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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 $\left( {{x_1}, \ldots ,{x_n}} \right) \in {\left[ {0,1} \right]^n}$ cyclic if there exist independent random variables ${U_1}, \ldots ,{U_n}$ with $P\left( {{U_i} = {U_j}} \right) = 0$ for $i \ne j$ such that $P\left( {{U_{i + 1}} > {U_i}} \right) = {x_i}$ for $i = 1, \ldots ,n - 1$ and $P\left( {{U_1} > {U_n}} \right) = {x_n}$. We call the tuple $\left( {{x_1}, \ldots ,{x_n}} \right)$ nontransitive if it is cyclic and in addition satisfies ${x_i} > {1 \over 2}$ for all $i$.

Let ${p_n}$ (resp. $p_n^ *$)denote the probability that a randomly chosen $n$-tuple $\left( {{x_1}, \ldots ,{x_n}} \right) \in {\left[ {0,1} \right]^n}$ is cyclic (resp. nontransitive). We determine ${p_3}$ and $p_3^ *$ exactly, while for $n \ge 4$ we give upper and lower bounds for ${p_n}$ that show that ${p_n}$ converges to $1$ as $n \to \infty$. We also determine the distribution of the smallest, middle, and largest elements in a cyclic triple.

## Citation

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  