On cyclic and nontransitive probabilities

Vuksanovic, Pavle; Hildebrand, A. J.

doi:10.2140/involve.2021.14.327

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On cyclic and nontransitive probabilities

Pavle Vuksanovic and A. J. Hildebrand

Vol. 14 (2021), No. 2, 327–348

DOI: 10.2140/involve.2021.14.327

Abstract

Motivated by classical nontransitivity paradoxes, we call an $n$ -tuple $(x_{1}, \dots, x_{n}) \in {[0, 1]}^{n}$ cyclic if there exist independent random variables $U_{1}, \dots, U_{n}$ with $P (U_{i} = U_{j}) = 0$ for $i \neq j$ such that $P (U_{i + 1} > U_{i}) = x_{i}$ for $i = 1, \dots, n - 1$ and $P (U_{1} > U_{n}) = x_{n}$ . We call the tuple $(x_{1}, \dots, x_{n})$ nontransitive if it is cyclic and in addition satisfies $x_{i} > \frac{1}{2}$ for all $i$ .

Let $p_{n}$ (resp. $p_{n}^{*}$ ) denote the probability that a randomly chosen $n$ -tuple $(x_{1}, \dots, x_{n}) \in {[0, 1]}^{n}$ is cyclic (resp. nontransitive). We determine $p_{3}$ and $p_{3}^{*}$ exactly, while for $n \geq 4$ 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.

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Keywords

nontransitivity paradoxes, probability, game theory, social choice theory

Mathematical Subject Classification

Primary: 60C05, 91A60

Milestones

Received: 30 August 2020

Accepted: 29 November 2020

Published: 6 April 2021

Communicated by Anant Godbole

Authors

Pavle Vuksanovic
Department of Mathematics University of Illinois Urbana, IL United States

A. J. Hildebrand
Department of Mathematics University of Illinois Urbana, IL United States

Vol. 14, No. 2, 2021