Noise Sensitivity and Chaos in Social Choice Theory

Kalai, Gil

doi:10.1007/978-3-642-13580-4_8

Cited by 12 publications

(17 citation statements)

References 16 publications

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“…In particular, as the number of faces grows, the dice can behave transitively, i.e., such that a triple of random dice is transitive with high probability. At the other end of the spectrum, there can be behavior that we call, borrowing the term from Kalai's paper on social choice [17], chaotic: in that regime, three dice are intransitive with probability 1 approaching 1/4. Some (mostly) experimental results were presented by Conrey, Gabbard, Grant, Liu and Morrison [7].…”

Section: Intransitive Dice: Transitivity Of Non-uniform Dicementioning

confidence: 99%

“…The above-mentioned work by Kalai [17] calls the situation when Y is a random tournament social chaos. He considers impartial culture model (without conditioning) and an arbitrary monotone odd function f : {−1, 1} n → {−1, 1} for pairwise elections (the setting we considered so far corresponds to f = Maj n ).…”

Section: Condorcet Paradox: Social Chaos For Close Majority Electionsmentioning

confidence: 99%

“…Under these assumptions, he proves that social chaos is equivalent to the asymptotic probability of Condorcet winner for three alternatives being equal to 3/4. [17] contains another equivalent condition for social chaos, stated in terms of noise sensitivity of function f for only two alternatives. It is interesting to compare it with the reduction from three to two dice in Lemma 2.1 of [32].…”

Section: Condorcet Paradox: Social Chaos For Close Majority Electionsmentioning

confidence: 99%

“…where (x, y) are 1/3-correlated. Another feature of the T ρ operator is that for noise sensitive functions (which [17] proved to be exactly those that result in chaotic elections without conditioning) the value |T ρ f (x)| is o(1) with high probability over x. If we decide to use |T ρ f (x)| as a measure of closeness, then this fact can be given the following (though by no means the only possible) interpretation: elections held according to a noise sensitive function are almost always close.…”

Section: Theorem 4 Under the Notation Above And The Event ε D As Defined Inmentioning

confidence: 99%

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The probability of intransitivity in dice and close elections

Hązła

Mossel

Ross

et al. 2020

Probab. Theory Relat. Fields

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We study the phenomenon of intransitivity in models of dice and voting. First, we follow a recent thread of research for n-sided dice with pairwise ordering induced by the probability, relative to 1/2, that a throw from one die is higher than the other. We build on a recent result of Polymath showing that three dice with i.i.d. faces drawn from the uniform distribution on $$\{1,\ldots ,n\}$$ { 1 , … , n } and conditioned on the average of faces equal to $$(n+1)/2$$ ( n + 1 ) / 2 are intransitive with asymptotic probability 1/4. We show that if dice faces are drawn from a non-uniform continuous mean zero distribution conditioned on the average of faces equal to 0, then three dice are transitive with high probability. We also extend our results to stationary Gaussian dice, whose faces, for example, can be the fractional Brownian increments with Hurst index $$H\in (0,1)$$ H ∈ ( 0 , 1 ) . Second, we pose an analogous model in the context of Condorcet voting. We consider n voters who rank k alternatives independently and uniformly at random. The winner between each two alternatives is decided by a majority vote based on the preferences. We show that in this model, if all pairwise elections are close to tied, then the asymptotic probability of obtaining any tournament on the k alternatives is equal to $$2^{-k(k-1)/2}$$ 2 - k ( k - 1 ) / 2 , which markedly differs from known results in the model without conditioning. We also explore the Condorcet voting model where methods other than simple majority are used for pairwise elections. We investigate some natural definitions of “close to tied” for general functions and exhibit an example where the distribution over tournaments is not uniform under those definitions.

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Section: Intransitive Dice: Transitivity Of Non-uniform Dicementioning

confidence: 99%

Section: Condorcet Paradox: Social Chaos For Close Majority Electionsmentioning

confidence: 99%

Section: Condorcet Paradox: Social Chaos For Close Majority Electionsmentioning

confidence: 99%

Section: Theorem 4 Under the Notation Above And The Event ε D As Defined Inmentioning

confidence: 99%

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The probability of intransitivity in dice and close elections

Hązła

Mossel

Ross

et al. 2020

Probab. Theory Relat. Fields

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show abstract

“…Bounds on the average and noise sensitivity of Boolean functions have direct applications in hardness of approximation [14,23], hardness amplification [31], circuit complexity [27], the theory of social choice [19], and quantum complexity [37]. In this paper, we focus on applications in learning theory, where it is known that bounds on the noise sensitivity of a class of Boolean functions yield learning algorithms for the class that succeed in harsh noise models (i. e., work in the agnostic model of learning) [18].…”

Section: Introduction 1backgroundmentioning

confidence: 99%

Untitled

Harsha¹,

Klivans²,

Meka³

2014

Theory of Comput.

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We give the first nontrivial upper bounds on the average sensitivity and noise sensitivity of polynomial threshold functions. More specifically, for a Boolean function f on n variables equal to the sign of a real, multivariate polynomial of total degree d, we prove • The average sensitivity of f is at most O(n 1−1/(4d+6)). (We also give a combinatorial proof of the bound O(n 1−1/2 d).) • The noise sensitivity of f with noise rate δ is at most O(δ 1/(4d+6)). Previously, only bounds for the degree d = 1 case were known (O(√ n) and O(√ δ), for average and noise sensitivity, respectively). We highlight some applications of our results in learning theory where our bounds immediately yield new agnostic learning algorithms and resolve an open problem of Klivans, O'Donnell and Servedio (FOCS'08).

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Inferring Rankings from First Order Marginals

Wolff

2021

Springer Proceedings in Mathematics &Amp; Statistics

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Noise Sensitivity and Chaos in Social Choice Theory

Cited by 12 publications

References 16 publications

The probability of intransitivity in dice and close elections

The probability of intransitivity in dice and close elections

Untitled

Inferring Rankings from First Order Marginals

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