Here is an introduction to nontransitive dice. The question is: given $n$-player with a $m$-sided dice each one, the what is the minimum of $m$ for a fixed $n$ to produce nontransitivity?

Here is some related posts:

What is the most extreme set 4 or 5 nontransitive n-sided dice?

`$\{1.01,5.02,8.99\}$`

and so on. This will give a large number of nontransitive dice for $m=3$. But this doesn't seem like the right case to consider. – Will Sawin Aug 24 '13 at 0:38