Generalized Intransitive Dice II: Partition Constructions

TL;DR

This paper explores how to construct collections of generalized dice that model any tournament structure on a set of players, using partition methods and bounds on the size of the dice.

Contribution

It introduces a method to model arbitrary tournaments with generalized dice using partitions of a specific set size, providing bounds on the necessary size.

Findings

Partition constructions can model any tournament.

Bound on dice size needed is N=3^{n-2}.

Generalized dice can represent complex 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} $. 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 $[Nn] $ 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 $N = 3^{n-2}$.