An upper bound for a cyclic sum of probabilities

Marengo, James E.; T., Kolt, Quinn; Farnsworth, David L.

doi:10.1016/j.spl.2020.108861

Cited by 3 publications

(2 citation statements)

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“…More formally, if we let A, B, C, D denote independent discrete random variables that are uniformly distributed over the values listed in (1.1), then these variables satisfy Another classic example of nontransitivity is given by the following set of three-sided dice which seems to have first appeared (in a different, but equivalent, context) in a paper by Moon and Moser [16]: There now exists a large body of research motivated by such nontransitivity phenomena; see [2,5,7,9,10,11,13,15] for some recent work on this subject.…”

Section: Introductionmentioning

confidence: 99%

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On cyclic and nontransitive probabilities

Vuksanovic,

Hildebrand

2020

Preprint

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Motivated by classical nontransitivity paradoxes, we call an n-tuple (x 1 , . . . , x n ) ∈ [0, 1] n cyclic if there exist independent random variables U 1 , . . . , U n withWe call the tuple (x 1 , . . . , x n ) nontransitive if it is cyclic and in addition satisfies x i > 1/2 for all i.Let p n (resp. p * n ) denote the probability that a randomly chosen n-tuple (x 1 , . . . , x n ) ∈ [0, 1] n is cyclic (resp. nontransitive). We determine p 3 and p * 3 exactly, while for n ≥ 4 we give upper and lower bounds for p n that show that p n converges to 1 as n → ∞. We also determine the distribution of the smallest, middle, and largest elements in a cyclic triple.

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Section: Introductionmentioning

confidence: 99%

“…Thus, it would be of interest to study the analogs of cyclic and nontransitive tuples if the indendence assumption on U i is dropped. We refer to Marengo et al [15] for some recent results in this direction, and to Suck [19] for an in-depth discussion of the differences between dependent and independent representations of the form (6.1).…”

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confidence: 99%