability of Consensus
Functions
• How hard is it to compute the winner of an
election?
• We know counting votes can be difficult
and time consuming.
• However:
• Bartholdi, Tovey and Trick (1989): There are
voting schemes where it can be computationally
intractable to determine who won an election.
65
Computational Intractability of Consensus
Functions
• So, is computational intractability necessarily
bad?
• Computational intractability can be a good thing
in an election: Designing voting systems where
it is computationally intractable to “manipulate”
the outcome of an election by “insincere voting”:
 Adding voters
 Declaring voters ineligible
 Adding candidates
 Declaring candidates ineligible
66
Computational Intractability of Consensus
Functions
• Given a set A of all possible candidates and a set
I of all possible voters.
• Suppose we know voter i’s ranking of all
candidates in A, for every voter i.
• Given a subset of I of eligible voters, a
particular candidate a in A, and a number k,
is there a set of at most k ineligible voters who
can be declared eligible so that candidate a is
the winner?
• Bartholdi, Tovey, Trick (1989): For some
consensus functions (voting rules), this is an NP-67
complete problem.
Computational Intractability of Consensus
Functions
• Given a set A of all possible candidates and a set
I of all possible voters.
• Suppose we know voter i’s ranking of all
candidates in A, for every voter i.
• Given a subset of I of eligible voters, a
particular candidate a in A, and a number k,
is there a set of at most k eligible voters who
can be declared ineligible so that candidate a
is the winner?
• Bartholdi, Tovey, Trick (1989): For some
consensus functions (voting rules), this is an NP-68
complete problem.
Computational Intractability of Consensus
Functions
• Software agents may be more likely to
manipulate than individuals (Conitzer and
Sandholm 2002):
Humans don’t think about manipulating
Computation can be tedious.
Software agents are good at running
algorithms
Software agents only need to have code for
manipulation written once.
All the more reason to develop
computational barriers to manipulation.
69
Computational Intractability of Consensus
Functions
• Stopping those software agents:
70
Computational Intractability of Consensus
Functions
• Conitzer and Sandholm (2002):
 Earlier results of difficulty of manipulation depend
on large number of candidates
 New results: manipulation possible with some
voting methods if smaller number (bounded number)
of candidates)
 In weighted voting, voters may have
different numbers of votes (as in US
presidential elections, where different
states (= voters) have different
numbers of votes). Here, manipulation is harder.
 Manipulation difficult when uncertainty about
71
others’ votes.
Computational Intractability of Consensus
Functions
• Conitzer and Sandholm (2006):
Try to find voting rules for which
manipulation is usually hard.
Why is this difficult to do?
One explanation: under one reasonable
assumption, it is impossible to find such rules.
72
Social Choice and CS: Outline
1.Consensus Rankings
2.Meta-search and Collaborative Filtering
3.Large Databases and Inference
4.Computational Intractability of Consensus
Functions
5.Electronic Voting
6.Software and Hardware Measurement
7.Power of a Voter
73
•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.
74
Electronic Voting
• Issues:
Correctness
Anonymity
Availability
Security
Privacy
75
Electronic Voting
Security Risks in Electronic Voting
• Threat of “denial of service attacks”
• Threat of penetration attacks involving a
delivery mechanism to transport a malicious
payload to target host (thru Trojan horse or
remote control program)
• Private and correct counting of votes
• Cryptographic challenges to keep votes private
• Relevance of work on secure multiparty
computation
76
Electronic Voting
Other CS Challenges:
• Resistance to “vote buying”
• Development of user-friendly interfaces
• Vulnerabilities of communication path between
the voting client (where you vote) and the server
(where votes are counted)
• Reliability issues: random hardware and
software failures
77
Social Choice and CS: Outline
1.Consensus Rankings
2.Meta-search and Collaborative Filtering
3.Large Databases and Inference
4.Computational Intractability of Consensus
Functions
5.Electronic Voting
6.Software and Hardware Measurement
7.Power of a Voter
78
Software & Hardware Measurement
• Theory of measurement developed by
mathematical social scientists
• Measurement theory studies ways to combine
scores obtained on different criteria.
• A statement involving scales of
measurement is considered meaningful if its
truth or falsity is unchanged under acceptable
transformations of all scales involved.
79
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: Is it meaningful to say that I weigh
more than a cicada?
• Yes.
• That is because if it is true in kilograms, then it
is also true in pounds, in grams, etc.
80
Software & Hardware Measurement
• Even meaningful to say I weigh twice as much as a
cicada.
81
Software & Hardware Measurement
• Is it meaningful to say that the temperature today is
twice as much as it was yesterday?
• No.
• Could be true in Fahrenheit, false in Centigrade.
82
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.
• Long literature on what averaging methods lead to
meaningful conclusions.
83
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
84
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
85
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
86
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
87
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 machine Z is best
88
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
89
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
90
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
91
Conclude that machine 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 as a
consensus method.
• Geometric mean is
n

 is(xi)
92
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 machine R is best
93
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 machine R is best
94
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
95
not.
Software and Hardware Measurement
• Message from measurement theory to computer
science (Operations Research, and other
disciplines):
Do not perform arithmetic operations on
data without paying attention to whether
the conclusions you get are meaningful.
96
Social Choice and CS: Outline
1.Consensus Rankings
2.Meta-search and Collaborative Filtering
3.Large Databases and Inference
4.Computational Intractability of Consensus
Functions
5.Electronic Voting
6.Software and Hardware Measurement
7.Power of a Voter
97
Power of a Voter
Shapley-Shubik Power Index
• Think of voting situations with many players
• Shapley-Shubik index measures the power
of each player in a multi-player “voting game.”
• Consider a game in which some coalitions
of players win and some lose, with no
subset of a losing coalition winning.
Martin
Shubik
Lloyd Shapley
98
Power of a Voter
Example: Shareholders of Company
Shareholder 1 holds 3 shares.
Shareholders 2, 3, 4, 5, 6, 7 hold 1 share each.
A majority of shares are needed to make a decision.
Coalition {1,4,6} is winning.
Coalition {2,3,4,5,6} is winning.
A coalition is winning iff it has at least 5 of the 1-share
players or it has the 3-share player and at least two
others.
99
Power of a Voter
Shapley-Shubik Power Index
• Consider a coalition forming at random, one
player at a time.
• A player i is pivotal if addition of i throws
coalition from losing to winning.
• Shapley-Shubik index of i = probability i is
pivotal if an order of players is chosen at random.
• Power measure applying to more general games
than voting games is called Shapley Value.
100
Power of a Voter
Example: Shareholders of Company
Shareholder 1 holds 3 shares.
Shareholders 2, 3, 4, 5, 6, 7 hold 1 share each.
A majority of shares are needed to make a decision.
Coalition {1,4,6} is winning.
Coalition {2,3,4,5,6} is winning.
Shareholder 1 is pivotal if he is 3rd, 4th, or 5th.
So shareholder 1’s Shapley value is 3/7.
Sum of Shapley values is 1 (since they are probabilities)
Thus, each other shareholder has Shapley value
(4/7)/6 = 2/21
101
Power of a Voter
Example: Australian Government
• The same game arises in the workings of the
government of Australia.
• There are six states and the federal
government.
• A measure passes if it has the support of at least 5
states or at least 2 states plus the federal government.
• It is not hard to see that the winning coalitions are
exactly the same as in a game in which the federal
government holds 3 votes and each state holds 1 vote
and 5 votes are needed to win.
• Thus, the federal government has power 3/7 and each
state in Australia has power 2/21.
102
.
Power of a Voter
Example: United Nations Security Council
•15 member nations
•5 permanent members
China, France,
Russia, UK, US
•10 non-permanent
•Permanent members
have veto power
•Coalition wins iff it has all 5 permanent members
and at least 4 of the 10 non-permanent members.
103
Power of a Voter
Example: United Nations Security Council
•What is the power of each
Member of the Security
Council?
•Fix non-permanent member i.
•i is pivotal in permutations in
which all permanent members
precede i and exactly 3 nonpermanent members do.
•How many such permutations are there?
104
Power of a Voter
Example: United Nations Security Council
•Choose 3 non-permanent members preceding i.
•Order all 8 members preceding i.
•Order remaining 6 non-permanent members.
•Thus the number of such permutations is:
C(9,3) x 8! x 6! = 9!/3!6! x 8! x 6! = 9!8!/3!
•The probability that i is pivotal = power of nonpermanent member =
9!8!/3!15! = .001865
•The power of a permanent member =
[1 – 10 x .001865]/5 = .1963.
•Permanent members have 100 times power of 105
non-permanent members.
Power of a Voter
•There are a variety of other power indices used
in game theory and political science (Banzhaf
index, Coleman index, …)
•Need calculate them for huge games
•Mostly computationally intractable
106
Power of a Voter: Allocation/Sharing
Costs and Revenues
• Shapley-Shubik power index and more
general Shapley value have been used
to allocate costs to different users in
shared projects.
Allocating runway fees in airports
Allocating highway fees to trucks of
different sizes
Universities sharing library facilities
Fair allocation of telephone calling
charges among users sharing
complex phone systems (Cornell’s
experiment)
107
Power of a Voter: Allocating/Sharing
Costs and Revenues
Allocating Runway Fees at Airports
• Some planes require longer runways.
Charge them more for landings. How
much more?
• Divide runways into meter-long
segments.
• Each month, we know how many
landings a plane has made.
• Given a runway of length y meters,
consider a game in which the players are
landings and a coalition “wins” if the
runway is not long enough for planes in
the coalition.
108
Power of a Voter: Allocating/Sharing
Costs and Revenues
Allocating Runway Fees at Airports
• A landing is pivotal if it is the first
landing added that makes a coalition
require a longer runway.
• The Shapley value gives the cost of
the yth meter of runway allocated to a
given landing.
• We then add up these costs over all
runway lengths a plane requires and
all landings it makes.
109
Power of a Voter: Allocating/Sharing
Costs and Revenues
Multicasting
• Applications in multicasting, e.g., sending
movies out to viewers.
• Unicast routing: Each packet sent from a
source is delivered to a single receiver.
• Sending it to multiple sites: Send multiple
copies and waste bandwidth.
• In multicast routing: Use a directed tree
connecting source to all receivers.
• At branch points, a packet is duplicated as
necessary.
110
Multicasting
111
Power of a Voter: Allocating/Sharing
Costs and Revenues
Multicasting
• Multicast routing: Use a directed tree
connecting source to all receivers.
• At branch points, a packet is duplicated as
necessary.
• Bandwidth is not directly attributable to a
single receiver.
• How to distribute costs among receivers?
• One idea: Use Shapley value.
112
Allocating/Sharing Costs and Revenues
• Feigenbaum, Papadimitriou, Shenker (2001):
no feasible implementation for Shapley value
in multicasting.
• Note: Shapley value is uniquely characterized
by four simple axioms.
• Sometimes we state axioms as general
principles we want a solution concept to have.
• Jain and Vazirani (1998): polynomial time
computable cost-sharing algorithm
 Satisfying some important axioms
 Calculating cost of optimum multicast tree within
factor of two of optimal.
113
Concluding Comment
• In recent years, interplay between CS
and biology has transformed major
parts of Bio 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 SS not
nearly as far along.
• Moreover: problems are spread over
many disciplines.
114
Concluding Comment
• However, CS-SS interplay has already
developed a unique momentum of its own.
• One can expect many more exciting outcomes
as partnerships between computer scientists
and social scientists expand and mature.
115
116