{"title":"Generalized intransitive dice: Mimicking an arbitrary tournament","authors":"E. Akin","doi":"10.3934/jdg.2020030","DOIUrl":null,"url":null,"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 $R$.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":" ","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2019-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Dynamics and Games","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/jdg.2020030","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 8
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 $R$.
期刊介绍:
The Journal of Dynamics and Games (JDG) is a pure and applied mathematical journal that publishes high quality peer-review and expository papers in all research areas of expertise of its editors. The main focus of JDG is in the interface of Dynamical Systems and Game Theory.