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Possible Probability and Irreducibility of Balanced Nontransitive Dice

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  • Injo Hur
  • Yeansu Kim
  • Antonio Di Crescenzo

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 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.

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  • Injo Hur & Yeansu Kim & Antonio Di Crescenzo, 2021. "Possible Probability and Irreducibility of Balanced Nontransitive Dice," Journal of Mathematics, Hindawi, vol. 2021, pages 1-8, January.
    • Handle: RePEc:hin:jjmath:6648248
    DOI: 10.1155/2021/6648248
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    Cited by:

    1. Gorbunova, A.V. & Lebedev, A.V., 2022. "Nontransitivity of tuples of random variables with polynomial density and its effects in Bayesian models," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 181-192.

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