Generalizations of Schütte’s property with applications to dice games

Abstract

In this work, we explore dice games of the following kind: k players choose a die from a collection of dice, and one final player chooses a die in response. The players then roll their dice and compare the results. Traditionally, these games have been represented by tournaments. Tournaments that give the last player an advantage are related to Sch ̈utte’s property, introduced by Erd ̋os in 1963. A tournament has Sch ̈utte’s property if for every k-set of vertices, there is another vertex directed toward that k-set. We examine various modifications of this property and the dice games associated with them. In Chapters 2 and 3, we provide background on Sch ̈utte’s property and dice sets that can realize a given tournament. We also explore computational techniques for searching for such tournaments and sets of dice. In Chapter 4, inspired by Grime’s dice, we generalize Sch ̈utte’s property to sets of tournaments and study the minimum number of vertices needed to form such sets. Additionally, we consider methods for realizing these sets with dice by rolling them multiple times and summing the results. In Chapter 5, we extend the concept of Sch ̈utte’s property to directed hypergraphs. We determine the exact values for the minimum order of hypergraphs with this property. In Chapter 6, we explore how to realize the relationships in such hypergraphs using dice. We demonstrate that any ordering of outcomes in three-wise rolls among n players can be achieved with a set of dice.