Complexity of manipulating
elections with few candidates
Vincent Conitzer and Tuomas Sandholm
Carnegie Mellon University
Computer Science Department
Outline
• Introduction
– Voting
– Manipulation
– Computational complexity as a barrier to manipulation
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•
•
•
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Problem specification & assumptions
Deterministic voting protocols
Randomized voting protocols
Uncertainty about others’ votes
Conclusion
Voting
• In multiagent systems, agents may have conflicting
preferences
• A preference aggregator often must choose one candidate
from the possible outcomes
–
–
–
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Deciding on a leader/coordinator/representative
Joint plans
Allocations of tasks/resources
…
• Voting is the most general preference aggregation method
– Applicable to any preference aggregation setting
– No side payments
Voting
Voter 1
“A > B > C”
Voter 2
“A > C > B”
Voting protocol chooses winner
(probably A here)
Voter 3
“B > A > C”
Manipulation in voting
• A voter is said to manipulate when it does not rank the
candidates truthfully
• Example:
– Voter prefers Nader to Gore to Bush
– But Nader is extremely unlikely to win
– So, voter ranks Gore first instead of Nader
• Why is manipulation bad?
– Protocol is designed to maximize social welfare
 Manipulation will cause a suboptimal outcome to be chosen
– If the protocol actually relies on manipulation to choose the right
outcome, there exists another nonmanipulable protocol that
chooses this same outcome (Revelation Principle)
Manipulation in voting
“Gore”
“Bush”
“Gore” “Bush”
Nader > Gore > Bush
• Voting truthfully (for Nader) might let Bush win,
certainly will not get Nader to win
• So, better to rank Gore first
Different protocols
• Plurality voting (only top candidate matters) is not the only option
• Example: Single Transferable Vote (STV)
– Each round, candidate with fewest (first) votes drops out
– When your candidate drops out, your vote transfers to your next most
preferred (remaining) candidate
– Now our voter can safely vote for Nader, then let the vote transfer to Gore
• STV is also manipulable
– Sometimes supporting a candidate hurts that candidate!
• Other protocols:
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–
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Borda (m points for the top candidate, m-1 for the second, …)
Copeland (maximum number of pairwise victories)
Maximin (maximum score in worst pairwise)
…
• Seminal result (Gibbard-Satterthwaite): all nondictatorial voting
protocols with >2 candidates are manipulable!
Software agents may manipulate more
• Human voters may not manipulate because:
– Do not consider the option of manipulation
– Insufficient understanding of the manipulability of the protocol
– Manipulation algorithms may be too tedious to run by hand
• For software agents, voting algorithms must be coded explicitly
– Rational strategic algorithms are preferred
– The voting algorithm needs to be coded only once
– Software agents are good at running algorithms
• Key idea: use computational complexity as a barrier to manipulation!
Overview
• Introduction
• Problem specification
– Prior research
– Small numbers of candidates
– Coalitions and weights
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•
•
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Deterministic voting protocols
Randomized voting protocols
Uncertainty about others’ votes
Conclusion
Prior research
• Individually manipulating Second-order Copeland is NP-complete
[Bartholdi, Tovey, Trick 1989]
• Individually manipulating STV is NP-complete [Bartholdi, Orlin 1991]
• Those results rely on the number of candidates (m) being unbounded
• We designed a recursive algorithm for individually manipulating
STV with O(m^1.62) calls (and usually much fewer) [Extended
version]
– Not too complex for realistic numbers of candidates
Manipulation complexity with few candidates
• Ideally, would like complexity results for constant number of
candidates
• But then manipulator can simply evaluate each possible vote
– assuming the others’ votes are known
• Even for coalitions, only polynomially many effectively different
votes
• However, if we place weights on votes, complexity may return…
Unbounded #candidates
Constant #candidates
Unweighted Weighted
voters
voters
Unweighted Weighted
voters
voters
Individual
manipulation
Can be
hard
Can be
hard
easy
easy
Coalitional
manipulation
Can be
hard
Can be
hard
easy
Potentially
hard
Why study weighted coalitional
manipulation?
• In large elections, usually impossible to manipulate
individually
• Many real world elections have weights
– E.g. electoral college
– Weights more likely with heterogeneous software agents
• Weighted coalitional manipulation is more realistic than
assuming unbounded #candidates
• We also derive other individual/unweighted results from
results in this setting
Overview
• Introduction
• Problem specification & assumptions
• Deterministic voting protocols
– Constructive manipulation
– Destructive manipulation
• Randomized voting protocols
• Uncertainty about others’ votes
• Conclusion
Constructive manipulation
• A coalition wants a certain candidate p to win
• Thrm.NP-complete for Borda, STV, Copeland, Maximin
– For 3, 3, 4, 4 candidates respectively
• Proof sketch for Borda:
– p is trailing the two other candidates, who are tied
– Naturally, colluders give p their first vote
– But must carefully divide their second votes across other
candidates
– This is basically doing PARTITION
Destructive manipulation
• A coalition wants a certain candidate h to not win
• Thrm. Easy for Borda, Copeland, Maximin, even with
unbounded numbers of candidates
– The algorithm relies on these methods being score-based and
monotonic (more support always helps)
• Thrm. NP-complete for STV (even with 4 candidates)
– Proof reduces constructive STV to destructive STV
• Sometimes, to get a candidate to not win, the coalition needs to get a
specific other candidate to the final round
Overview
•
•
•
•
•
•
Introduction
Problem specification & assumptions
Deterministic voting protocols
Randomized voting protocols
Uncertainty about others’ votes
Conclusion
Randomization can be used to make
manipulation hard
• Consider the Cup protocol:
– Candidates play an elimination tournament based on pairwise elections
b
c
b
a
b c
d
• Given the schedule (leaf labels), any type of manipulation is easy
even with unbounded #candidates
– For each node in tree, can build the set of candidates that would reach this
node for some vote by the coalition (from bottom up)
• Manipulating a subtree only requires commitment on the order of candidates in
that subtree
• Idea: randomize (uniformly) over schedules after votes received
• Theorem. Manipulating Randomized Cup is NP-complete
– Proof is complex & uses 7 candidates
Overview
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•
•
•
•
Introduction
Problem specification & assumptions
Deterministic voting protocols
Randomized voting protocols
Uncertainty about others’ votes
– Dropping the coalitional assumption
– Dropping the weights assumption
• Conclusion
Uncertainty about others’ votes
Even individual weighted manipulation can be hard
• So far we assumed that manipulator(s) know the others’ votes
– Unrealistic -> drop this assumption
• Theorem. Whenever constructive coalitional manipulation is hard
under certainty, individual manipulation is hard under uncertainty
– Holds even when manipulator’s vote is worthless
• i.e. we just wish to evaluate an election
– Even with very limited kinds of uncertainty
• Independence
• All votes either completely known or not at all
– Proof sketch. When manipulator’s vote is worthless, it is difficult to figure
out if a certain candidate has a chance of winning, because this requires a
constructive vote by the unknown voters
Uncertainty about others’ votes
Even individual unweighted manipulation can be hard
• Let’s drop the assumption of independence between voters
– Usually votes are highly correlated
– Identical software agents will vote identically
• Theorem. Whenever evaluating an election is hard with independent
weighted voters, it is hard with correlated unweighted voters
– Even with very limited kinds of correlation
• Perfect correlation or independence
– Proof sketch. Just replace a vote of weight k by k unweighted, perfectly
correlated voters
• So,
– because evaluation with independent weighted voters is hard for Borda,
Copeland, Maximin and STV,
– evaluation is hard for those protocols even for (correlated) unweighted voters
Overview
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•
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Introduction
Problem specification & assumptions
Deterministic voting protocols
Randomized voting protocols
Uncertainty about others’ votes
Conclusion
What do our results suggest?
• All of the protocols we discussed are computationally
more manipulation-proof than Plurality
• Among these, STV seems inherently least manipulable
• Randomizing the protocol can make manipulation hard
• Manipulation is computationally difficult because
usually there is uncertainty about others’ votes
Critique
• NP-hardness is a worst-case measure
– It may not prevent all (or even most) instances from
being manipulable
• Future research
– Average-case complexity
– Cryptographic manipulation-proofness