the shield that never was: societies with single-peaked preferences are more open to manipulation...

The Shield that Never Was:Societies with Single-Peaked Preferences are More Open to Manipulation and Control

Piotr FaliszewskiAGH University of Science and

Technology, Krakow

Jörg RotheHeinrich-Heine-Universität

Düsseldorf

Lane A. HemaspaandraUniversity of Rochester

Edith HemaspaandraRochester Institute of

Technology

Moscow, SCW 2010

1

Outline Introduction

Computational Foundations of Social Choice, an ESF Project Elections and Single-Peaked Preferences. Thanks, Toby! Control and Manipulation

Overview of Results Control: Single-Peakedness Removing NP-Hardness Shields Manipulation: Single-Peaked Preferences

NP-Hardness Shields: Removing them Leaving them in Place Erecting them

A Dichotomy Result for 3-Candidate Scoring Protocols

A Sample Proof Sketch

2

3

CFSC Project Participants

Principal Investigators: Felix Brandt (München) Ulle Endriss (Amsterdam) Jeffrey Rosenschein (Jerusalem) Jörg Rothe (Düsseldorf) Remzi Sanver (Instanbul)

Associated Partners: Vincent Conitzer (Duke University) Edith Elkind (Singapore/Southampton) Edith Hemaspaandra (Rochester) Lane Hemaspaandra (Rochester) Jerome Lang (Paris/Toulouse) Jean-François Laslier (Paris) Nicolas Maudet (Paris)

AI TCS

AI LOG

AI

TCS

ECON

AI ECON

TCS

TCS LOG

TCS

AI LOG

ECON

AI

4

What did the Düsseldorf Group do in 2009?

This is Nadja Betzler from Jena,not Magnus Roos from D’dorf.

ClaudiaDoro Gábor

Jörg

5

What did the Düsseldorf Group do in 2009?

Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Jérôme Yann

NicolasJean-François The Shield that Never Was: Societies

with Single-Peaked Preferences are MoreOpen to Manipulation and Control.

TARK’09

Introduction Computational Social Choice

Applications in AI Multiagent systems Multicriteria decision making Meta search-engines Planning

Applications in social choice theory and political science Computational barrier to prevent cheating in elections

Manipulation Control Bribery

Computational agents can systematically

analyze an election to find the optimal

behavior.

6

Introduction



behavior.

Using the power of NP-hardness, vulcans have created complexity shields to

protect elections against many types of manipulation and procedural control.

7

Introduction



behavior.

Using the power of NP-hardness, vulcans have created complexity shields to

protect elections against many types of manipulation and procedural control.

Our Main Theme: Complexity shields may

evaporate in single-peaked societies

7

Elections An election is a pair (C,V) with

candidate set C = {c1, ..., cm}:

and a list of votes V = (v1, ..., vn):

Each vote vi is represented via its preferences over C: Either linear orders:

> > > >

Or approval vectors: (1,1,0,0,1)

An election system aggregates the preferences and outputs the set of winners.

Hi v7, I hope you are not one of those awful people who

support Mr. Smith!

Hi, my name is v7.

How will they aggregate our

votes?!

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Election Systems Approval (any number of candidates): Every

vote is an approval vector from All candidates with the most points are winners. Example:

C1,0

v1 1 1 0 0 1

v2 0 1 1 0 0

v3 1 1 0 0 1

v4 0 0 0 1 0

v5 1 0 0 1 1

v6 1 0 0 0 1

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C1,0

v1 1 1 0 0 1

v2 0 1 1 0 0

v3 1 1 0 0 1

v4 0 0 0 1 0

v5 1 0 0 1 1

v6 1 0 0 0 1

∑ 4 3 1 2 4

9



Winners:

C1,0

v1 1 1 0 0 1

v2 0 1 1 0 0

v3 1 1 0 0 1

v4 0 0 0 1 0

v5 1 0 0 1 1

v6 1 0 0 0 1

∑ 4 3 1 2 4

9


vote is an approval vector from All candidates with the most points are winners. Scoring protocols for m candidates are specified

by scoring vectors with where each voter‘s i-th candidate gets points: m-candidate plurality:

m-candidate j-veto:

Borda: Plurality (any number of candidates): Veto (any number of candidates):

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

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

m

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

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

)0,1,...,1(

C1,0

i

9

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).

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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


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


Single-Peaked Preferences

Preference that is inconsistent with linear order of candidates

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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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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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If each vote vi in V is an approval vector over C, this means that for each triple of candidates c, d, and e:

c L d L e implies that for each i,if vi approves of both c and e then vi approves of d.

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If each vote vi in V is an approval vector over C, this means that for each triple of candidates c, d, and e:

c L d L e implies that for each i,if vi approves of both c and e then vi approves of d.

Fulkerson & Gross (1965); Booth & Lueker (1976): Given a collection V of approval vectors 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.

10

Control and Manipulation The bad guy wants to make someone win (constructive)

or prevent someone from winning (destructive). The bad guy knows everybody else’s votes. In control, the chair modifies an election‘s structure by:

Adding candidates (limited/unlimited number) Deleting candidates Partition of candidates with/without runoff Adding/deleting voters Partition of voters

In manipulation, a coalition of agents change their votes to obtain their desired effect. Both nonmanipulators and manipulators are weighted. In the single-peaked case, both nonmanipulators and

manipulators are single-peaked w.r.t. the same order L. See Bartholdi, Tovey & Trick (1989; 1992), Conitzer, Sandholm

& Lang (2007), Hemaspaandra, Hemaspaandra & Rothe (2007).

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12

Control Results: Approval Theorem 2: For the single-peaked case, approval voting is

vulnerable to constructive control by adding voters and constructive control by deleting voters,

in the unique-winner and the nonunique-winner model, for the standard and the succinct input model.

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For comparison: Among all types of control by adding/deleting either candidates or voters, the above two cases are the only two resistances in the general case.

(Hemaspaandra, Hemaspaandra & Rothe, AAAI’05; Artificial Intelligence 2007)

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For comparison:

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Approval Voting (general case)

constructive destructiveAdding Candidates (limited) Immune Vulnerable

Adding Candidates (unlimited) Immune Vulnerable

Deleting Candidates Vulnerable Immune

Adding Voters Resistant Vulnerable

Deleting Voters Resistant Vulnerable

Control Results: Plurality Theorem 3: For the single-peaked case, plurality voting is

vulnerable to constructive and destructive control by adding candidates, adding an unlimited number of candidates, and deleting candidates

in the unique-winner and the nonunique-winner model.

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in the unique-winner and the nonunique-winner model. For comparison:

For each of these six types of candidate control plurality voting is resistant in the general case, but is vulnerable to the four types of control involving adding/deleting voters.

(Bartholdi, Tovey & Trick, 1992; Hemaspaandra, Hemaspaandra & Rothe, 2007)

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in the unique-winner and the nonunique-winner model. For comparison:

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Plurality (general case)

constructive destructiveAdding Candidates (limited) Resistant Resistant

Adding Candidates (unlimited) Resistant Resistant

Deleting Candidates Resistant Resistant

Adding Voters Vulnerable Vulnerable

Deleting Voters Vulnerable Vulnerable







15

Manipulation: Removing NP-Hardness Shields

Theorem 4: For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) for each of the following election systems is in P:

The scoring protocol , i.e., 3-candidate Borda. Each of the scoring protocols , . Veto.

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

ji

ji

16


Theorem 4: For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) for each of the following election systems is in P:

The scoring protocol , i.e., 3-candidate Borda. Each of the scoring protocols , . Veto.

For comparison: 3-candidate Borda, Veto, and the „ “ cases of , , are NP-complete in the general case (and the rest is in P).

(Hemaspaandra & Hemaspaandra, 2007; Procaccia & Rosenschein, 2007; Conitzer, Sandholm & Lang, 2007).

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

ji

ji

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

ji

16


Theorem 5: For the single-peaked case, the constructive coalition weighted manipulation problem for m-candidate 3-veto is in P for m in {3,4,6,7,8,…} and is resistant (indeed, NP-complete) for m=5 candidates.

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Theorem 5: For the single-peaked case, the constructive coalition weighted manipulation problem for m-candidate 3-veto is in P for m in {3,4,6,7,8,…} and is resistant (indeed, NP-complete) for m=5 candidates.

For comparison: m-candidate 3-veto is in P for m in {3,4} and is resistant (indeed, NP-complete) for five or more candidates.

(Hemaspaandra & Hemaspaandra; Journal of Computer and System Sciences 2007).

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Manipulation: Leaving them in Place

Theorem 6: For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) is resistant (indeed, NP-complete) for the scoring protocol and the scoring protocol ,

i.e., 4-candidate Borda.

)0,1,3()0,1,2,3(

19

Manipulation: Leaving them in Place

Theorem 6: For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) is resistant (indeed, NP-complete) for the scoring protocol and the scoring protocol ,

i.e., 4-candidate Borda.

For comparison: These problems are known to be NP-complete also in the general case.

(Hemaspaandra & Hemaspaandra, 2007)

These results are particularly inspired by Walsh (2007) who proved the same for Single Transferable Voting.

)0,1,3()0,1,2,3(

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Manipulation: Erecting NP-Hardness Shields

Can restricting to single-peaked preferences ever erect a complexity shield?

General case

Single-peakedcase

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Manipulation: Erecting NP-Hardness Shields

Can restricting to single-peaked preferences ever erect a complexity shield?

Theorem 7: There exists an election system, whose votes are approval vectors, for which constructive size-3-coalition unweighted manipulation is in P for the general case but is NP-complete in the single-peaked case.

General case

Single-peakedcase

20

Manipulation: A Dichotomy Result

Theorem 8: Consider a 3-candidate scoring protocol

For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) is resistant (indeed, NP-complete) when

and is in P otherwise.

),,,( 321 .321

02 3231

21







22

A Sample Proof Sketch Theorem 2: For the single-peaked case, approval voting is



23




We focus on: constructive control by adding voters in the unique-winner model for the succinct input model.

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We give a poly-time algorithm that, given collections V and W of votes over candidate set C and single-

peaked w.r.t. order L, a designated candidate p in C, and an addition limit k,

decides if by adding at most k votes from W we can make p the unique winner.

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1

1

473

95

2

number of approvals from voters in V for candidates that are

votes in W that can be added (withmultiplicities)

24


1

1

473

95

2


votes in W that can be added (withmultiplicities) Which vote

types from W should we add? Especially if they are incomparable?

24


1

1

473

95

2


votes in W that can be added (withmultiplicities) We‘ll handle

this by a „smart greedy“

algorithm.

24


1

1

473

95

2


votes in W that can be added (withmultiplicities) Why are F, C,

B, c, f, and j dangerous but the remaining candidates can be ignored?

24


1

1

473

95

2


votes in W that can be added (withmultiplicities) First, each

added vote will be an interval

including p. So drop all others.

24


1

1

473

2


votes in W that can be added (withmultiplicities) First, each

added vote will be an interval

including p. So drop all others.

24


1

1

473

2


votes in W that can be added (withmultiplicities) Now, if adding

votes from W causes p to

beat c then p must also beat

a and b.

24


1

1

473

2


votes in W that can be added (withmultiplicities) Thus, c is a

dangerous rival for p

but a and b can safely

be ignored.

24


1

1

473

2


votes in W that can be added (withmultiplicities) Likewise, f is

dangerous but d and e

can safely be ignored.

24


1

1

473

2


votes in W that can be added (withmultiplicities) Likewise, j is

dangerous but g, h, and i can safely be

ignored.

24


1

1

473

2


votes in W that can be added (withmultiplicities) Hey, why do

you do that step by step?Just say j is dangerous and ignore a, …, i.

24


1

1

473

2


votes in W that can be added (withmultiplicities) No! Look what

happens if we add 6 votes of the type with multiplicity 7!

24


1

1

413

2votes in W that can be added (withmultiplicities) No! Look what

happens if we add 6 votes of the type with multiplicity 7!

24


1

1

473

2


votes in W that can be added (withmultiplicities) OK, that‘s not

illogical. But how does your „smart greedy“ algorithm work?

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Smart Greedy Algorithm OK, first I need more space for that!

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Smart Greedy Algorithm OK, first I need more space for that! In smart greedy, we eat through all dangerous

rivals to the right of p starting with the leftmost: c. To become the unique winner, p must beat c. Only votes in W whose right endpoints fall in [p,c)

can help. Let B be the set of those votes. Choose votes from B starting with the rightmost left

endpoint. This is a perfectly safe strategy!

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Smart Greedy Algorithm

1

1

473

2votes in W that can be added (withmultiplicities)

25


1

1

2votes in B that can be added (withmultiplicities)

25


1

0


25


1

First rival

defeated


25

Smart Greedy Algorithm OK, first I need more space for that! In smart greedy, we eat through all dangerous

rivals to the right of p starting with the leftmost: c. To become the unique winner, p must beat c. Only votes in W whose right endpoints fall in [p,c)

can help. Let B be the set of those votes. Choose votes from B starting with the rightmost left

endpoint. This is a perfectly safe strategy! Iterate. If you run out of dangerous candidates on the right

of p, mirror image the societal order (i.e., reverse L) and finish off the remaining dangerous candidates until you either succeed or reach the addition limit.

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Summary and Future Directions Single-peakedness

removes many complexity shields against control and manipulation

leaves others in place can even erect complexity shields

When choosing election systems for single-peaked electorates, one must not rely on such shields.

26

Summary and Future Directions Single-peakedness

removes many complexity shields against control and manipulation

leaves others in place can even erect complexity shields

When choosing election systems for single-peaked electorates, one must not rely on such shields.

Do such shield removals hold in two-dimensional (or k-dimensional) analogues of our unidimensional single-peakedness?

Can our results be extended to „very nearly“ single-peaked societies?

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COMSOC 2010

Third International Workshop on Computational Social ChoiceDüsseldorf, Germany, September 13–16, 2010

Important Dates•Paper submission deadline: May 15, 2010 •Notification of authors: July 1, 2010 •Camera-ready copies due: July 15, 2010 •Early registration deadline: July 15, 2010 •LogICCC Tutorial day: September 13, 2010 •Workshop dates: September 14–16, 2010

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Thank you!

Stop it!No more

questions please!

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the shield that never was: societies with single-peaked preferences are more open to manipulation...

Documents

election systemsapproval

number of candidates

complexity shields

types of manipulation

procedural control

optimal behavior

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