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2021 On cyclic and nontransitive probabilities
Pavle Vuksanovic, A. J. Hildebrand
Involve 14(2): 327-348 (2021). DOI: 10.2140/involve.2021.14.327

Abstract

Motivated by classical nontransitivity paradoxes, we call an n-tuple (x1,,xn)[0,1]n cyclic if there exist independent random variables U1,,Un with P(Ui=Uj)=0 for ij such that P(Ui+1>Ui)=xi for i=1,,n1 and P(U1>Un)=xn. We call the tuple (x1,,xn) nontransitive if it is cyclic and in addition satisfies xi>12 for all i.

Let pn (resp. pn)denote the probability that a randomly chosen n-tuple (x1,,xn)[0,1]n is cyclic (resp. nontransitive). We determine p3 and p3 exactly, while for n4 we give upper and lower bounds for pn that show that pn converges to 1 as n. We also determine the distribution of the smallest, middle, and largest elements in a cyclic triple.

Citation

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Pavle Vuksanovic. A. J. Hildebrand. "On cyclic and nontransitive probabilities." Involve 14 (2) 327 - 348, 2021. https://doi.org/10.2140/involve.2021.14.327

Information

Received: 30 August 2020; Accepted: 29 November 2020; Published: 2021
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.2140/involve.2021.14.327

Subjects:
Primary: 60C05, 91A60

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.14 • No. 2 • 2021
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