Balanced Nontransitive Dice: Existence and Probability

Abstract

A triple $(A,B,C)$ of dice is called nontransitive if each of $P(A<B)$, $P(B<C)$, and $P(C<A)$ is greater than $\frac12$ and called balanced if $P(A<B)=P(B<C)=P(C<A)$. From the result of Trybuła, it is known that $P(A<B)$ is less than $\frac{-1+\sqrt{5}}{2}$, the golden ratio, for every balanced nontransitive triple $(A,B,C)$ of dice. Schaefer asked whether this upper bound is tight, and Hur and Kim conjectured that the upper bound can be reduced to $\frac12+\frac19$. In this paper, we characterize all possible probabilities $P(A<B)$ for balanced nontransitive triples $(A,B,C)$ of dice. Precisely, we prove that, for every rational $\frac12 <q<\frac{-1+\sqrt{5}}{2}$, there exists a balanced nontransitive triple $(A,B,C)$ of dice with $P(A<B)=q$, which disproves Hur and Kim's conjecture and answers Schaefer's question.
 We also characterize all triples $(m,n,\ell)$ of positive integers such that there exists a balanced nontransitive triple $(A,B,C)$ of dice, where $A$, $B$, and $C$ are $m$-sided, $n$-sided, and $\ell$-sided dice, respectively. This generalizes Schaefer and Schweig's result showing the existence of a balanced nontransitive triple of $n$-sided dice for every $n\ge 3$.