How “impossible” is it to design a Voting rule?

Angelina VidaliUniversity of Athens

individuals (voters)

alternatives

N = f1;:: : ;ngA = fa1; : : : ;amg

The setting

ui (a) > ui (b) Voter i prefers a to b

voter i

ui

u = (u1; : : : ;un)

a preference profile

abc

Voting rule vs Social welfare function

• A Voting rule

is a mapping

• it chooses the “winning alternative”

f : U ! Aset of all (strict) preference

profiles uset of

alternatives

• A Social welfare function is a mapping

• it chooses the “social ranking”

set of all (weak) preference

profiles u

Gibbard-Satterthwaite ‘70s Arrow ‘50s

f : U ! U¤

• a beats b in a pairwise election if the majority of voters prefers a to b

• a is a Condorcet winner if a beats any other candidate in a pairwise election

4

Condorcet rule

c

b

a

a

c

b

b

a

c

5

a

a

a

b

b

b

c

c

c

a

cb

the Condorcet paradox

voter 1 voter 2

voter 3

Some voting rules

• Plurality vote: a single winner is chosen by having more votes than any other individual representative.

• Borda vote: The Kth-ranked alternative gets a score of N-K. All scores are summed and the candidate with the highest total score wins.

• Instant Runoff Voting: If no candidate receives a majority of first preference rankings, the candidate with the fewest number of votes is eliminated and that candidate's votes redistributed to the voters' next preferences among the remaining candidates. This process is repeated until one candidate has a majority of votes.

A manipulable voting rule

• If individual i reports a false preference profile instead of his true preference the outcome of the elections is better for him.

voter i

KKE

ΠΑΣΟΚ

ΝΔ

If I vote ΚΚΕ the outcome f(u) will be either ΝΔ or ΠΑΣΟΚif I vote ΠΑΣΟΚ the outcome will be ΠΑΣΟΚ.It is better for me to report ΠΑΣΟΚ.

non-manipulable=strategyproof

ui (f (u0i ;u¡ i )) > ui (f (u))

The voter who lies determines the winner in a tie!

• Ties is the most tricky part…

voter 1

KKE

ΠΑΣΟΚ

ΝΔ

voter 2

KKE

ΠΑΣΟΚ

ΝΔ KKE

ΠΑΣΟΚ

ΝΔ KKE

ΠΑΣΟΚ

ΝΔ

voter 3 voter 4

KKE

ΠΑΣΟΚ

ΝΔ

voter 5

A dictator

The dictators top alternative is the outcome of the elections.

ui (f (u)) ¸ ui (a) for all a 2 A and u 2 U

Neutral

• The names of the candidates don’t matter.i.e. f commutes with permutations of [m]

Example:• b is the winnerif all voters a b in their preference

profiles • then a is the winner.

f (¼(u)) = ¼(f (u))

Monotonicity“If our preference for the government increases

it is reelected”

• Let f be a strategyproof voting rule, f(u)=a. • As long as, for all voters, the alternatives that

were worse than a in u, remain worse in v the allocation remains the same.

…even if one of the black elements moves below a

as long as the red elements stay below athe outcome remains a

u =

0

BBBBB@

a1 ¢¢¢

a2...

aa3a4

1

CCCCCA

0

BBBBB@

a1 ¢¢¢

a...

a3a2a4

1

CCCCCA

Pareto Optimality

• “If everybody prefers a to b then b is not elected.”

Pareto Optimality follows from Monotonicity

u =

0

BBBBBB@

a...

...... a

... a... a b

b... a b

...... b b

......

1

CCCCCCA

Gibbard (‘73)-Satterthwaite (‘75) theorem

• If the number of alternatives • then a voting rule that is strategyproof and onto

is dictatorial.

n ¸ 3

• follows from Arrow’s impossibility theorem (1951)

using the correspondence between:

Independence of Irrelevant Alternatives and strategyproofness

Independence of Irrelevant Alternatives

The social relative ranking of two alternatives

a, b depends only on their relative ranking.

If one candidate dies the choice to be made

among the set S of surviving candidates should be

independent of the the preferences of individuals

for candidates not in S.

[See: Social Choice and Individual Values, K.J. Arrow p.26]

a voting method that violates IIA

• The alternative with the highest

weighted sum of votes wins.

voter 1 voter 2

cd

voter 3

cd a

weight4321

At firsta is chosena:4+4+2=10b:7c:8d:6but if b leaves:a:10c:10d:7we get a tie between a and c

b

ba

cd

ba

see: www.scorevoting.net

The proof of G-S theorem forn=2 voters

So a wins in both u and v. voter 1 becomes a dictator for a.

c cannot win (Pareto Optimality)

Assume a wins (w.l.o.g.)

Then b wins (monotonicity) cdc

u =

0

@a bb ac c

1

A

v =

0

@a bb cc a

1

A c cannot win (Pareto Optimal.)

Suppose b wins

The proof of G-S theorem forn=2 voters (2)

• Repeat for every pair of alternatives {a,b}

• A1={x| player 1 is a dictator for x}

• A2={y| player 2 is a dictator for y}

• because: if it had two distinct elements then one of them should belong to A1 or A2.

• Finally some Ai= all the element belong to

Aj and j is the dictator.

jA n(A1 \ A2)j · 1

;

Towards a Quantitative version

Gibbard-Satherwaite theorem:

“Every non-trivial (=non dictatorial) voting

rule is strategically vulnerable.”

How often? For what fraction of profiles does such a

manipulation exist?

Impartial culture assumption

• Voters vote independently and randomly

• We draw independently and uniformly a random ranking for each voter

• possible rankings for voter i: m!

• P(each ranking)= 1/m!

Manipulation power of a voter: Mi(f)

The manipulation power Mi( f ), of voter i on thesocial choice function f, is the probability that :if voter i reports a chosen uniformly at randomthis is a profitable manipulation of f for voter i .

u0i

$i$

What is the probability I can gain something by just drawing one of the m alternatives randomly and reporting this instead of my true preference?

individual i

ε-strategyproof

P [f (u0i ;u¡ i ) > f (u)] · ²

The manipulation power Mi( f ), of voter i on the social choice function f, satisfies

M i (f ) < ²

=

δ-far from dictarorship

• The distance between two functions f,g is

• f is δ-far from dictatorship, if for any dictatorship g

¢ (f ;g) = Pu2U[f (u) 6= g(u)]

¢ (f ;g) > ±

Quantitative version of G-S theorem[Friedgut-Kalai-Nisan FOCS’08]

For every >0 if f is a voting rule for n voters is• neutral• among 3 alternatives• -far from dictatorship, then one of the voters has a non-negligible

manipulation power of . ( 1n )

²

²

Quantitative version of G-S theorem[Xia-Conitzer EC’08]

- a list of assumptions…• homogeneity • anonymity • non-imposition• a canceling out condition • there exists a stable profile that is still stable after

one given alternative is uniformly moved to different positions

+ (they argue that many known voting rules satisfy them)+ for arbitrarily many alternatives and players

Quantitative version of G-S theorem2 voters

[Dobzinski-Proccacia WINE’08]

For if a voting rule f for 2 voters is

• Pareto optimal (annoying condition!)• among at least 3 alternatives• with manipulation power < then f is -far from dictatorship.

²

² < 132m9

16m8²

no neutrality assumption here!m can also be greater than 3

Some open problems

• Quantitative version of G-S theorem for more than 2 voters (with less conditions!)

• What about the impartial culture assumption? is it plausible?

• Find quantitative versions of known mechanism design results: straptegyproof ε-strategyproof

Endnote

"Most systems are not going to work badly all of the time, all I proved is that all can work badly at times."

K. J. Arrow

…or do they work badly most of the time???