manipulation and control for approval voting and other voting systems

Manipulation and Controlfor Approval Voting andOther Voting Systems

Jörg Rothe

Oxford Meeting for COST Action IC1205on Computational Social Choice

April 16, 2013

IntroductionSocial Choice Theory

voting theory preference aggregation judgment aggregation

Theoretical 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

Theoretical Computer Science artificial intelligence algorithm design computational complexity theory

- worst-case/average-case complexity- optimization, etc.

Software agents can

systematically analyze

elections to find optimal

strategies

Computational Social Choice

computational barriers to prevent • manipulation• control• bribery


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 NP-hardness complexity

shields enough? Or do they evaporate for

single-peaked electorates?

NP-Hardness Shields to Protect Elections

NP-hardness shields

Manipulation & Control inSingle-peaked Electorates

Elections & Voting Systems

Manipulation & Control

Proof Sketch: CCAV in Approval

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 Hi, I am Ramon Llull. In 1299, I

came up with the voting system

that these guys now study!

Llull/Copeland Rule For FIFA World Championships or UEFA European Championships: Simply use = 1/3 as the tie value.

Difference between the Llull and the Copeland rule?What happens if the head-to-head contest ends with a tie? Llull: Both get 1 point Copeland0: Both get 0 points Copeland0.5: Both get half a point Copeland: Both get points, for a rational , 0<<1

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.


NP-hardness shields





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

NP-Hardness Shields for Control

References: Control J. Bartholdi, C. Tovey, and M. Trick: How Hard is it to Control an

Election? Mathematical and Computer Modelling, 1992. E. Hemaspaandra , L. Hemaspaandra, and J. Rothe: Anyone but

Him: The Complexity of Precluding an Alternative. Artificial Intelligence, 2007. (AAAI-2005)

P. Faliszewski, E. Hemaspaandra , L. Hemaspaandra, and J. Rothe: Llull and Copeland Voting Computationally Resist Bribery and Constructive Control. Journal of Artificial Intelligence Research, 2009. (AAAI-2007; AAIM-2008)

G. Erdélyi, M. Nowak, and J. Rothe: SP-AV Fully Resists Constructive Control and Broadly Resists Destructive Control. Mathematical Logic Quarterly, 2009. (MFCS-2008)

G. Erdélyi and J. Rothe: Control Complexity in Fallback Voting. Proceedings of CATS-2010.

G. Erdélyi, L. Piras, and J. Rothe: The Complexity of Voter Partition in Bucklin and Fallback Voting: Solving Three Open Problems. Proceedings of AAMAS-2011.

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





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


NP-hardness shields





A Sample Proof Sketch

F E D C B A p a b c d e f g h i j k012345678

dangerousto be ignored

1

1

473

9 5

2

number of approvals from voters in V for candidates that are

votes in W that can be added (withmultiplicities)




1

1

473

9 5

2


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

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




1

1

473

9 5

2


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

this by a „smart greedy“

algorithm.




1

1

473

9 5

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?




1

1

473

9 5

2


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

added vote will be an interval

including p. So drop all others.




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.




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.




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.




1

1

473

2


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

dangerous but d and e

can safely be ignored.




1

1

473

2


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

dangerous but g, h, and i can safely be

ignored.




1

1

473

2


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

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




1

1

473

2


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

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


1

1

413

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

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




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?

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

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

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!

cp,

Smart Greedy Algorithm

1

1

473

2 votes in W that can be added (withmultiplicities)






1

1

2 votes in B that can be added (withmultiplicities)


1

0



1

First rival

defeated


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

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.

cp,

Thank you very much!

That‘s typical for you humans! Please wait

until the talk ist finished before you

start asking questions!

manipulation and control for approval voting and other voting systems

Documents

plurality voting pv

nphardness complexity

approval vectors

hardness vulcans

approvalnphardness shields

approvals winners

power of np

candidates strictlycopeland