Tied dice. $II$. Some Asymptotic Results

Abstract

A dice family $D(n,a,b,s)$ includes all lists $(x_1,\dots ,x_n)$ of integers with $n\ge 1$, $a\le x_1\le \dots \le x_n\le b$, and $\sum x_i=s$. Given two dice $X$ and $Y$, we compare the number of pairs $(i,j)$ with $x_i{y}_j$. If the second number is larger, then $X$ is $stronger$ than $Y$, and if the two numbers are equal, then $X$ and $Y$ are $tied$. In previous work, it has been observed that the density of ties in $D(n,a,b,s)$ is generally lower than one might expect. In this note, we provide more information about this observation by calculating the asymptotic proportion of ties in certain kinds of dice families. Many other properties of dice families remain to be determined.