Transcript When Are Elections with Few Candidates Hard to Manipulate

When Are Elections with Few Candidates
Hard to Manipulate
V. Conitzer, T. Sandholm, and J. Lang
Subhash Arja
CS 286r
October 29, 2008
Motivation
• Avoid coalitional manipulation by a group of
weighted voters in regular voting protocols.
• Study constructive manipulation and
destructive manipulation
• Find the exact number of candidates that
makes manipulation hard
• Expand this result to manipulation by an
individual in un-weighted and uncertain
setting.
Outline
• Prior Work
• Background information
– Voting protocols
– Conditions
• Proof of easiness for constant number of candidates
• Proof of NP-hard through reduction of Partition
problem
• Extend to case of destructive manipulation and
weighted voters
• Conclusion and future work
Applications
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Political elections
Surveys
Shareholder meetings
Allocation of goods and resources
Any application where the goal to achieve
optimal preference aggregation.
Prior Work
• Number of candidates and voters was
unbounded
• Method 1: Voters’ preferences are restricted
– Problem: protocol designer cannot guarantee that the
agents’ preferences fall within restrictions
• Method 2: randomization approach
– Problem: too much noise, could introduce
manipulation possibilities
• Method 3: make it hard to manipulate so agents
will be unlikely to succeed.
Voting Protocols
• Positional scoring protocols
– α = (α1, α2,…, αm) where α1 ≥ α2 ≥ … ≥ αm
– For each voter, a candidate receives α1 if it is ranked first by the
voter, etc. The score of a candidate is the total number of points
he receives.
– Examples: Borda, plurality, veto
• Maximin
– For two distinct candidates, i and j, N(i,j) = number of voters
who prefer i to j. The score of i is s(i) = minj≠i N(i,j)
• Copeland
– For two distinct candidates, i and j, let C(i,j) = 1 if N(i,j) > N(j,i),
C(i,j) = 0 if N(i,j) = N(j,i), and C(i,j) = -1 if N(i,j) < N(j,i)
– Score of candidate: s(i) = ∑j≠i C(i,j)
Voting Protocols
• Single Transferable Vote (STV)
– Series of m-1 rounds.
– In each round, candidate with the lowest number of
voters ranking it first among the remaining candidates
is eliminated.
– Votes for that candidate transfer to the next
remaining candidate.
• Plurality with Run-off
– All candidates except two with the highest plurality
scores.
– Scores are transferred to the winners and second
round determines the winner.
Voting Protocols
• Cup
– Balanced binary tree with each leaf representing
each candidate.
– Each non-leaf node is assigned the winner of its
children.
– In randomized version, assignment of candidates
to leaves is chosen at random after voters have
voted.
Conditions
• Complete information – hardness results
directly imply hardness for the incomplete
information setting
• Coalitional Manipulation – individuals have a
small effect on the outcome
• Weighted voters – the case of un-weighted
voters is easy.
• Constructive and Destructive
Easiness Results
• Plurality protocol: constructive manipulation can be
solved in polynomial time (any number of candidates)
– Proof: manipulators check if p will win if they vote for p.
Otherwise, they cannot make p win.
• Cup protocol: constructive manipulation can be solved
in polynomial time for any number of candidates
– Proof: key claim is that a candidate can win a sub-election
iff it can win one of its children and beat the potential
winners of the sibling child.
– Coalition ranks all candidates in p’s half above those in h’s
half. h = potential winner of other half.
– When p and h win their halves, p will defeat h in the final.
Easiness Results
• Copeland protocol: easy if there are 3 candidates
and all manipulators vote identically
– Proof: involves four cases
1. Weights of manipulators’ votes are greater than weights
for nonmanipulators votes for both candidates other than
p.
–
Any configuration of votes where p ranks first wins the
election.
2. Weights of manipulators’ votes are equal to that of
nonmanipulators for one candidate and greater for the
other candidate (K > Ds(a,p) and K = Ds(b,p) )
–
It is proven that all manipulators must vote in the order (p,a,b)
Easiness Results
3.
4.
Same case as above but reverse “a” and “b”
Weights of manipulators’ votes are less than weights of
nonmanipulators’ votes. (K < Ds(a,p) and K < Ds(b,p))
–
p cannot be guaranteed to win, therefore there is no successful
manipulation
• Maximin protocol: easy with 3 candidates and all
manipulators vote identically.
– Proof: all manipulators set p as rank 1 and two cases involve the
ranking of other two candidates.
– When all of the coalition votes same way for other two
candidates with p as rank 1, p will win.
• Randomized cup protocol: easy if there are six candidates
and all manipulators vote identically.
– Proof: divide candidates into two sets – B = candidates that
defeat p and G = candidates that p defeats
Easiness Results
– Have to make sure that p doesn’t face an
opponent from set B.
– Rest of the proof goes over how the manipulators
should choose the order of candidates within B
and within G.
Hardness Results
• Basic idea: proved P results for protocols with l
candidates and now prove each is NP-complete for l+1
candidates.
• Partition problem: given a set of S of integers,
determine two disjoint subsets S1 and S2 where sum(S1)
= sum(S2)
– NP-complete problem.
– Use reduction from the Partition problem to show that a
protocol is NP-complete.
– All the proofs use this idea with variations.
• S = non-manipulators’ votes, T = weights of
manipulators’ votes, and K = total weight in T.
Hardness Results
• Any positional scoring rule other than the plurality protocol
is NP-complete for 3 candidates
- Proof: one half of the partition in T are (p,a,b) and the other half
is (p,b,a) => this makes p the winner
- This only happens in the case where the total weight of the
voters voting (p,a,b) equals the total weight of the manipulators
voting (p,b,a).
• Copeland Protocol: manipulation is NP-complete for 4
candidates.
– Proof: p wins if two other candidates tie, and the third loses.
– This only happens if the combined weight of the manipulators’
votes maintain this tie => requires a partition.
Hardness Results
• Maximin protocol: manipulation is NPcomplete for 4 candidates
• STV protocol: manipulation is NP-complete for
3 candidates
• Plurality with Runoff: NP-complete for 3
candidates
Destructive Manipulation
• Destructive manipulation can never be harder
than constructive manipulation.
• Can be done in polynomial time for veto,
Borda, Copeland, and maximin protocols
– Proof: each colluder places candidate h at the
bottom and order the other candidates in any
order.
– A total of m-1 winner determinations are done to
and each winner determination is in P.
Destructive Manipulation
• STV Protocol: with 3 candidates, manipulation
is NP-complete
– Proof: reduce partition to case where three
candidates are a, b, and h.
– Show that in T for every ki there is a vote of weight
2ki
• Plurality with runoff: with 3 candidates,
manipulation is NP-complete
– Proof: coincides with the STV protocol for 3
candidates.
Uncertainty about others’ votes
• Only the distribution over the other voters is
known
– Restricted probability distributions.
• Overall conclusions:
– With weighted voters, whenever coalitional
manipulation is hard, evaluating a candidate’s
probability to win is hard when there is uncertainty.
• Individual manipulation is also hard
– An individual cannot find the strategically optimal
vote for him to make.
Uncertainty about others’ votes
• Approval protocol: each candidate either
approves or disapproves of a candidate
– Easy for constructive, non-weighted case.
– In weighted case, manipulation is NP-hard
Un-weighted Voters
• Special case of weighted voting where each
vote is assigned the same weight.
• General conclusion:
– For every protocol that is hard in the weighted
case, it is also hard in the un-weighted case.
Conclusions
Constructive CW-Manipulation
# of candidates
2
3
4,5,6
≥7
Borda
P
NP-complete
NP-complete
NP-complete
Veto
P
NP-complete
NP-complete
NP-complete
STV
P
NP-complete
NP-complete
NP-complete
Plurality w/ runoff
P
NP-complete
NP-complete
NP-complete
Copeland
P
P
NP-complete
NP-complete
Maximin
P
P
NP-complete
NP-complete
Randomized Cup
P
P
P
NP-complete
Regular cup
P
P
P
P
plurality
P
P
P
P
Adopted from V. Conitzer, et al.
Conclusions
Destructive CW-Manipulation
Number of Candidates
2
3
STV
P
NP-complete
Plurality with runoff
P
NP-complete
Borda
P
P
Veto
P
P
Copeland
P
P
Maximin
P
P
Regular cup
P
P
Plurality
P
P
Adopted from V. Conitzer, et al.
Future Work
• Ideal case: make all or most instances hard to manipulate.
• Prove hardness of protocols that are more restricted (e.g.
auctions)
• “Can manipulation be made hard for most instances?”
• It is too much to ask for every instance hard to manipulate.
• Combine some amount of randomization with
computational complexity.
• Pivotal voters will not benefit or lose from the chosen
candidate.
– Pivotal voters could be “banished”.
– Achieve a middle ground between making voting truthfully a
dominant strategy and altering the definition of the voting rule.
Future Work
• Make the voting rule itself hard to execute.
– Simulations become complex and manipulation is
thwarted.
– Disadvantage: determining the election winner
also becomes difficult.