/
Projektseminar Projektseminar

Projektseminar - PowerPoint Presentation

giovanna-bartolotta
giovanna-bartolotta . @giovanna-bartolotta
Follow
383 views
Uploaded On 2017-04-16

Projektseminar - PPT Presentation

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

control single voters manipulation single control manipulation voters peaked borda amp voting preferences hardness shields 2009 score elections systems

Share:

Link:

Embed:

Download Presentation from below link

Download Presentation The PPT/PDF document "Projektseminar" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.


Presentation Transcript

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!

That‘s

typical

for you humans! Please wait until the talk ist finished before you start asking questions!

Related Contents


Next Show more