The probability of intransitivity in dice and close elections

Hązła, Jan; Mossel, Elchanan; Ross, Nathan; Zheng, Guangqu

doi:10.1007/s00440-020-00994-7

Cited by 7 publications

(8 citation statements)

References 30 publications

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“…Thinking of CV cond as a random variable over the choice of balanced B and C, we bound its second and fourth moments and apply the Paley-Zygmund inequality to conclude that indeed sometimes it is bounded away from zero. In the proof we employ a modification of the argument used in [HMRZ20]: If B and C are not balanced, estimating the moments is an elementary calculation. To account for the conditioning on balanced B and C, we employ yet another precise local central limit theorem to estimate changes in the relevant moments.…”

Section: Proof Strategymentioning

confidence: 99%

“…To start with, as in Section 2.3 we will argue that throughout the whole proof we can assume faces come from a distribution different than [0, 1]. This time, for consistency with [HMRZ20] and to simplify some calculations we take the faces to be uniform in…”

Section: Proof Strategy For Lemmas 16 and 17mentioning

confidence: 99%

“…, n} and sample dice with n iid faces from this distribution. Some thought reveals that this does not generate intransitivity (see, e.g., Theorem 6 in [HMRZ20] for details). The reason is that with high probability, in a small set of dice sampled from this model, A beats B if and only if n i=1 a i > n i=1 b i .…”

Section: Introductionmentioning

confidence: 99%

“…Perhaps surprisingly, it turns out that both balancedness and uniformity are important. In particular, a follow-up work [HMRZ20] showed that if dice are balanced with faces iid from any (continuous) distribution that is not uniform (and also in some other models with correlated Gaussian faces), then the intransitivity does not show up and three dice are transitive with high probability.…”

Section: Introductionmentioning

confidence: 99%

“…The proof of Theorem 3 combines the main ideas present in two preceding works [Pol17a] and [HMRZ20] and can roughly be divided into two parts inspired by these papers. However, there is a considerable number of items that need to be fleshed out and some care is required to get the details right.…”

Section: Introductionmentioning

confidence: 99%

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Intransitive dice tournament is not quasirandom

Cornacchia¹,

Hązła²

2020

Preprint

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We settle a version of the conjecture about intransitive dice posed by Conrey, Gabbard, Grant, Liu and Morrison in 2016 and Polymath in 2017. We consider generalized dice with n faces and we say that a die A beats B if a random face of A is more likely to show a higher number than a random face of B. We study random dice with faces drawn iid from the uniform distribution on [0, 1] and conditioned on the sum of the faces equal to n/2. Considering the "beats" relation for three such random dice, Polymath showed that each of eight possible tournaments between them is asymptotically equally likely. In particular, three dice form an intransitive cycle with probability converging to 1/4. In this paper we prove that for four random dice not all tournaments are equally likely and the probability of a transitive tournament is strictly higher than 3/8.

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Section: Proof Strategymentioning

confidence: 99%

Section: Proof Strategy For Lemmas 16 and 17mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Intransitive dice tournament is not quasirandom

Cornacchia¹,

Hązła²

2020

Preprint

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The intransitive dice kernel: $$\frac{\mathbbm {1}_{x\ge y}-\mathbbm {1}_{x\le y}}{4} - \frac{3(x-y)(1+xy)}{8}$$

Sah,

Sawhney

2024

Probab. Theory Relat. Fields

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Answering a pair of questions of Conrey, Gabbard, Grant, Liu, and Morrison, we prove that a triplet of dice drawn from the multiset model are intransitive with probability $$1/4+o(1)$$ 1 / 4 + o ( 1 ) and the probability a random pair of dice tie tends toward $$\alpha n^{-1}$$ α n - 1 for an explicitly defined constant $$\alpha $$ α . This extends and sharpens the recent results of Polymath regarding the balanced sequence model. We further show the distribution of larger tournaments converges to a universal tournamenton in both models. This limit naturally arises from the discrete spectrum of a certain skew-symmetric operator (given by the kernel in the title acting on $$L^2([-1,1])$$ L 2 ( [ - 1 , 1 ] ) ). The limit exhibits a degree of symmetry and can be used to prove that, for instance, the limiting probability that $$A_i$$ A i beats $$A_{i+1}$$ A i + 1 for $$1\le i\le 4$$ 1 ≤ i ≤ 4 and that $$A_5$$ A 5 beats $$A_1$$ A 1 is $$1/32+o(1)$$ 1 / 32 + o ( 1 ) . Furthermore, the limiting tournamenton has range contained in the discrete set $$\{0,1\}$$ { 0 , 1 } . This proves that the associated tournamenton is non-quasirandom in a dramatic fashion, vastly extending work of Cornacchia and Hązła regarding the continuous analogue of the balanced sequence model. The proof is based on a reduction to conditional central limit theorems (related to work of Polymath), the use of a “Poissonization” style method to reduce to computations with independent random variables, and the systematic use of switching-based arguments to extract cancellations in Fourier estimates when establishing local limit-type estimates.

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Construction of voting situations concordant with ranking patterns

Santis

Spizzichino

2023

Decisions Econ Finan

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Referring to a standard context of voting theory, and to the classic notion of voting situation, here we show that it is possible to observe any arbitrary set of elections’ outcomes, no matter how paradoxical it may appear. In this respect, we consider a set of candidates $$1, 2, \ldots , m $$ 1 , 2 , … , m and, for any subset A of $$\{1, 2, \ldots , m \}$$ { 1 , 2 , … , m } , we fix a ranking among the candidates belonging to A. We wonder whether it is possible to find a population of voters whose preferences, expressed according to the Condorcet’s proposal, give rise to that family of rankings. We will show that, whatever be such family, a population of voters can be constructed that realize all the rankings of it. Our conclusions are similar to those coming from D. Saari’s results. Our results are, however, constructive and allow for the study of quantitative aspects of the wanted voters’ populations.

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The probability of intransitivity in dice and close elections

Cited by 7 publications

References 30 publications

Intransitive dice tournament is not quasirandom

Intransitive dice tournament is not quasirandom

The intransitive dice kernel: $$\frac{\mathbbm {1}_{x\ge y}-\mathbbm {1}_{x\le y}}{4} - \frac{3(x-y)(1+xy)}{8}$$

Construction of voting situations concordant with ranking patterns

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