On cyclic and nontransitive probabilities

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 $i\ne j$ 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.~$p_n^{\ast }$) 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 $p_3^{\ast }$ 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.

Accepted for publication in Involve

References in corpus (1)

• Intransitive Dice