Complexity of manipulating elections with few candidates

Vincent Conitzer and Tuomas Sandholm

Carnegie Mellon UniversityComputer Science Department

Outline• Introduction

– Voting – Manipulation– Computational complexity as a barrier to manipulation

• 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– 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

“A > B > C” “A > C > B” “B > A > C”

Voting protocol chooses winner (probably A here)

Voter 1 Voter 2 Voter 3

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:– 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

• 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…

Unweightedvoters

Weightedvoters

Individualmanipulation

Coalitionalmanipulation

Can behard easy

easy

easy

Constant #candidatesUnbounded #candidates

Can behard

Can behard

Can behard

Potentiallyhard

Unweightedvoters

Weightedvoters

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

• 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

a b

b

c

b

d

c

Overview• 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’ votesEven 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• 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