How Hard Is It To Manipulate Voting

1
How Hard Is It To Manipulate Voting?

• Edith Elkind, U. of Warwick
• Helger Lipmaa, Tartu U.

2
Short Bio

• High school diploma, School ? 15, Tallinn, 1993
• M.Sc., Moscow State University, Department of
  Mathematics, 1998
• Ph.D, Princeton University, 2005
• Now Postdoctoral researcher, U.
  of Warwick, UK
• Research interests algorithmic game theory,
  voting, algorithms, complexity

3
Bib Info

• Small Coalitions Cannot Manipulate Voting,
  Financial Cryptography 05
• Hybrid Voting Protocols And Hardness of
  Manipulation, ISAAC 05, to appear

4
What Is Manipulation?

• In a small country far, far away there is an
  election coming up

5
Manipulation Example

• 99 voters, 3 candidates (Red, Blue, Green).
• 49 voters R gt B gt G.
• 48 voters B gt R gt G.
• 2 voters (Edith and Helger) G gt B gt R.
• Aggregation rule Plurality
• each voter casts a vote for one candidate.
• the candidate with the largest number of votes
  wins.
• draws are resolved by a coin toss.

6
What Will Edith and Helger Do?
R 49 votes B 48 votes
G gt B gt R
If I vote for G, R will get elected, so Id
rather vote for B
If Edith and Helger vote B gt G gt R, they can
guarantee that B is elected
7
Why Manipulation Is Bad

• Aggregation rules are designed with certain
  social welfare criteria in mind.
• Misrepresentation of preferences results in a
  suboptimal choice w.r.t. these criteria.
• Also, election results do not reflect true
  distribution of preferences in the society
• maybe, in fact, in 2000 20 of the U.S.
  population prefered Nader to Gore to Bush?

8
What If We Change Aggregation Rule?

• Single Transferable Vote
• This time, Edith and Helger are better off voting
  honestly, but this will not always be the case
• Other popular voting schemes (Borda, Copeland)
  suffer from the same problem

1st round
R gt B gt G
49 votes
B gt R gt G
48 votes
G gt B gt R
2 votes
9
Formal Setup

• n voters
• m candidates c1, , cm
• Preference of a voter i
  a permutation pi of c1, , cm
  (best to worst).
• Aggregation rule S
  p1, , pn ? cj.

10
Voting Schemes Examples

• Borda a candidate gets
• m points for each voter who ranks him 1st,
• m-1 point for each voter who ranks him 2nd, etc.
• Copeland
• candidate that wins the largest of pairwise
  elections
• Maximin
• cs score against d of voters that prefer c to
  d
• cs of points min score in any pairwise
  election.
• many, many others

11
Voting Schemes Properties

• Pareto-optimality if everyone prefers a to b, b
  does not win
• Condorcet-consistency if there is a candidate
  that wins every pairwise election, this candidate
  wins
• Majority if there is a candidate that is ranked
  first by a majority of voters, this candidate
  wins
• Monotonicity it is impossible to cause a winning
  candidate to lose by moving it up in ones vote

Arrows theorem there is no perfect scheme
12
Manipulation Definition

• A voter i can manipulate
  a voting scheme S if there is
• a preference vector
• p (p1,,pi, ,pn)
• a permutation pi s.t.
• S(p1,,pi, ,pn) gti S(p).

Theorem (Gibbard-Satterthwaite, 1971) every
non-dictatorial aggregation rule with 3
candidates is manipulable.
13
How Do We Get Around The Impossibility Result?

• We cannot make manipulation impossible
• But we can try to make it hard!
• How do you manipulate Plurality?
• vote for your favorite candidate among those
  tied for the top position.
• How do you manipulate Borda?
• rank your favorite feasible candidate highest,
  move his competitors to the bottom of your vote.
• How do you manipulate STV?
• try all m! possible ballots

14
What Is Known?

• 2nd order Copeland is NP-hard to manipulate
  (Bartholdi, Tovey, Trick 1989)
• STV is NP-hard to manipulate (Bartholdi, Orlin
  1991)
• these rules may not reflect the welfare goals
  (why is there so many voting rules out there?)
• Want a universal method to turn any voting
  protocol into a hard-to-manipulate one.

15
Adding a Preround (Conitzer-Sandholm03)
Carl
Diana
Ernest
Frank

• Retains some of the flavor of the original
  protocol.
• Is NP-hard to manipulate for many base
  protocols.
• Still, the outcome may be very different from
  the
• original protocol

16
Binary Cup
Do most voters prefer A to B?
R1
F
R2
C
F
R3
C
Binary Cup itself is easy to manipulate.
17
Our Work Hybrid Protocols

• Protocols with a preround can be viewed as
  hybrids of BC and other protocols
• how about other hybrids?
• Hyb(Xk, Y) execute k steps of X, then apply Y to
  the remaining candidates.
• Hyb(Pluralityk, Borda) eliminate k candidates
  with the lowest Plurality scores, then compute
  Borda scores w.r.t. survivors.
• Observation
• Hyb(Plurality1, , Pluralitym) STV.

a b c d e
3 2 1
18
New Results

• New protocols that are hard to manipulate
• Hyb(Xk, STV), Hyb(STVk, Y)
• (for any reasonable X, Y)
• Hyb(Bordak, Plurality), Hyb(Maximink, Plurality)
• Generally, Hyb(Xk, X) ? X (and may be much harder
  to manipulate)
• manipulating Hyb(Bordak, Borda) is NP-hard
• Extensions to utlity-based voting (voters rate
  candidates rather that rank them)
• manipulating Hyb(HighScorek, HighScore) is NP-hard

19
Proof Idea

• Set Cover G g1, , gn, S s1, , sm, each
  si is a subset of G. Is there a cover of size K,
  i.e, C si1, , siK s.t. each gi is contained
  in some sj?
• Set Cover is NP-hard.
• Can we reduce Set Cover
• to protocol manipulation?

Suppose someone can manipulate Hyb (Xk, Y). We
want to show that he can also find a set cover.
20
Proof Idea (continued)
manipulators prefered candidate

• Candidates g1, , gn, s1, , sm, p, d1, , dt
• Voters
• ( p, ) A ballots
• for each gi
• (gi, , ) A - 1 ballot
• (sj1, , sjr, gi, ) s.t. gi is in sj1, , sjr
  B ballots
• some ballots of the form (dj, , )
• k m K
• s1, , sm are tied for the last place under Xk
• manipulators vote decides which K candidates
  survive the preround

21
Proof Idea (continued)

• p wins iff none of gi gains votes after Xk
• gi gains votes iff all sj s.t. gi is in sj are
  eliminated
• no gi gains votes iff surviving sj cover all gi
• manipulation is successful iff manipulator can
  find a set cover of size K

22
Worst-Case vs. Average-Case Hardness

• Is 3-Coloring hard on a random graph?
• likely to contain just say
  no
• poly-time, works for almost all graphs
• We embed a problem that is sometimes hard into
  some class of preference profiles
• Want a reduction that
• works for (almost) all preference profiles
• reduces from a problem that is (almost) always
  hard
• We show how to do (2) using one-way functions
  (1) is an open problem.

23
One-Way Functions

• f is a one-way function (OWF) if it is
• easy to compute for any x, f(x) is polynomial
  time computable.
• hard to invert
• x is drawn at random from 0, 1n
• we are given y f(x)
• have to guess x, or any z s.t. y f(z)
• average-case hardness any poly-time algorithm
  has negligible chance of success (over the choice
  of x).

x
OWF
24
OWF Properties

• It is not known if OWF exist (implies P ? NP).
• Multiplication of large primes is conjectured to
  be a OWF (factoring is believed to be hard)
• OWF are widely used in cryptography
• adversarys task must be hard almost always,
  not just sometimes.
• Can we make manipulation as hard as inverting
  OWF?
• sort of we embed inverting OWF into some
  preference profiles.

25
Recall Preround (CS03)
Carl
Diana
Ernest
Frank

• Retains some of the flavor of the original
  protocol.
• Is NP-hard to manipulate for many base
  protocols.
• Still, the outcome may be very different from
  the
• original protocol

26
Our Scheme Preround With a Twist

• Idea use votes themselves to select preround
  schedule.

27
Key Properties

• No trusted source of randomness is needed.
• Manipulating this protocol is as hard as
  inverting the underlying one-way function
• even for a large coalition of conspiring would-be
  manipulators
• Proof proceeds by constructing a vector of honest
  voters preferences s.t. there is a unique
  preround schedule that makes manipulators
  protégé a winner
• still a worst-case construction

28
Average-Case Hardness?

• Can we make manipulation hard with high
  probability over honest voters preferences?
• Do we want to make assumptions about the
  distribution of preference profiles?
• In real life, not all preference profiles are
  equally likely

29
Hyb(BC, Plurality) Easy for Any Preround Schedule

• Candidates a, b, c1, , c2t1, p
• k1 voters p gt a gt b gt C, k n/3
• k voters a gt b gt p gt C
• n-2k-2 voters C gt p gt a gt b
• Manipulator C gt a gt b gt p
• Under any preround schedule
• p survives
• a cj survives
• either a or b survives
• In the final round, p competes against a or b
• manipulator is better off voting a gt b gt p gt C

30
Open Problems

• Average-case hardness seems hard to achieve using
  a preround. Other approaches?
• What is the maximum fraction of manipulators we
  can tolerate?
• for most base protocols, we can prove security
  against 1/6 of all voters. Is it optimal?
• Our results are for specific protocols. More
  general proofs?