Motivated by classical nontransitivity paradoxes, we call an -tuple cyclic if there exist independent random variables with for such that for and . We call the tuple nontransitive if it is cyclic and in addition satisfies for all .
Let (resp. )denote the probability that a randomly chosen -tuple is cyclic (resp. nontransitive). We determine and exactly, while for we give upper and lower bounds for that show that converges to as . We also determine the distribution of the smallest, middle, and largest elements in a cyclic triple.
"On cyclic and nontransitive probabilities." Involve 14 (2) 327 - 348, 2021. https://doi.org/10.2140/involve.2021.14.327