Approximately classic judgement aggregation

Nehama, Ilan

doi:10.1007/s10472-013-9358-6

Cited by 14 publications

(14 citation statements)

References 48 publications

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“…A quantitative Gibbard-Satterthwaite theorem [16,36] was proved for m = 3 by Friedgut et al [13], and the theorem was subsequently developed in [5,17,43], and the general case was resolved by Mossel and Racz [27]. In judgement aggregation, Nehama [30] and Filmus et al [6] developed quantitative characterizations of AND-homomorphism as oligarchy, whose worst-case version was due to List and Pettit [22,23]. Xia [40] proved a smoothed version of the ANR impossibility theorem on anonymity and neutrality, whose worst-case version was due to Moulin [28].…”

Section: Quantitative and Smoothed Impossibilitymentioning

confidence: 99%

A Smoothed Impossibility Theorem on Condorcet Criterion and Participation

Xia¹

2021

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In 1988, Moulin [29] proved an insightful and surprising impossibility theorem that reveals a fundamental incompatibility between two commonly-studied axioms of voting: no resolute voting rule (which outputs a single winner) satisfies Condorcet Criterion and Participation simultaneously when the number of alternatives m is at least four. In this paper, we prove an extension of this impossibility theorem using smoothed analysis: for any fixed m ≥ 4 and any voting rule r, under mild conditions, the smoothed likelihood for both Condorcet Criterion and Participation to be satisfied is at most 1 − Ω(n −3 ), where n is the number of voters that is sufficiently large. Our theorem immediately implies a quantitative version of the theorem for i.i.d. uniform distributions, known as the Impartial Culture in social choice theory.

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Section: Quantitative and Smoothed Impossibilitymentioning

confidence: 99%

A Smoothed Impossibility Theorem on Condorcet Criterion and Participation

Xia¹

2021

Preprint

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“…We can ask a similar question about the various results listed in Section 2. Nehama [40] showed that if we allow ε to depend on n, then we can choose ε = Θ(δ 3 /n). Theorem 1.1 eliminates the dependence on n in return for an exponential dependence on δ.…”

Section: Open Questionsmentioning

confidence: 99%

“…If ε = 0, it is not hard to check that f is either constant or an AND of a subset of the coordinates. Nehama [40] showed that when ε > 0, f must be O((nε) 1/3 )-close to a constant function or to an AND (in other words, Pr[f = g] = O((nε) 1/3 ), where g is constant or an AND). The main result in this paper implies, as a corollary, a similar statement, in which the distance between f, g vanishes with ε, without any dependence on n.…”

Section: Introductionmentioning

confidence: 99%

AND Testing and Robust Judgement Aggregation

Filmus¹,

Lifshitz²,

Minzer³

et al. 2019

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A function f : {0, 1}n → {0, 1} is called an approximate AND-homomorphism if choosing x, y ∈ {0, 1}n randomly, we have that f (x ∧ y) = f (x) ∧ f (y) with probability at least 1 − ε, where x ∧ y = (x 1 ∧ y 1 , . . . , x n ∧ y n ). We prove that if f : {0, 1} n → {0, 1} is an approximate AND-homomorphism, then f is δ-close to either a constant function or an AND function, where δ(ε) → 0 as ε → 0. This improves on a result of Nehama, who proved a similar statement in which δ depends on n.Our theorem implies a strong result on judgement aggregation in computational social choice. In the language of social choice, our result shows that if f is ε-close to satisfying judgement aggregation, then it is δ(ε)-close to an oligarchy (the name for the AND function in social choice theory). This improves on Nehama's result, in which δ decays polynomially with n.Our result follows from a more general one, in which we characterize approximate solutions to the eigenvalue equation Tf = λg, where T is the downwards noise operator Tf (x) = E y [f (x ∧ y)], f is [0, 1]-valued, and g is {0, 1}-valued. We identify all exact solutions to this equation, and show that any approximate solution in which Tf and λg are close is close to an exact solution.

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“…In work integrating the classical axiomatic method with ideas from computer science, Nehama (2013) showed that relaxing the cornerstones of most impossibility theorems, consistency and independence, to approximate variants of these desiderata does not allow us to significantly improve on known negative results.…”

Section: Bibliographic Notes and Further Readingmentioning

confidence: 99%