Projektseminar Computational Social Choice -Eine Einführung-

Projektseminar Computational Social Choice

-Eine Einführung-

Jörg Rothe & Lena SchendSS 2012, HHU Düsseldorf

4. April 2012

IntroductionSocial Choice Theory

voting theory preference aggregation judgment aggregation

Computer Science artificial intelligence algorithm design computational 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 strategies

IntroductionSocial Choice Theory

voting theory preference aggregation judgment aggregation

Computational Social Choice

Computer Science artificial intelligence algorithm design computational 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

strategies


With the power of NP-hardness vulcans have constructed complexity shields to protect elections against many

types of manipulation and control.


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"?

NP-Hardness Shields Evaporating?

NP-hardness shields

single-peaked electorates

junta distributions

approximation

experimental analysis

Elections An 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 preferences

Voting SystemsApproval Voting (AV) votes are approval vectors in C1,0

v1 1 1 0 1v2 0 1 0 0v3 1 1 0 1v4 0 0 1 0v5 1 0 1 1v6 1 0 0 1

Voting SystemsApproval Voting (AV) votes are approval vectors in winners: all candidates with the most approvals

v1 1 1 0 1v2 0 1 0 0v3 1 1 0 1v4 0 0 1 0v5 1 0 1 1v6 1 0 0 1∑ 4 3 2 4

C1,0

Voting SystemsApproval Voting (AV) votes are approval vectors in winners: all candidates with the most approvals

winners:

v1 1 1 0 1v2 0 1 0 0v3 1 1 0 1v4 0 0 1 0v5 1 0 1 1v6 1 0 0 1∑ 4 3 2 4

C1,0

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

),...,,( 21 m m ...21

i

Borda: Plurality Voting (PV):

k-Approval (m-k-Veto): Veto (Anti-Plurality):

)0,...,0,1(1

m

)0,...,0,1,...,1( kmk

)0,...,2,1( mm

)0,1,...,1(

- 4:0 2:2 3:1

0:4 - 1:3 2:2

2:2 3:1 - 2:2

1:3 2:2 2:2 -

Voting SystemsPairwise Comparison

v1: > > > v3: > > >v2: > > > v4: > > >

Condorcet: beats all other candidates

strictlyCopeland : 1 point for

victory points for tie

Maximin: maximum of theworst pairwise comparison

0,1α

1α

α

1α

Voting SystemsRound-based: Single Transferable Vote (STV)

v1: > > > v2: > > >v3: > > > v4: > > >

Round 1over

eliminate cand. with lowestplurality score

Round 2over


Final Round

over



v1: > > v2: > > v3: > > v4: > >

Round 1over


Round 2over


Final Round

over



v1: v2: v3: v4:

Round 1over


Round 2over


Final Round

over

… and the winner is…

Voting SystemsLevel-based: Bucklin Voting (BV)

v1: > > >v2: > > > v3: > > > v4: > > >v5: > > >

5 voters => strict majority threshold is 3

Lvl 1 1 2 2 0


v1: > > >v2: > > > v3: > > > v4: > > >v5: > > >


Lvl 1 1 2 2 0Lvl 2 2 2 3 3


v1: > > >v2: > > > v3: > > > v4: > > > Level 2 Bucklinv5: > > > winners:


Lvl 1 1 2 2 0Lvl 2 2 2 3 3

Voting SystemsLevel-based: Fallback Voting (FV) combines AV and BV

Candidates:

v: { , } | { , }

v: > | { , }

Bucklin winners are fallback winners. If no Bucklin winner exists (due to disapprovals),

then approval winners win.

War on Electoral ControlAV

winners:

"chair": knows all preferences

v1 1 1 0 1v2 0 1 0 0v3 1 1 0 1v4 0 0 1 0v5 1 0 1 1v6 1 0 0 1∑ 4 3 2 4

War on Electoral ControlAV winner:

"chair": knows all preferences and can change the

structure of an election

v1 1 1 0 1v2 0 1 0 0v3 1 1 0 1v4 0 0 1 0v5 1 0 1 1v6 1 0 0 1∑ 2 3 1 2

War on Electoral ControlAV winner:

"chair": knows all preferences and can change the

structureOther types of control: of an election adding/partitioning voters deleting/adding/partitioning candidates

v1 1 1 0 1v2 0 1 0 0v3 1 1 0 1v4 0 0 1 0v5 1 0 1 1v6 1 0 0 1∑ 2 3 1 2

NP-Hardness Shields for Control

Resistance = NP-hardness, Vulnerability = P, Immunity, and Susceptibility

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 ManipulationCopeland : winner

v1: > > > v3: > > >v2: > > > v4: > > >

I like Spock but I don‘t

want him to be the

captain!!21

Copeland : winner

v1: > > > v3: > > >v2: > > > v4: > > >

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!!21

Copeland : winners

v1: > > > v3: > > >v2: > > > v4: > > >

Here: unweighted voters, single manipulator

. Other types: - coalitional

manipulation - weighted voters

Cope-land

Score- 3:1 2:2 2:2 2

1:3 - 1:3 1:3 0

2:2 3:1 - 2:2 2

2:2 3:1 2:2 - 2

War on Manipulation21

I like Spock but I don‘t

want him to be the

captain!!

NP-Hardness Shields for Manipulation

Results due to Conitzer, Sandholm, Lang (J.ACM 2007)


NP-hardness shields


junta distributions

approximation


Junta Distributionsof Procaccia and Rosenschein (JAAMAS 2007) are omitted here, as they are a rather technical concept.


NP-hardness shields


junta distributions

approximation


Experiments Manipulation testing (heuristic) algorithms for manipulation problem

at hand on given elections sample real elections generate random elections:

Impartial Culture (IC) Polya-Eggenberger (PE)• voters vote independently

• all preferences are equally likely

• voters are highly correlated

v1 v2 v3 ...

Walsh (IJCAI 2009; ECAI 2010)

Experiments Manipulation Results for STV

Single 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 voters Coalitional Manipulation:

larger coalitions are more likely to be successful again: computational costs are low for up to 128

candidates/voters Results for Veto (weighted)

if manipulators‘ weights are too big/small => trivial even in critical region: computational costs are low only correlated voters increase computational costs

Walsh (IJCAI 2009; ECAI 2010)


NP-hardness shields


junta distributions

approximation


Approximating ManipulationBefore:

Is manipulation possible? ?

Approximating ManipulationBefore:

Is manipulation possible?

Now: How many manipulators are needed?

(min!)

Approximation Algorithms: efficient algorithms do not always find optimal solution can be analyzed both theoretically and

experimentally

??

Approximating Borda

3x > > > > > >

2x > > > > > >

Borda winner

manipulators prefer

B-Score 5 0 18 19 20 21 22

Approximating BordaAlgorithm for Borda-CCUM : "Reverse"

m1 > > > > > >

B-Score 5 0 18 19 20 21 22

Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)


m1 > > > > > >

B-Score 11 5 22 22 22 22 22



m1 > > > > > >

m2 > > > > > >

B-Score 11 5 22 22 22 22 22



m1 > > > > > >

m2 > > > > > >

B-Score 17 10 26 25 24 23 22



m1 > > > > > >

m2 > > > > > >

m3 > > > > > >

B-Score 17 10 26 25 24 23 22



m1 > > > > > >

m2 > > > > > >

m3 > > > > > >

B-Score 23 15 26 26 26 26 26



m1 > > > > > >

m2 > > > > > >

m3 > > > > > >

m4 > > > > > >

B-Score 23 15 26 26 26 26 26



m1 > > > > > >

m2 > > > > > >

m3 > > > > > >

m4 > > > > > >

B-Score 29 20 30 29 28 27 26



m1 > > > > > >

m2 > > > > > >

m3 > > > > > >

m4 > > > > > >

m5 > > > > > >

B-Score 29 20 30 29 28 27 26



m1 > > > > > >

m2 > > > > > >

m3 > > > > > >

m4 > > > > > >

m5 > > > > > >

B-Score 35 25 30 30 30 30 30


Approximating BordaOptimal solution: 4 manipulators

m1 > > > > > >

m2 > > > > > >

m3 > > > > > >

m4 > > > > > >

"Reverse" needs one manipulator more than optimal

B-Score 29 20 28 28 28 28 28


Approximation Results Maximin:

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 incomparable experimental comparison:

ØØ

IC model 76% 83% 99%PE model 76% 43% 99%

Ø

>

>

>

Zuckerman, Lev & Rosenschein (AAMAS 2011) Davies, Katsirelos, Narodytska & Walsh (AAAI 2011)


NP-hardness shields


junta distributions

approximation


Single-Peaked Preferences 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

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


Single-Peaked Preferences

Single-peaked preference consistent with linear order of candidates


A voter‘s > > > preference curve on galactic taxes


Single-Peaked Preferences

Preference that is inconsistent with this linear order of candidates



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.



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.


Single-peaked w.r.t. this order?

v1 1 1 0 0 1 nov2 0 1 1 0 0 yesv3 1 1 0 0 1 nov4 0 0 0 1 0 yesv5 1 0 0 1 1 nov6 1 0 0 0 1 no

Single-Peaked Approval Vectors

Removing NP-hardness shields: 3-candidate Borda veto every scoring protocol for -candidate 3-veto,

Leaving them in place: STV (Walsh AAAI 2007) 4-candidate Borda 5-candidate 3-veto

Erecting 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

ji )0,...,0,1,...,1( ji

6mm

Constructive Coalitional Weighted Manipulation

Removing NP-hardness shields: Approval

Constructive control by adding voters Constructive control by deleting voters

Plurality constructive control by adding candidates destructive control by adding candidates constructive control by deleting candidates destructive 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 Electorates

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 and 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.

),,,( 321 :321 02 3231


NP-hardness shields


junta distributions

approximation


Experiments Control same approach as for manipulation testing (heuristic) algorithms for control problem at

hand on given elections sample real elections generate random elections:

Impartial Culture (IC) Two Mainstreams (TM)• voters vote independently


• voters are correlated

Experiments Control same approach as for manipulation testing (heuristic) algorithms for control problem at

hand on given elections sample real elections generate random elections:

Impartial Culture (IC) Two Mainstreams (TM)• voters vote independently


• voters are correlated

v1 v2 v3 v4 ...

Experiments ControlObservations: destructive control shows more yes-instances (up to

100%) and lower computational costs DCPV-TP in FV


100%) and lower computational costs CCPV-TP in FV


100%) and lower computational costs FV and BV show similar tendencies voter control in PV has lower computational costs deleting/adding voters show similar tendencies for constructive control: voter control shows more

yes-instances than candidate control as expected: more yes-instances in the IC model

than in the TM model

Thank you very much!

That‘s typical for you humans! Please wait

until the talk ist finished before you

start asking questions!

Projektseminar Computational Social Choice -Eine Einführung-

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