Intransitive dice tournament is not quasirandom

TL;DR

This paper proves that intransitive dice are not quasirandom by showing that for four random dice, the likelihood of transitive tournaments exceeds the expected 3/8, contrasting prior results for three dice.

Contribution

It establishes that the distribution of tournaments among four random dice deviates from quasirandom behavior, providing new insights into their probabilistic structure.

Findings

Probability of transitive tournaments exceeds 3/8 for four dice.

Intransitive dice are not quasirandom.

Distribution of tournaments differs from the three-dice case.

Abstract

We settle a version of the conjecture about intransitive dice posed by Conrey, Gabbard, Grant, Liu and Morrison in 2016 and Polymath in 2017. We consider generalized dice with $n$ faces and we say that a die $A$ beats $B$ if a random face of $A$ is more likely to show a higher number than a random face of $B$. We study random dice with faces drawn iid from the uniform distribution on $[0,1]$ and conditioned on the sum of the faces equal to $n/2$. Considering the "beats" relation for three such random dice, Polymath showed that each of eight possible tournaments between them is asymptotically equally likely. In particular, three dice form an intransitive cycle with probability converging to $1/4$. In this paper we prove that for four random dice not all tournaments are equally likely and the probability of a transitive tournament is strictly higher than $3/8$.