Possible probability and irreducibility of balanced non-transitive dice

TL;DR

This paper constructs irreducible balanced non-transitive dice for any number of sides, explores their probability distributions, and shows probabilities can approach 1/2 or be slightly above it.

Contribution

It provides a method to construct irreducible balanced non-transitive dice for any n and analyzes their probability bounds.

Findings

Probability can be arbitrarily close to 1/2.

Constructed a balanced non-transitive set with probability approximately 1/2 + 1/9.12.

Established existence of irreducible balanced non-transitive dice for all n.

Abstract

We construct irreducible balanced non-transitive sets of $n$-sided dice for any positive integer $n$, which was raised in \cite[Question 5.2]{SS17}. One main tool of the construction is to study so-called fair sets of dice. Furthermore, we also study the distribution of the probabilities of balanced non-transitive sets of dice. For a lower bound, we show that the probability could be arbitrarily close to $\frac{1}{2}$ and for a upper bound, we construct a balanced non-transitive set of dice whose probability is $\frac{1}{2} + \frac{13-\sqrt{153}}{24} \approx \frac{1}{2} + \frac{1}{9.12}.$