# Balanced Non-Transitive Dice II: Tournaments

## TL;DR

This paper explores balanced non-transitive dice, proving their existence for all sizes and demonstrating how to construct sets that emulate any tournament's directed graph structure.

## Contribution

It establishes the existence of balanced non-transitive dice for all sizes and provides a method to construct dice sets matching any tournament's structure.

## Key findings

- Balanced non-transitive dice exist for all n,m ≥ 3
- Construction method for dice sets matching any tournament
- Relation 'is a better die than' can replicate any tournament

## Abstract

We further study sets of labeled dice in which the relation "is a better die than" is non-transitive. Focusing on sets with an additional symmetry we call "balance," we prove that sets of $n$ such $m$-sided dice exist for all $n,m \geq 3$. We then show how to construct a set of $n$ dice such that the relation behaves according to the direction of the arrows of any tournament (complete directed graph) on $n$ vertices.