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How Hard Is It To Manipulate Voting?

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

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

Bib Info

- Small Coalitions Cannot Manipulate Voting,

Financial Cryptography 05 - Hybrid Voting Protocols And Hardness of

Manipulation, ISAAC 05, to appear

What Is Manipulation?

- In a small country far, far away there is an

election coming up

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.

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

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?

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

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.

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

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

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.

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

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.

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

Binary Cup

Do most voters prefer A to B?

R1

F

R2

C

F

R3

C

Binary Cup itself is easy to manipulate.

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

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

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.

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

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

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.

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

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.

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

Our Scheme Preround With a Twist

- Idea use votes themselves to select preround

schedule.

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

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

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

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?