Approximately Strategy-Proof Voting Eleanor BirrellRafael Pass Cornell University

u Charlie (A) = 1 u Charlie (B) =.9 u Charlie (C) =.2 The Model … σ Alice = {A,B,C}σ Bob = {C, A, B}σ Charlie = {A,C,B}σ Zelda = {C,B,A} ABC σ Charlie (A) > σ Charlie (B) σ Charlie (B) > σ Charlie (C) Goal: Voters honestly report their preference σ f Goal: f is strategy-proof Bad News: Only if f is dictatorial or binary. [Gibb73, Gibb77, Satt75] Goal: f is strategy-proof Bad News: Only if f is dictatorial or binary. [Gibb73, Gibb77, Satt75] u i (j) Є [0,1] Goal: f is strategy-proof Bad News: Only if f is trivial. [Gibb73, Gibb77, Satt75] 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) 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)

σ Alice σ Bob σ Charlie σ Zelda AB f C u Charlie (A) = 1 u Charlie (B) =.9 u Charlie (C) =.2 Strategy Proof: By lying (mis-reporting their preference σ i ), no voter can improve their utility u i. ε-Strategy Proof: By lying (mis-reporting their preference σ i ), no voter can improve their utility u i 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 σ Alice σ Bob σ Charlie σ Zelda … σ' Bob σ‘ Zelda 5 2 4

Is ε-Strategy Proof Voting Possible? ε = o (1/n)ε = ω (1/n) δ = βnNoYes 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 = 2 d = 3 d = 4 d = 5

f ε-Strategy Proof Voting: A Construction AB C {A, B, C} {A, C, B} {C, A, B} {C, B, A} Distance: d f ( f(σ), j) Proportional Probability: Pr [ f’ ( σ ) = j ] ξ ACB 1 ε/3 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. CandidateVotes Obama69,498,215 McCain59,948,240 Nader738,720 Baldwin199,437 McKinney161,680 CandidateVotes Carpenter6,582 Fishpaw5,865 Cole4,500 Sweeney1,988 Carlson1,837

This is Asymptotically Optimal h(σ):= i=1 i=n … …… j=1 j=k j=1j=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

Summary ε = 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