Generalized Intransitive Dice: Mimicking an Arbitrary Tournament

TL;DR

This paper demonstrates that for any tournament graph, one can construct a set of large-sided dice whose pairwise beating relations exactly replicate the tournament's directions, generalizing intransitive dice concepts.

Contribution

The paper proves the existence of large-sided dice sets that realize any arbitrary tournament as their beating relation structure.

Findings

Existence of dice sets matching any tournament for large N

Construction method for dice realizing arbitrary tournaments

Extension of intransitive dice to generalized tournament structures

Abstract

A generalized $N$-sided die is a random variable $D$ on a sample space of $N$ equally likely outcomes taking values in the set of positive integers. We say of independent $N$ sided dice $D_i, D_j$ that $D_i$ beats $D_j$, written $D_i \to D_j$, if $Prob(D_i > D_j) > \frac{1}{2} $. Examples are known of intransitive $6$-sided dice, i.e. $D_1 \to D_2 \to D_3$ but $D_3 \to D_1$. A tournament of size $n$ is a choice of direction $i \to j$ for each edge of the complete graph on $n$ vertices. We show that if $R$ is tournament on the set $\{ 1, \dots, n \}$, then for sufficiently large $N$ there exist sets of independent $N$-sided dice $\{ D_1, \dots, D_n \}$ such that $D_i \to D_j$ if and only if $i \to j$ in $R$.