An upper bound for a cyclic sum of probabilities

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In the study of nontransitive dice, seemingly paradoxically, the probability that one die rolls higher than another is not transitive. We prove a more general result which implies that a cyclic sum of ordering probabilities for n random variables can be much greater than 1, and that each probability can be much greater than 1n. In particular, we establish an upper bound for this sum and give necessary and sufficient conditions under which it is attained. We also prove that, given an absolutely continuous univariate probability distribution and ε>0, there are random variables X1,X2,…,Xn, each of which has this distribution, and for which each of the probabilities PrX1>X2,PrX2>X3,…,PrXn−1>Xn, and PrXn>X1 exceeds 1−ε.