On cyclic and nontransitive probabilities

TL;DR

This paper investigates the probabilities of cyclic and nontransitive probability tuples, providing exact results for triples and bounds for larger n, revealing that cyclicity becomes almost certain as n grows.

Contribution

It precisely determines the probabilities for 3-tuples and offers bounds for larger n, advancing understanding of cyclic and nontransitive probability structures.

Findings

Exact probability for 3-tuples being cyclic and nontransitive.

Bounds showing probability of cyclic n-tuples approaches 1 as n increases.

Distribution of extremal elements in cyclic triples.

Abstract

Motivated by classical nontransitivity paradoxes, we call an $n$-tuple $(x_1,\dots,x_n) \in[0,1]^n$ \textit{cyclic} if there exist independent random variables $U_1,\dots, U_n$ with $P(U_i=U_j)=0$ for $i\not=j$ 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 $(x_1,\dots,x_n)$ \textit{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 $(x_1,\dots,x_n)\in[0,1]^n$ is cyclic (resp.~nontransitive). We determine $p_3$ and $p_3^*$ exactly, while for $n\ge4$ 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.