Possible Probability and Irreducibility of Balanced Nontransitive Dice

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 1/2+1/9, and we construct a balanced nontransitive set of dice whose probability is 1/2+13−153/24≈1/2+1/9.12