Computational Social Choice Eine Einführung Jörg Rothe amp Lena Schend SS 2012 HHU Düsseldorf 4 April 2012 Introduction Social Choice Theory voting theory preference ID: 538016
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Slide1
Projektseminar Computational Social Choice -Eine Einführung-
Jörg
Rothe & Lena
Schend
SS 2012,
HHU Düsseldorf
4. April 2012Slide2
IntroductionSocial Choice Theoryvoting
theory
preference
aggregationjudgment aggregationComputer Scienceartificial intelligencealgorithm designcomputational complexity theory - worst-case/average-case complexity - optimization, etc.
voting in multiagent systems multi-criteria decision making meta search, etc. ...
Software
agents
can
systematically
analyze
elections
to
find optimal
strategiesSlide3
IntroductionSocial Choice Theoryvoting
theory
preference
aggregationjudgment aggregationComputational Social Choice Computer Scienceartificial intelligencealgorithm designcomputational complexity theory - worst-case/average-case
complexity - optimization, etc.
computational barriers to prevent manipulation control bribery
Software
agents
can
systematically
analyze
elections
to
find optimal
strategiesSlide4
Computational Social Choice
With
the
power of NP-hardness vulcans have constructed complexity shields to protect elections against many types of manipulation and
control. Slide5
Computational Social Choice
With
the
power of NP-hardness vulcans have constructed complexity shields to protect elections against many types of manipulation and
control. Question:
Are worst-case complexity shields enough? Or do they evaporate on "typical elections"?Slide6
NP-Hardness Shields Evaporating?
NP-
hardness
shieldssingle-peaked electoratesjunta distributionsapproximation
experimental analysisSlide7
ElectionsAn election is a pair (C,V) with
a finite
set
C of candidates:a finite list V of voters.Voters are represented by their preferences over
C:either by linear orders: > > > or
by approval vectors: (1,1,0,1)Voting system: determines winners from the preferencesSlide8
Voting SystemsApproval Voting (AV) votes are
approval
vectors in
v1
1101v20100
v
3
1
1
0
1
v
4
0
0
1
0
v
5
1
0
1
1
v
6
1
0
0
1Slide9
Voting SystemsApproval Voting (AV) votes are
approval
vectors in winners: all candidates with the most approvals
v1
1101v20100
v
3
1
1
0
1
v
4
0
0
1
0
v
5
1
0
1
1
v
6
1
0
0
1
∑
4
3
2
4Slide10
Voting SystemsApproval Voting (AV) votes are
approval
vectors in winners: all candidates with the most approvals
winners:
v11101v2
0
1
0
0
v
3
1
1
0
1
v
4
0
0
1
0
v
5
1
0
1
1
v
6
1
0
0
1
∑
4
3
2
4Slide11
Voting SystemsPositional Scoring Rules (for m candidates
)
defined
by scoring vector with each voter gives points to the candidate on position i winners: all candidates with
maximum score
Borda
:
Plurality
Voting
(PV):
k
-
Approval
(
m-k
-Veto): Veto (Anti-
Plurality
):Slide12
-
4:0
2:2
3:1
0:4-1:3
2:22:23:1-
2:21:32:22:2-Voting SystemsPairwise
Comparison v
1
: > > > v
3
: > > >
v
2
: > > > v
4
: > > >
Condorcet:
beats
all
other
candidates
strictly
Copeland :
1
point
for
victory
points
for
tie
Maximin
:
maximum
of
the
worst
pairwise
comparison
Slide13
Voting SystemsRound-based: Single Transferable Vote (STV)
v
1
: > > > v
2: > > >v3: > > > v4: > > >Slide14
Voting SystemsRound-based: Single Transferable Vote (STV)
v
1
: > > v
2: > > v3: > > v4: > > Slide15
Voting SystemsRound-based: Single Transferable Vote (STV)
v
1
: v
2: v3: v4: Slide16
Voting SystemsLevel-based: Bucklin Voting (BV)v1
: > > >
v
2
: > > > v3: > > > v4: > > >v5: > > >5 voters => strict majority threshold is 3
Lvl
1
1
2
2
0Slide17
Voting SystemsLevel-based: Bucklin Voting (BV)v1
: > > >
v
2
: > > > v3: > > > v4: > > >v5: > > >5 voters => strict majority threshold is 3
Lvl
1
1
2
2
0
Lvl
2
2
2
3
3Slide18
Voting SystemsLevel-based: Bucklin Voting (BV)v1
: > > >
v
2
: > > > v3: > > > v4: > > > Level 2 Bucklinv5: > > > winners: 5 voters => strict majority threshold is 3
Lvl
1
1
2
2
0
Lvl
2
2
2
3
3Slide19
Voting SystemsLevel-based: Fallback Voting (FV)combines AV and BVCandidates
:
v: { , } | { , }
v: > | { , }
Bucklin winners are fallback winners.If no Bucklin winner exists (due to disapprovals), then approval winners win.Slide20
War on Electoral Control AV winners:
"
chair
": knows all preferences
v111
01v20100v
3
1
1
0
1
v
4
0
0
1
0
v
5
1
0
1
1
v
6
1
0
0
1
∑
4
3
2
4Slide21
War on Electoral Control AV winner:
"
chair
": knows all preferences and can change the structure of an election
v
11101v2
01
0
0
v
3
1
1
0
1
v
4
0
0
1
0
v
5
1
0
1
1
v
6
1
0
0
1
∑
2
3
1
2Slide22
War on Electoral Control AV winner:
"
chair
": knows all preferences and can change the structureOther types of
control: of an electionadding/partitioning votersdeleting/adding/partitioning candidates
v111
01
v
2
0
1
0
0
v
3
1
1
0
1
v
4
0
0
1
0
v
5
1
0
1
1
v
6
1
0
0
1
∑
2
3
1
2Slide23
NP-Hardness Shields for ControlR
esistance = NP-
hardness
,
Vulnerability = P, Immunity, and SusceptibilitySlide24
Cope-land
Score
-
4:0
2:23:12.5
0:4-1:32:20.5
2:23:1-2:221:32:2
2:2-1
War on Manipulation
Copeland :
winner
v
1
: > > > v
3
: > > >
v
2
: > > > v
4
: > > >
I
like
Spock but I
don‘t
want
him
to
be
the
captain
!!Slide25
Copeland : winner v1: > > > v3: > > >v2: > > >
v
4
: > > >
assumption: . v4 knows the other voters‘ votes v4 lies to
make his most preferred
candidate win Cope-land Score
-
4:0
2:2
3:1
2.5
0:4
-
1:3
2:2
0.5
2:2
3:1
-
2:2
2
1:3
2:2
2:2
-
1
War on Manipulation
I
like
Spock but I
don‘t
want
him
to
be
the
captain
!!Slide26
Copeland : winners v1: > > > v3: > > >v2: > > >
v
4
: > > >
Here: unweighted voters, single manipulator . Other types: - coalitional manipulation - weighted
voters
Cope-land Score-3:12:22:22
1:3
-
1:3
1:3
0
2:2
3:1
-
2:2
2
2:2
3:1
2:2
-
2
War on Manipulation
I
like
Spock but I
don‘t
want
him
to
be
the
captain
!!Slide27
NP-Hardness Shields for Manipulation
Results
due
to
Conitzer, Sandholm, Lang (J.ACM 2007)Slide28
NP-Hardness Shields Evaporating?
NP-
hardness
shieldssingle-peaked electoratesjunta distributionsapproximation
experimental analysisSlide29
Junta Distributionsof Procaccia and Rosenschein (JAAMAS 2007) are omitted here
,
as
they are a rather technical concept.Slide30
NP-Hardness Shields Evaporating?
NP-
hardness
shieldssingle-peaked electoratesjunta distributionsapproximation
experimental analysisSlide31
Experiments Manipulationtesting (heuristic) algorithms for manipulation problem at
hand
on
given
electionssample real electionsgenerate random elections: Impartial Culture (IC) Polya-Eggenberger (PE)
voters
vote
independently
all
preferences
are
equally
likely
voters
are
highly
correlated
v
1
v
2
v
3
...
Walsh (IJCAI 2009; ECAI 2010)Slide32
Experiments Manipulationtesting (heuristic) algorithms for manipulation problem at
hand
on
given
electionssample real electionsgenerate random elections: Impartial Culture (IC) Polya-Eggenberger (PE)
voters
vote
independently
all
preferences
are
equally
likely
voters
are
highly
correlated
v
1
v
2
v
3
...
Walsh (IJCAI 2009; ECAI 2010)Slide33
Experiments Manipulationtesting (heuristic) algorithms for manipulation problem at
hand
on
given
electionssample real electionsgenerate random elections: Impartial Culture (IC) Polya-Eggenberger (PE)
voters
vote
independently
all
preferences
are
equally
likely
voters
are
highly
correlated
v
1
v
2
v
3
...
Walsh (IJCAI 2009; ECAI 2010)Slide34
Experiments Manipulationtesting (heuristic) algorithms for manipulation problem at
hand
on
given
electionssample real electionsgenerate random elections: Impartial Culture (IC) Polya-Eggenberger (PE)
voters
vote
independently
all
preferences
are
equally
likely
voters
are
highly
correlated
v
1
v
2
v
3
...
Walsh (IJCAI 2009; ECAI 2010)Slide35
Experiments Manipulationtesting (heuristic) algorithms for manipulation problem at
hand
on
given
electionssample real electionsgenerate random elections: Impartial Culture (IC) Polya-Eggenberger (PE)
voters
vote
independently
all
preferences
are
equally
likely
voters
are
highly
correlated
v
1
v
2
v
3
...
Walsh (IJCAI 2009; ECAI 2010)Slide36
Experiments Manipulationtesting (heuristic) algorithms for manipulation problem at
hand
on
given
electionssample real electionsgenerate random elections: Impartial Culture (IC) Polya-Eggenberger (PE)
voters
vote
independently
all
preferences
are
equally
likely
voters
are
highly
correlated
v
1
v
2
v
3
...
Walsh (IJCAI 2009; ECAI 2010)Slide37
Experiments ManipulationResults for STVSingle Manipulation:for up to 128 candidates/
voters
manipulation
has low computational costs (for all voter distributions)chance of successful manipulation decreases with increasing number of nonmanipulative votersCoalitional Manipulation:larger coalitions are
more likely to be successfulagain: computational costs are low
for up to 128 candidates/votersResults for Veto (weighted)if manipulators‘ weights are too big/small => trivialeven in critical region: computational costs are lowonly correlated voters increase
computational costs
Walsh (IJCAI 2009; ECAI 2010)Slide38
NP-Hardness Shields Evaporating?
NP-
hardness
shieldssingle-peaked electoratesjunta distributionsapproximation
experimental analysisSlide39
Approximating ManipulationBefore: Is manipulation possible?
?Slide40
Approximating ManipulationBefore: Is manipulation
possible
?
Now
: How many manipulators are needed? (min!)Approximation Algorithms:efficient algorithmsdo not always find optimal solutioncan be analyzed both
theoretically and experimentally
??Slide41
Approximating Borda3x > > > > > >
2x > > > > > >
Borda
winner manipulators prefer
B-Score
5
0
18
19
20
21
22Slide42
Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m
1
> > > > > >
B-Score
5
0
18
19
20
21
22
Zuckerman
,
Procaccia
& Rosenschein (
Artificial
Intelligence
2009)Slide43
Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m
1
> > > > > >
B-Score
11
5
22
22
22
22
22
Zuckerman
,
Procaccia
& Rosenschein (
Artificial
Intelligence
2009)Slide44
Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m
1
> > > > > >
m2 > > > > > >
B-Score
11
5
22
22
22
22
22
Zuckerman
,
Procaccia
& Rosenschein (
Artificial
Intelligence
2009)Slide45
Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m
1
> > > > > >
m2 > > > > > >
B-Score
17
10
26
25
24
23
22
Zuckerman
,
Procaccia
& Rosenschein (
Artificial
Intelligence
2009)Slide46
Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m
1
> > > > > >
m2 > > > > > >m3 > > > > > >
B-Score
17
10
26
25
24
23
22
Zuckerman
,
Procaccia
& Rosenschein (
Artificial
Intelligence
2009)Slide47
Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m
1
> > > > > >
m2 > > > > > >m3 > > > > > >
B-Score
23
15
26
26
26
26
26
Zuckerman
,
Procaccia
& Rosenschein (
Artificial
Intelligence
2009)Slide48
Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m
1
> > > > > >
m2 > > > > > >m3 > > > > > >m4 > > > > > >
B-Score
23
15
26
26
26
26
26
Zuckerman
,
Procaccia
& Rosenschein (
Artificial
Intelligence
2009)Slide49
Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m
1
> > > > > >
m2 > > > > > >m3 > > > > > >m4 > > > > > >
B-Score
29
20
30
29
28
27
26
Zuckerman
,
Procaccia
& Rosenschein (
Artificial
Intelligence
2009)Slide50
Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m
1
> > > > > >
m2 > > > > > >m3 > > > > > >m4 > > > > > >m5 > > > > > >
B-Score
29
20
30
29
28
27
26
Zuckerman
,
Procaccia
& Rosenschein (
Artificial
Intelligence
2009)Slide51
Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m
1
> > > > > >
m2 > > > > > >m3 > > > > > >m4 > > > > > >m5 > > > > > >
B-Score
35
25
30
30
30
30
30
Zuckerman
,
Procaccia
& Rosenschein (
Artificial
Intelligence
2009)Slide52
Approximating BordaOptimal solution: 4 manipulators
m
1
> > > > > >
m2 > > > > > >m3 > > > > > >m4 > > > > > > "Reverse" needs one manipulator
more than optimal
B-Score
29
20
28
28
28
28
28
Zuckerman
,
Procaccia
& Rosenschein (
Artificial
Intelligence
2009)Slide53
Approximation ResultsMaximin: factor 2 (twice number of optimal manipulators)factor
5/3 (not
better
than 3/2 unless P=NP) Borda: Reverse: additional 1 Largest Fit unbounded additional number Average Fit of manipulators , , and are theoretically incomparableexperimental comparison:
Ø
Ø
IC model
76%
83%
99%
PE model
76%
43%
99%
Ø
>
>
>
Zuckerman
, Lev & Rosenschein (AAMAS 2011)
Davies,
Katsirelos
,
Narodytska
& Walsh (AAAI 2011)Slide54
NP-Hardness Shields Evaporating?
NP-
hardness
shieldssingle-peaked electoratesjunta distributionsapproximation
experimental analysisSlide55
Single-Peaked PreferencesA collection V of votes is said to
be
single-
peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls).
A voter‘s preference curve on galactic
taxes low galactic taxes high galactic taxesSlide56
A collection V of votes is said to be single-
peaked
if
there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls). A
voter‘s > > > preference curve on galactic taxes
low galactic taxes high galactic taxes
Single-Peaked Preferences
Single-
peaked
preference
consistent
with
linear
order
of
candidatesSlide57
A collection V of votes is said to be single-
peaked
if
there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls). A
voter‘s > > > preference curve on galactic taxes
low galactic taxes high galactic taxes
Single-
Peaked
Preferences
Preference
that
is
inconsistent
with
this
linear
order
of
candidatesSlide58
Single-Peaked PreferencesA collection V of votes is
said
to
be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises
or just falls).If each vote vi
in V is a linear order >i over C, this means that for each triple of candidates c, d, and e:(c L d L e or
e L d L c) implies that
for
each
i,
if
c >
i
d
then
d >
i
e.Slide59
Single-Peaked PreferencesA collection V of votes is said to
be
single-
peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls).
If each vote vi in V is
a linear order >i over C, this means that for each triple of candidates c, d, and e:(c L d L e or e L d L c)
implies that
for
each
i,
if
c >
i
d
then
d >
i
e.
Bartholdi & Trick (1986); Escoffier, Lang & Öztürk (2008):
Given
a
collection
V
of
linear
orders
over
C, in
polynomial
time
we
can
produce
a linear
order
L
witnessing
V‘s
single-
peakedness
or
can
determine
that V is
not single-peaked.Slide60
A collection V of votes is said
to
be
single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“
rises to a peak and then falls (or just rises
or just falls).Single-peaked w.r.t. this order?
v1
1
1
0
0
1
no
v
2
0
1
1
0
0
yes
v
3
1
1
0
0
1
no
v
4
0
0
0
1
0
yes
v
5
1
0
0
1
1
no
v
6
1
0
0
0
1
no
Single-
Peaked
Approval
VectorsSlide61
Removing NP-hardness shields:3-candidate Bordaveto
every
scoring
protocol for -candidate 3-veto,Leaving them in place:STV (Walsh AAAI 2007)4-candidate Borda
5-candidate 3-vetoErecting NP-hardness shields:Artificial
election system with approval votes, for size-3-coalition unweighted manipulationResults due to Faliszewski, Hemaspaandra, Hemaspaandra & Rothe (Information &
Computation 2011)
General
Single-
peaked
Constructive
Coalitional
Weighted
ManipulationSlide62
Removing NP-hardness shields:ApprovalConstructive control
by
adding votersConstructive control by deleting votersPluralityconstructive control by adding candidatesdestructive control by adding candidates
constructive control by deleting candidatesdestructive control by deleting candidates
Results due to Faliszewski, Hemaspaandra, Hemaspaandra & Rothe (2011)Brandt, Brill, Hemaspaandra & Hemaspaandra (AAAI 2010) achieved similar
results
for
other
voting
systems
as
well
(e.g.,
for
systems
satisfying
the
weak
Condorcet
criterion
)
and
also
for
constructive
control
by
partition
of
voters
.
General
Single-
peaked
Control
for
Single-
Peaked
ElectoratesSlide63
More
Results
on Single-
Peaked
Preferences
Faliszewski
, Hemaspaandra, Hemaspaandra & Rothe (2011) also prove a dichotomy result for the scoring protocol
CCWM is
NP-
complete
if
a
nd
in P
otherwise
.
Brandt, Brill,
Hemaspaandra
&
Hemaspaandra
(AAAI 2010)
generalize
this
dichotomoy
to
scoring
protocols
with
any
fixed
number
of
candidates
.
Mattei
(ADT 2011)
empirically
investigates
huge
data
sets
from
real-
world elections
(drawn from
the
Netflix
Prize
)
and
observes
that single-peaked
preferences
very rarely occur
in
practice
.
Faliszewski
,
Hemaspaandra
&
Hemaspaandra
(TARK 2011) study manipulative
attacks in nearly
single-peaked electorates
.Slide64
NP-Hardness Shields Evaporating?
NP-
hardness
shieldssingle-peaked electoratesjunta distributionsapproximation
experimental analysisSlide65
Experiments Controlsame approach as for manipulationtesting (heuristic)
algorithms
for
control problem at hand on given electionssample real electionsgenerate random elections: Impartial Culture (IC) Two Mainstreams (TM)
voters vote independently all preferences are
equally likely voters are correlated Slide66
Experiments Controlsame approach as for manipulationtesting (heuristic)
algorithms
for
control problem at hand on given electionssample real electionsgenerate random elections: Impartial Culture (IC) Two Mainstreams (TM)
voters vote independently all preferences are
equally likely voters are correlated Slide67
Experiments Controlsame approach as for manipulationtesting (heuristic)
algorithms
for
control problem at hand on given electionssample real electionsgenerate random elections: Impartial Culture (IC) Two Mainstreams (TM)
voters vote independently all preferences are
equally likely voters are correlated v1 v2 v3 v4 ... Slide68
Experiments ControlObservations:destructive control shows more yes-instances (up
to
100%)
and
lower computational costsDCPV-TP in FVSlide69
Experiments ControlObservations:destructive control shows more yes-instances (up
to
100%)
and
lower computational costsCCPV-TP in FVSlide70
Experiments ControlObservations:destructive control shows more yes-instances (up
to
100%)
and
lower computational costsFV and BV show similar tendenciesvoter control in PV has lower computational costsdeleting/adding voters show similar tendenciesfor constructive
control: voter control shows more yes-instances than candidate controlas expected:
more yes-instances in the IC model than in the TM modelSlide71
Thank you very much!
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