Approximately Strategy-Proof Voting

Eleanor Birrell Rafael PassCornell University

uCharlie (A) = 1 uCharlie (B) = .9uCharlie (C) = .2

The Model

…σAlice = {A,B,C} σBob = {C, A, B} σCharlie = {A,C,B} σZelda = {C,B,A}

A B C

σCharlie (A) > σCharlie (B) σCharlie (B) > σCharlie (C)

Goal: Voters honestly report their preference σ

fGoal: f is strategy-proof Goal: f is strategy-proof

Bad News: Only if f is dictatorial or binary. [Gibb73, Gibb77, Satt75]

ui(j) Є [0,1]

Goal: f is strategy-proof

Bad News: Only if f is trivial. [Gibb73, Gibb77, Satt75]

Circumventing Gibbard-Satterthwaite

• Hard to manipulate?– BTT89, FKN09, IKM10

• Randomized Approximations?– CS06, Gibb77, Proc10

• Restricted preferences?– Moul80

• Relaxed Problem?

ε - Strategy Proof: By lying, no voter can improve their utility very much

δ - Approximations: f’ returns an outcome that is close to f(σ)

σAlice σBob σCharlie σZelda

A B

f

C

uCharlie (A) = 1 uCharlie (B) = .9uCharlie (C) = .2

Strategy Proof: By lying (mis-reporting their preference σi), no voter can improve their utility ui .

ε-Strategy Proof: By lying (mis-reporting their preference σi), no voter can improve their utility ui by more than ε.

ɛ-Strategy-Proof Voting

Strategy Proof:

ε-Strategy Proof:

δ - Approximations

Defining “Close”Defining Approximation

• f’ is a δ-approx. of f if the outcome of f’ is always close to that of f .

• Distance depends on both input and output:

f’(x) = f(y) s.t. Δ(x,y) < δ

σAlice σBob σCharlie σZelda

…σ'Bob σ‘Zelda

A B C0

2

4

6 5

2

4

Is ε-Strategy Proof Voting Possible?

ε = o (1/n) ε = ω (1/n)

δ = βn No Yes

Theorem 1:

Theorem 2:

ε-Strategy Proof Voting: A Construction

Deterministic Rule ( f ): Approximation ( f’ ):

d = 5

d = 2

d = 1

d = 3

d = 4

d = 1

d = 2d = 3

d = 4

d = 5

f

ε-Strategy Proof Voting: A Construction

A

B

C

{A, B, C}

{A, C, B}

{C, A, B}

{C, B, A}0 1 2 3

0

0.2

0.4

0.6

0.8

1

Distance: df( f(σ), j) Prop

ortio

nal P

roba

bilit

y: P

r [ f’

(σ) =

j ]

ξ

A C B1

ε/3

𝛿 Note: Only works

for

How Good is This?• Every voting rule has a .05-strategy-proof 650-approx.• And a . 01-strategy-proof 3,250-approx.• And a .005-strategy-proof 6,500-approx.• And a .001-strategy-proof 32,500-approx.• And a .0005-strategy-proof 65,000-approx.

Candidate Votes Obama 69,498,215 McCain 59,948,240 Nader 738,720 Baldwin 199,437 McKinney 161,680

Candidate Votes Carpenter 6,582 Fishpaw 5,865 Cole 4,500 Sweeney 1,988 Carlson 1,837

This is Asymptotically Optimal

h(σ):=

i=1 i=n…

… …j=1 j=kj=1 j=k

Return g(σ)Select player i:

Select rank j:

Prob: kε(k-1) kε(k-k) kε(k-1) kε(k-k) 1 - n∑kε(k-j)j

Punish Deviating

0-strategy proof trivial trivial

0-strategy proof prob. dist. over trivial rules. [Gibb77]

ε-strategy proof prob. dist. over trivial rules (ε = o(1/n)).

ε = o(1/n) no good ε-strategy proof approx of Plurality.

trival no good approx.Reduction: ε-SP to 0-SP

p p

1 - np

Summaryε = o (1/n) ε = ω (1/n)

δ = βn

Thank you!

• A new technique for circumventing Gibbard-Satterthwaite• Extensions• Small elections? • Uncertainty in inputs?

YesNo