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How Hard Is It To Manipulate Voting

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Title: 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?
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