Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Possible probability and irreducibility of balanced non-transitive dice   We construct irreducible balanced non-transitive sets of $n$-sided dice for
any positive integer $n$, which was raised in \cite[Question 5.2]{SS17}. 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
non-transitive sets of dice. For a lower bound, we show that the probability
could be arbitrarily close to $\frac{1}{2}$ and for a upper bound, we construct
a balanced non-transitive set of dice whose probability is $\frac{1}{2} +
\frac{13-\sqrt{153}}{24} \approx \frac{1}{2} + \frac{1}{9.12}.$

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The categories of this paper are: q-alg
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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: acc-phys
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: q-bio
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The categories of this paper are: cond-mat
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The categories of this paper are: chem-ph
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The categories of this paper are: math-ph
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The categories of this paper are: nlin