projektseminar computational social choice -eine einführung- jörg rothe & lena schend ss 2012,...
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Projektseminar Computational Social Choice
-Eine Einführung-
Jörg Rothe & Lena SchendSS 2012, HHU Düsseldorf
4. April 2012
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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
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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
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Computational Social Choice
With the power of NP-hardness vulcans have constructed complexity shields to protect elections against many
types of manipulation and control.
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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"?
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NP-Hardness Shields Evaporating?
NP-hardness shields
single-peaked electorates
junta distributions
approximation
experimental analysis
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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
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Voting SystemsApproval Voting (AV) votes are approval vectors in C1,0
v1 1 1 0 1
v2 0 1 0 0
v3 1 1 0 1
v4 0 0 1 0
v5 1 0 1 1
v6 1 0 0 1
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Voting SystemsApproval Voting (AV) votes are approval vectors in winners: all candidates with the most approvals
v1 1 1 0 1
v2 0 1 0 0
v3 1 1 0 1
v4 0 0 1 0
v5 1 0 1 1
v6 1 0 0 1
∑ 4 3 2 4
C1,0
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Voting SystemsApproval Voting (AV) votes are approval vectors in winners: all candidates with the most approvals
winners:
v1 1 1 0 1
v2 0 1 0 0
v3 1 1 0 1
v4 0 0 1 0
v5 1 0 1 1
v6 1 0 0 1
∑ 4 3 2 4
C1,0
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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(
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- 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
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Voting SystemsRound-based: Single Transferable Vote (STV)
v1: > > > v2: > > >
v3: > > > v4: > > >
Round 1over
eliminate cand. with lowestplurality score
Round 2over
eliminate cand. with lowestplurality score
Final Roundover
eliminate cand. with lowestplurality score
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Voting SystemsRound-based: Single Transferable Vote (STV)
v1: > > v2: > >
v3: > > v4: > >
Round 1over
eliminate cand. with lowestplurality score
Round 2over
eliminate cand. with lowestplurality score
Final Roundover
eliminate cand. with lowestplurality score
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Voting SystemsRound-based: Single Transferable Vote (STV)
v1: v2:
v3: v4:
Round 1over
eliminate cand. with lowestplurality score
Round 2over
eliminate cand. with lowestplurality score
Final Roundover
… and the winner is…
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Voting SystemsLevel-based: Bucklin Voting (BV)
v1: > > >
v2: > > >
v3: > > >
v4: > > >
v5: > > >
5 voters => strict majority threshold is 3
Lvl 1 1 2 2 0
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Voting SystemsLevel-based: Bucklin Voting (BV)
v1: > > >
v2: > > >
v3: > > >
v4: > > >
v5: > > >
5 voters => strict majority threshold is 3
Lvl 1 1 2 2 0
Lvl 2 2 2 3 3
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Voting SystemsLevel-based: Bucklin Voting (BV)
v1: > > >
v2: > > >
v3: > > >
v4: > > > Level 2 Bucklin
v5: > > > winners:
5 voters => strict majority threshold is 3
Lvl 1 1 2 2 0
Lvl 2 2 2 3 3
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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.
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War on Electoral ControlAV
winners:
"chair": knows all preferences
v1 1 1 0 1
v2 0 1 0 0
v3 1 1 0 1
v4 0 0 1 0
v5 1 0 1 1
v6 1 0 0 1
∑ 4 3 2 4
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War on Electoral ControlAV winner:
"chair": knows all preferences and can change the
structure of an election
v1 1 1 0 1
v2 0 1 0 0
v3 1 1 0 1
v4 0 0 1 0
v5 1 0 1 1
v6 1 0 0 1
∑ 2 3 1 2
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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 1
v2 0 1 0 0
v3 1 1 0 1
v4 0 0 1 0
v5 1 0 1 1
v6 1 0 0 1
∑ 2 3 1 2
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NP-Hardness Shields for Control
Resistance = NP-hardness, Vulnerability = P, Immunity, and Susceptibility
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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
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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
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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 Manipulation2
1
I like Spock but I don‘t
want him to be the
captain!!
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NP-Hardness Shields for Manipulation
Results due to Conitzer, Sandholm, Lang (J.ACM 2007)
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NP-Hardness Shields Evaporating?
NP-hardness shields
single-peaked electorates
junta distributions
approximation
experimental analysis
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Junta Distributionsof Procaccia and Rosenschein (JAAMAS 2007) are omitted here, as they are a rather technical concept.
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NP-Hardness Shields Evaporating?
NP-hardness shields
single-peaked electorates
junta distributions
approximation
experimental analysis
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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NP-Hardness Shields Evaporating?
NP-hardness shields
single-peaked electorates
junta distributions
approximation
experimental analysis
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Approximating ManipulationBefore:
Is manipulation possible? ?
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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
?
?
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Approximating Borda
3x > > > > > >
2x > > > > > >
Borda winner
manipulators prefer
B-Score 5 0 18 19 20 21 22
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Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m1 > > > > > >
B-Score 5 0 18 19 20 21 22
Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)
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Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m1 > > > > > >
B-Score 11 5 22 22 22 22 22
Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)
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Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m1 > > > > > >
m2 > > > > > >
B-Score 11 5 22 22 22 22 22
Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)
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Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m1 > > > > > >
m2 > > > > > >
B-Score 17 10 26 25 24 23 22
Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)
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Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m1 > > > > > >
m2 > > > > > >
m3 > > > > > >
B-Score 17 10 26 25 24 23 22
Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)
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Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m1 > > > > > >
m2 > > > > > >
m3 > > > > > >
B-Score 23 15 26 26 26 26 26
Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)
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Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m1 > > > > > >
m2 > > > > > >
m3 > > > > > >
m4 > > > > > >
B-Score 23 15 26 26 26 26 26
Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)
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Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m1 > > > > > >
m2 > > > > > >
m3 > > > > > >
m4 > > > > > >
B-Score 29 20 30 29 28 27 26
Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)
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Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m1 > > > > > >
m2 > > > > > >
m3 > > > > > >
m4 > > > > > >
m5 > > > > > >
B-Score 29 20 30 29 28 27 26
Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)
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Approximating BordaAlgorithm for Borda-CCUM : "Reverse"
m1 > > > > > >
m2 > > > > > >
m3 > > > > > >
m4 > > > > > >
m5 > > > > > >
B-Score 35 25 30 30 30 30 30
Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)
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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
Zuckerman, Procaccia & Rosenschein (Artificial Intelligence 2009)
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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)
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NP-Hardness Shields Evaporating?
NP-hardness shields
single-peaked electorates
junta distributions
approximation
experimental analysis
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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
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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 candidates
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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 candidates
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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).
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.
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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).
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.
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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
v2 0 1 1 0 0 yes
v3 1 1 0 0 1 no
v4 0 0 0 1 0 yes
v5 1 0 0 1 1 no
v6 1 0 0 0 1 no
Single-Peaked Approval Vectors
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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
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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
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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
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NP-Hardness Shields Evaporating?
NP-hardness shields
single-peaked electorates
junta distributions
approximation
experimental analysis
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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
• all preferences are equally likely
• voters are correlated
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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
• all preferences are equally likely
• voters are correlated
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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
• all preferences are equally likely
• voters are correlated
v1 v2 v3 v4 ...
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Experiments ControlObservations: destructive control shows more yes-instances (up to
100%) and lower computational costs DCPV-TP in FV
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Experiments ControlObservations: destructive control shows more yes-instances (up to
100%) and lower computational costs CCPV-TP in FV
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Experiments ControlObservations: destructive control shows more yes-instances (up to
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
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