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Approximately Strategy-Proof Voting

Approximately Strategy-Proof Voting. Eleanor Birrell Rafael Pass Cornell University. The Model. …. u Charlie (A) = 1 u Charlie (B) = .9 u Charlie (C) = .2 . σ Alice = {A,B,C}. σ Bob = {C, A, B}. σ Charlie = {A,C,B}. σ Zelda = {C,B,A}.

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Approximately Strategy-Proof Voting

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  1. Approximately Strategy-Proof Voting Eleanor Birrell Rafael Pass Cornell University

  2. 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]

  3. 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)

  4. 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

  5. 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

  6. Is ε-Strategy Proof Voting Possible? Theorem 1: Theorem 2:

  7. 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

  8. 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

  9. 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.

  10. 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

  11. Summary Yes No • A new technique for circumventing Gibbard-Satterthwaite • Extensions • Small elections? • Uncertainty in inputs? Thank you!

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