Balanced Non-Transitive Dice II: Tournaments

We further study sets of labeled dice in which the relation "is a better die than" is non-transitive. Focusing on sets with an additional symmetry we call "balance," we prove that sets of $n$ such $m$-sided dice exist for all $n,m \geq 3$. We then show how to construct a set of $n$ dice such that the relation behaves according to the direction of the arrows of any tournament (complete directed graph) on $n$ vertices.