Avoiding manipulation in elections through computational complexity

1
Avoiding manipulation in elections through
computational complexity
Vincent Conitzer Computer Science
Department Carnegie Mellon University Guest
lecture 15-892, 2005
2
Introduction
Bartholdi, Tovey, Trick 1989 Bartholdi, Orlin
1991
3
Voting

• In multiagent systems, agents may have
  conflicting preferences
• Based on reported 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

4
Voting
Voter 1
Voter 2
Voter 3
B gt A gt C
A gt C gt B

• A gt B gt C

VOTING RULE
Winner (probably A here)
5
Manipulation in voting

• A voter is said to manipulate when it does not
  rank the candidates according to its true
  preferences
• Example not ranking your most preferred
  candidate first because that candidate has no
  chance anyway
• Why is manipulation bad?
• Protocol is designed to maximize (some measure
  of) social welfare with respect to the reported
  preferences
• Manipulation will cause a suboptimal outcome to
  be chosen
• Also if the protocol actually relies on
  manipulation to choose the right outcome
• then there exists another nonmanipulable
  protocol that will lead to the same outcome
  (Revelation Principle)

6
Manipulation in voting

• Kerry

Bush
Bush
Kerry
Kerry
Nader gt Kerry gt Bush

• Voting truthfully (for Nader) might let Bush
  win, certainly will not get Nader to win
• So, better rank Kerry first

MANIPULATION!!!
7
Some well-known protocols (rules)

• Plurality candidate with the most votes wins
• Borda candidate gets m-1 points for each vote,
  but also m-2 points for each second place in a
  vote, m-3 for each third place,
• Maximin
• From the complete rankings, for each pair of
  candidates, we can deduce how each voter would
  have voted with only these two candidates
• This defines (m choose 2) pairwise elections
• In Maximin, winner is the candidate with the best
  score in her worst pairwise
• Single Transferable Vote (STV)
• Each round, candidate with fewest votes drops out
• When your candidate drops out, your vote
  transfers to your next most preferred (remaining)
  candidate
• Continue until one candidate remains
• Note now our voter can safely vote for Nader,
  then let the vote transfer to Kerry
• Still manipulable in other cases

Seminal result (Gibbard-Satterthwaite) all
nondictatorial voting protocols with gt2
candidates are manipulable!
8
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 (strategic) 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!
9
Computational manipulation problem

• The simplest version of the manipulation problem
  (defined relative to a protocol)
• CONSTRUCTIVE-MANIPULATION. We are given the
  (unweighted) votes of the other candidates, and a
  candidate c. We are asked if we can cast our
  (single) vote to make c win.
• E.g. for the Borda protocol
• Voter 1 votes A gt B gt C Voter 2, B gt A gt C
  Voter 3, C gt A gt B
• Borda scores are now A 4, B 3, C 2
• Can we make B win?
• Answer YES. Vote B gt C gt A (Borda scores A 4,
  B 5, C 3)
• Utility-theoretically, the special case where the
  manipulator has utility 1 for c and 0 for
  everyone else

10
Prior research

• Theorem. CONSTRUCTIVE-MANIPULATION is NP-complete
  for the second-order Copeland rule. Bartholdi,
  Tovey, Trick 1989
• Theorem. CONSTRUCTIVE-MANIPULATION is NP-complete
  for the STV rule. Bartholdi, Orlin 1991
• All the other rules are easy to manipulate (in P)

11
Universal voting protocol tweaks to make
manipulation hard
Conitzer, Sandholm IJCAI-2003
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Tweaks for protocols

• Hardness of manipulation is one factor in
  choosing a voting protocol, but
• Many existing protocols have other nice
  properties
• Each attempts to maximize a certain notion of
  welfare
• It would be nice to be able to tweak protocols
• Change the protocol slightly so that
• Hardness of manipulation is increased
  (significantly)
• (Most of) the original protocols properties
  still hold
• It would also be nice to have a single, universal
  tweak for all (or many) protocols
• A preround turns out to be such a tweak!
• And it introduces hardness far beyond previous
  results

13
Adding a preround to the protocol

• A preround proceeds as follows
• Pair the candidates
• Each candidate faces its opponent in a pairwise
  election
• The winners proceed to the original protocol

Original protocol
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Preround example (with Borda)
STEP 1 A. Collect votes and B. Match
candidates (no order required)

• Voter 1 AgtBgtCgtDgtEgtF
• Voter 2 DgtEgtFgtAgtBgtC
• Voter 3 FgtDgtBgtEgtCgtA

Match A with B Match C with F Match D with E
A vs B A ranked higher by 1,2 C vs F F ranked
higher by 2,3 D vs E D ranked higher by all
STEP 2 Determine winners of preround
Voter 1 AgtDgtF Voter 2 DgtFgtA Voter 3 FgtDgtA
STEP 3 Infer votes on remaining candidates
STEP 4 Execute original protocol (Borda)
A gets 2 points F gets 3 points D gets 4 points
and wins!
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Matching first, or vote collection first?

• Match, then collect

A vs C, B vs D.
A vs C, B vs D.
D gt C gt B gt A

• Collect, then match (randomly)

A vs C, B vs D.
A gt C gt D gt B
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Could also interleave

• Elicitor alternates between
• (Randomly) announcing part of the matching
• Eliciting part of each voters vote

A vs F
B vs E
C gt D
A gt E


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Main result how hard is manipulation?

• Manipulation hardness differs depending on the
  order/interleaving of preround matching and vote
  collection
• Theorem. NP-hard if preround matching is done
  first
• Theorem. P-hard if vote collection is done first
• Theorem. PSPACE-hard if the two are interleaved
  (for a complicated interleaving protocol)
• In each case, the tweak introduces the hardness
  for any protocol satisfying certain sufficient
  conditions
• All of Plurality, Borda, Maximin, STV satisfy the
  conditions in all cases, so they are hard to
  manipulate with the preround

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NP-hard with preround matching first

• Theorem. (Sufficient condition) Suppose that in a
  protocol we can construct a set of votes for the
  other voters s.t.
• There is a set of candidates K that could
  possibly defeat our preferred candidate p in the
  final round (original protocol)
• However, there is another set of candidates L of
  nemeses of K
• For each candidate c_k in K, there are some
  candidates in L such that if even one of these
  candidates continues, c_k will not defeat p in
  the final round
• The candidates in L are matched in pairs (c_l
  and c_-l), and each pair is tied in their
  pairwise election
• Then manipulation is NP-hard!
• Proof idea Suppose each pair of candidates in L
  faces each other in the preround. We have to
  choose between nemeses such that each candidate
  in K gets at least one nemesis ? SATISFIABILITY.

19
Still not done

• Now we still have to show that protocols meet
  this complicated condition
• Simple example Plurality
• Assume that for each c_k in K, each of its
  nemeses would steal some of its votes in
  Plurality
• That is, c_k is ranked below (only) its nemeses
  in these votes
• Then, any of the nemeses going on to the final
  round would push c_k below p

20
Conclusions on tweaks

• We have shown that it is possible to tweak
  protocols
• A tweak preserves many of the protocols
  properties
• in order to drastically increase computational
  hardness of manipulation
• If manipulation is computationally hard, it is
  less likely to occur
• The tweak we introduced is a preround
• One round of pairwise elimination
• This makes manipulation NP-hard, P-hard, or even
  PSPACE-hard depending on whether scheduling the
  preround is done before, after, or during vote
  collection
• First results where manipulation is more than
  NP-hard

21
Hardness of manipulationwith few candidates
Conitzer, Sandholm AAAI-2002 Conitzer, Lang,
Sandholm TARK-2003
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What if there are few candidates?

• The previous results rely on the number of
  candidates (m) being unbounded
• We designed a recursive algorithm for
  individually manipulating STV with O(1.62m) calls
  (and usually much fewer)
• E.g. 20 candidates 1.6220 15500
• Sometimes the candidate space is much larger
• Voting over allocations of goods/tasks
  California
• But what if it is not?
• A typical election for a representative will only
  have a few

23
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 voters
Unweighted voters
Weighted voters
Weighted voters
Individual manipulation
Can be hard
Can be hard
easy
easy
Coalitional manipulation
Can be hard
Can be hard
Potentially hard
easy
24
Why study weighted coalitional manipulation?

• In large elections, usually an effective
  individual manipulation does not exist
• Many real world elections have weights
• E.g. electoral college
• Weights more likely with heterogeneous software
  agents
• Weighted coalitional manipulation may be more
  realistic than assuming unbounded number of
  candidates
• Theorem Whenever weighted coalitional
  manipulation is hard under certainty, individual
  weighted manipulation is hard under uncertainty
• Theorem Whenever evaluating an election is hard
  with independent weighted voters, it is hard with
  correlated unweighted voters

25
Constructive manipulation now becomes

• We are given the weighted votes of the others
  (with the weights)
• And we are given the weights of members of our
  coalition
• Can we make our preferred candidate c win?
• E.g. another Borda example
• Voter 1 (weight 4) AgtBgtC, voter 2 (weight 7)
  BgtAgtC
• Manipulators one with weight 4, one with weight
  9
• Can we make C win?
• Yes! Solution weight 4 voter votes CgtBgtA, other
  CgtAgtB
• Borda scores A 24, B 22, C 26

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Destructive manipulation

• Exactly the same, except
• Instead of a preferred candidate
• We now have a hated candidate
• Our goal is to make sure that the hated candidate
  does not win (whoever else wins)
• Utility-theoretically hated candidate gives
  utility 0, everyone else utility 1

27
Some other well-known rules

• Veto candidate with the fewest vetoes wins
• Copeland the candidate with the most pairwise
  election victories wins
• Cup candidates are arranged in a tennis-style
  tournament and defeat each other based on
  pairwise elections
• Plurality with runoff the two candidates with
  the highest plurality score advance to the
  runoff winner is winner of pairwise election

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A simple example of hardness

• We want given the other voters votes
• it is NP-hard to find votes for the
  manipulators to achieve their objective
• Simple example veto rule, constructive
  manipulation, 3 candidates
• Suppose, from the given votes, p has received
  2K-1 more vetoes than a, and 2K-1 more than b
• The manipulators combined weight is 4K (every
  individual has weight a multiple of 2)
• The only way for p to win is if the manipulators
  veto a with 2K weight, and b with 2K weight
• But this is doing PARTITION gt NP-complete!

29
What does it mean for a protocol to be easy to
manipulate?

• Given the other voters votes
• there is a polynomial-time algorithm to find
  votes for the manipulators to achieve their
  objective
• If the protocol is easy to run, it is easy to
  check whether a vector of votes for the
  manipulators is successful
• Lemma Suppose the protocol satisfies (for some
  number of candidates)
• If there is a successful (constructive,
  destructive) manipulation
• Then there is a successful (constructive,
  destructive) manipulation where all manipulators
  vote identically.
• Then the protocol is easy to manipulate
• Simply check all possible orderings of the
  candidates (constant)

30
Example Maximin with 3 candidates is easy to
manipulate constructively

• Recall candidates Maximin score worst score
  in any pairwise election
• 3 candidates p, a, b. Manipulators want p to win
• Suppose there exists a vote vector for the
  manipulators that makes p win
• WLOG can assume that all manipulators rank p
  first
• So, they either vote p gt a gt b or p gt b gt a
• Case I as worst pairwise is against b, bs
  worst against a
• One of them would have a maximin score of at
  least half the vote weight, and win (or be tied
  for first) gt cannot happen
• Case II one of a and bs worst pairwise is
  against p
• Say it is a then can have all the manipulators
  vote p gt a gt b
• Will not affect p or as score, can only decrease
  bs score

31
Why do we care about the exact number of
candidates required for hardness?

• If your election has 3 candidates, and the
  protocol becomes hard to manipulate only at 4
  candidates, that hardness is of little use to
  you!
• If you do not know beforehand how many candidates
  the election will have, the lower the hardness
  occurs, the better the chances you will get the
  hardness
• The minimal number of candidates for hardness can
  be used to compare and quantify the relative
  hardness of manipulation across protocols
• Hardness of manipulation is only one factor in
  deciding on a protocol, so it is good to know how
  much harder one protocol is to manipulate than
  another

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Results for constructive manipulation
33
Results for destructive manipulation
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Worst-case hardness

• All of these are worst-case measures
• It may not prevent all (or even most) instances
  from being manipulable
• It would be nice if we had some sort of average
  case hardness
• In the works impossibility result
• Impossible to make a constant fraction of
  instances hard to manipulate in class of
  protocols

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Thank you for your attention!
36
Uncertainty about others votes

• So far we assumed that manipulator(s) know the
  others votes
• Unrealistic -gt drop this assumption
• Theorem. Whenever constructive weighted
  coalitional manipulation is hard under certainty,
  individual weighted manipulation is hard under
  uncertainty
• Holds even when manipulators 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 manipulators 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

37
Uncertainty about others votes

• Lets drop the assumption of independence between
  voters
• Usually votes are highly correlated
• Also, 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, Veto, STV, Plurality
  with runoff, Copeland, Maximin and Randomized
  Cup,
• evaluation is hard for those rules even for
  (correlated) unweighted voters

38
Randomization can be used to make manipulation
hard

• Consider the Cup rule
• 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
  (manipulation is easy at 6 !)

39
Hardness of manipulation andthe revelation
principle
Conitzer, Sandholm LOFT-2004
40
Computational criticisms of the revelation
principle

• The revelation principle says nothing about the
  computational implications of using direct,
  truthful mechanisms
• Does restricting oneself to such mechanisms lead
  to computational hassles? YES
• If the participating agents have computational
  limits, does restricting oneself to such
  mechanisms lead to a loss in the objective (e.g.
  social welfare)? YES
• Even if the center is computationally unbounded!

41
Criticizing one-step mechanisms

• Theorem. There are settings where
• Executing the optimal single-step mechanism
  requires an exponential amount of communication
  and computation
• There exists an entirely equivalent two-step
  mechanism that only requires a linear amount of
  communication and computation
• Holds both for dominant strategies and Bayes-Nash
  implementation

42
Criticizing truthful mechanisms

• Theorem. There are settings where
• Executing the optimal truthful (in terms of
  social welfare) mechanism is NP-complete
• There exists an insincere mechanism, where
• The center only carries out polynomial
  computation
• Finding a beneficial insincere revelation is
  NP-complete for the agents
• If the agents manage to find the beneficial
  insincere revelation, the insincere mechanism is
  just as good as the optimal truthful one
• Otherwise, the insincere mechanism is strictly
  better (in terms of s.w.)
• Holds both for dominant strategies and Bayes-Nash
  implementation
• Theorem. The same holds under an oracle model
  (replace NP-complete with requiring an
  exponential number of queries)
• Holds both for dominant strategies and Bayes-Nash
  implementation

43
Is there a systematic approach?

• The previous result is for a very specific
  setting
• How do we take such computational issues into
  account in general in mechanism design?
• What is the correct tradeoff?
• Cautious make sure that computationally
  unbounded agents would not make the mechanism
  worse than the best truthful mechanism (like
  previous result)
• Aggressive take a risk and assume agents are
  probably somewhat bounded
• What kind of mechanism design approaches can
  help?
• Classical attempt to theoretically characterize
  mechanisms that take maximal advantage of
  computational issues
• Automated compute the mechanism on the fly for
  the setting at hand