Possible Probability and Irreducibility of Balanced Nontransitive Dice

Abstract

We construct irreducible balanced nontransitive sets of $n$ -sided dice for any positive integer $n$ . One main tool of the construction is to study so-called fair sets of dice. Furthermore, we also study the distribution of the probabilities of balanced nontransitive sets of dice. For a lower bound, we show that the winning probability can be arbitrarily close to $1/2$ . We hypothesize that the winning probability cannot be more than $\left(1/2\right)+\left(1/9\right)$ , and we construct a balanced nontransitive set of dice whose probability is $\left(1/2\right)+\left(13-\sqrt{153}|/|24\right)\approx \left(1/2\right)+\left(1/9.12\right)$ .