#### Vol. 14, No. 2, 2021

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On cyclic and nontransitive probabilities

### Pavle Vuksanovic and A. J. Hildebrand

Vol. 14 (2021), No. 2, 327–348
##### Abstract

Motivated by classical nontransitivity paradoxes, we call an $n$-tuple $\left({x}_{1},\dots ,{x}_{n}\right)\in {\left[0,1\right]}^{n}$ cyclic if there exist independent random variables ${U}_{1},\dots ,{U}_{n}$ 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 $i=1,\dots ,n-1$ and $P\left({U}_{1}>{U}_{n}\right)={x}_{n}$. We call the tuple $\left({x}_{1},\dots ,{x}_{n}\right)$ nontransitive if it is cyclic and in addition satisfies ${x}_{i}>\frac{1}{2}$ for all $i$.

Let ${p}_{n}$ (resp. ${p}_{n}^{\ast }$) denote the probability that a randomly chosen $n$-tuple $\left({x}_{1},\dots ,{x}_{n}\right)\in {\left[0,1\right]}^{n}$ 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.

##### Keywords
nontransitivity paradoxes, probability, game theory, social choice theory
##### Mathematical Subject Classification
Primary: 60C05, 91A60