Generalized Intransitive Dice II: Partition Constructions

Ethan Akin; Julia Saccamano

paper

Generalized Intransitive Dice II: Partition Constructions

arXiv:1905.01750

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 $P r o b (D_{i} > D_{j}) > \frac{1}{2}$ . A collection of dice ${D_{i} : i = 1, \dots, n}$ models a tournament on the set $[n] = {1, 2, \dots, n}$ , i.e. a complete digraph with $n$ vertices, when $D_{i} \to D_{j}$ if and only if $i \to j$ in the tournament. By using $n$ -fold partitions of the set $[N n]$ with each set of size $N$ we can model an arbitrary tournament on $[n]$ . A bound on the required size of $N$ is obtained by examples with $N = 3^{n - 2}$ .