Generalized Intransitive Dice: Mimicking an Arbitrary Tournament

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 $R$.