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
Computational Social Choice
With the power of NP-hardness vulcans have constructed complexity shields to protect elections against many
types of manipulation and control.
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 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
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
eliminate cand. with lowestplurality score
Final Round
over
eliminate cand. with lowestplurality score
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 Round
over
eliminate cand. with lowestplurality score
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 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
Voting SystemsLevel-based: Bucklin Voting (BV)
v1: > > >v2: > > > v3: > > > v4: > > >v5: > > >
5 voters => strict majority threshold is 3
Lvl 1 1 2 2 0Lvl 2 2 2 3 3
Voting SystemsLevel-based: Bucklin Voting (BV)
v1: > > >v2: > > > v3: > > > v4: > > > Level 2 Bucklinv5: > > > winners:
5 voters => strict majority threshold is 3
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 to Protect Elections
NP-hardness shields
Manipulation & Control inSingle-peaked Electorates
Elections & Voting Systems
Manipulation & Control
Proof Sketch: CCAV in Approval
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 to Protect Elections
NP-hardness shields
Manipulation & Control inSingle-peaked Electorates
Elections & Voting Systems
Manipulation & Control
Proof Sketch: CCAV in Approval
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
low galactic taxes high galactic taxes
Single-Peaked Preferences
Single-peaked preference consistent with linear order of candidates
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
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.
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.
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 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
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 to Protect Elections
NP-hardness shields
Manipulation & Control inSingle-peaked Electorates
Elections & Voting Systems
Manipulation & Control
Proof Sketch: CCAV in Approval
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)
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) Which vote
types from W should we add? Especially if they are incomparable?
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) We‘ll handle
this by a „smart greedy“
algorithm.
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) Why are F, C,
B, c, f, and j dangerous but the remaining candidates can be ignored?
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) First, each
added vote will be an interval
including p. So drop all others.
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
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities) First, each
added vote will be an interval
including p. So drop all others.
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
2
number of approvals from voters in V for candidates that are
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.
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
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities) Thus, c is a
dangerous rival for p
but a and b can safely
be ignored.
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
2
number of approvals from voters in V for candidates that are
votes in V‘ that can be added (withmultiplicities) Likewise, f is
dangerous but d and e
can safely be ignored.
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
2
number of approvals from voters in V for candidates that are
votes in V‘ that can be added (withmultiplicities) Likewise, j is
dangerous but g, h, and i can safely be
ignored.
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
2
number of approvals from voters in V for candidates that are
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.
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
2
number of approvals from voters in V for candidates that are
votes in V‘ that can be added (withmultiplicities) No! Look what
happens if we add 6 votes of the type with multiplicity 7!
A Sample Proof Sketch
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!
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
2
number of approvals from voters in V for candidates that are
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)
F E D C B A p a b c d e f g h i j k012345678
dangerousto be ignored
F E D C B A p a b c d e f g h i j k012345678
dangerousto be ignored
Smart Greedy Algorithm
1
1
2 votes in B that can be added (withmultiplicities)
Smart Greedy Algorithm
1
0
2 votes in B that can be added (withmultiplicities)
Smart Greedy Algorithm
1
First rival
defeated
1 votes in B that can be added (withmultiplicities)
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!