Generalized Intransitive Dice: Mimicking an Arbitrary Tournament

Ethan Akin

paper

Generalized Intransitive Dice: Mimicking an Arbitrary Tournament

arXiv:1901.09477

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 $P r o b (D_{i} > D_{j}) > \frac{1}{2}$ . 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 $R$ .