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This paper explores strategy-proof voting mechanisms, focusing on their design and implications. It defines ε-strategy-proofness, a concept that guarantees that voters cannot significantly improve their utility through misreporting preferences. The study discusses the limitations of traditional voting systems under the Gibbard-Satterthwaite theorem, presenting new techniques for circumventing its constraints. The results indicate a range of approximations for practical applications, aiming for greater resilience against manipulation in electoral processes. The research also considers the implications for small elections and uncertain inputs, striving for robust democratic methodologies.
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Approximately Strategy-Proof Voting Eleanor Birrell Rafael Pass Cornell University
The Model … uCharlie(A) = 1 uCharlie(B) = .9 uCharlie(C) = .2 σAlice = {A,B,C} σBob = {C, A, B} σCharlie = {A,C,B} σZelda = {C,B,A} σCharlie (A) > σCharlie (B) σCharlie (B) >σCharlie (C) ui(j) Є[0,1] f Goal: f is strategy-proof Goal: Voters honestly report their preference σ A B C 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 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)
ɛ-Strategy-Proof Voting uCharlie(A) = 1 uCharlie(B) = .9 uCharlie(C) = .2 σAlice σBob σCharlie σZelda f Strategy Proof: By lying (mis-reporting their preference σi), no voter can improve their utility ui. Strategy Proof: ε-Strategy Proof: By lying (mis-reporting their preference σi), no voter can improve their utility uiby more than ε. ε-Strategy Proof: A B C
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 σ'Bob σCharlie σZelda σ‘Zelda 5 4 2
Is ε-Strategy Proof Voting Possible? Theorem 1: Theorem 2:
ε-Strategy Proof Voting: A Construction Deterministic Rule ( f ): Approximation ( f’ ): d = 1 d = 1 d = 2 d = 2 d = 3 d = 3 d = 4 d = 4 d = 5 d = 5
Note: Only works for ε-Strategy Proof Voting: A Construction {A, B, C} Proportional Probability: Pr [ f’(σ) = j ] {C, A, B} ξ {A, C, B} f C ε/3 A B {C, B, A} Distance: df( f(σ), j) A C B 1
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.
This is Asymptotically Optimal ε-strategy proof prob. dist. over trivial rules (ε = o(1/n)). ε = o(1/n) no good ε-strategy proof approx of Plurality. 0-strategy proof prob. dist. over trivial rules. [Gibb77] h(σ):= Reduction: ε-SP to 0-SP trival no good approx. Punish Deviating Return g(σ) i=1 i=n Select player i: … p p j=1 j=k Select rank j: … … j=1 j=k 1 - np kε(k-1) kε(k-k) kε(k-k) kε(k-1) 1 - n∑kε(k-j) Prob: j 0-strategy proof trivial trivial
Summary Yes No • A new technique for circumventing Gibbard-Satterthwaite • Extensions • Small elections? • Uncertainty in inputs? Thank you!
