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

Motivated by classical nontransitivity paradoxes, we call an n-tuple (x1,,xn) [0,1]n cyclic if there exist independent random variables U1,,Un with P(Ui=Uj) = 0 for ij such that P(Ui+1>Ui) = xi for i = 1,,n 1 and P(U1>Un) = xn. We call the tuple (x1,,xn) nontransitive if it is cyclic and in addition satisfies xi > 1 2 for all i.

Let pn (resp. pn) denote the probability that a randomly chosen n-tuple (x1,,xn) [0,1]n is cyclic (resp. nontransitive). We determine p3 and p3 exactly, while for n 4 we give upper and lower bounds for pn that show that pn converges to 1 as n . We also determine the distribution of the smallest, middle, and largest elements in a cyclic triple.

nontransitivity paradoxes, probability, game theory, social choice theory
Mathematical Subject Classification
Primary: 60C05, 91A60
Received: 30 August 2020
Accepted: 29 November 2020
Published: 6 April 2021

Communicated by Anant Godbole
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