A Direct Construction of Non-Transitive Dice Sets

TL;DR

This paper presents a direct method to construct sets of non-transitive dice that realize any given tournament, using a number of sides proportional to the number of vertices, based on a fundamental graph theory theorem.

Contribution

It introduces a new, straightforward construction method for non-transitive dice sets that can realize any tournament with optimal side count, relying solely on standard graph theory results.

Findings

Construction works for any tournament with n vertices

Number of sides is proportional to n, optimal among known methods

Relies only on a standard graph theory theorem

Abstract

In this paper, we give a direct construction for a set of dice realizing any given tournament $T$. The construction for a tournament with $n$ vertices requires a number of sides on the order of $n$, which appears to be the best general construction to date. Our construction relies only on a standard theorem from graph theory.