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doi: 10.3934/jdg.2021005

Generalized intransitive dice II: Partition constructions

Ethan Akin and  Julia Saccamano

Mathematics Department, The City College, 137 Street and Convent Avenue, New York City, NY 10031, USA

Received  September 2020 Published  March 2021

A generalized $ N $-sided die is a random variable $ D $ on a sample space of $ N $ equally likely outcomes taking values in the set of positive integers. We say of independent $ N $-sided dice $ D_i, D_j $ that $ D_i $ beats $ D_j $, written $ D_i \to D_j $, if $ Prob(D_i > D_j) > \frac{1}{2} $. A collection of dice $ \{ D_i : i = 1, \dots, n \} $ models a tournament on the set $ [n] = \{ 1, 2, \dots, n \} $, i.e. a complete digraph with $ n $ vertices, when $ D_i \to D_j $ if and only if $ i \to j $ in the tournament. By using regular $ n $-fold partitions of the set $ [Nn] $ to label the $ N $-sided dice we can model an arbitrary tournament on $ [n] $ and $ N $ can be chosen to be less than or equal to $ N = 3^{n-2} $.

Keywords: Intransitive dice, nontransitive dice, digraph, tournament, partitions, regular partitions, mimicking a tournament, modeling a tournament.
Mathematics Subject Classification: 05C20, 05C25, 05C62, 05C70.
Citation: Ethan Akin, Julia Saccamano. Generalized intransitive dice II: Partition constructions. Journal of Dynamics & Games, doi: 10.3934/jdg.2021005
References:
[1]

E. Akin, Rock, paper, scissors, etc. - topics in the theory of regular tournaments, arXiv: 1806.11241v1, (2018), v4(2020). Google Scholar

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E. Akin, Generalized intransitive dice: Mimicking an arbitrary tournament, J. of Dynamics and Games, 8 (2021), 1-20.  doi: 10.3934/jdg.2020030.  Google Scholar

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B. Conrey, J. Gabbard, K. Grant, A. Liu and K. E. Morrison, Intransitive Dice, Math. Mag., 89 (2016), 133-143.  doi: 10.4169/math.mag.89.2.133.  Google Scholar

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F. Harary and L. Moser, The theory of round robin tournaments, Amer. Math. Monthly, 73 (1966), 231-246.  doi: 10.1080/00029890.1966.11970749.  Google Scholar

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J. W. Moon, Topics on Tournaments, Holt, Rinehart and Winston, New York, NY, 1968, Reprinted, Dover Publications, Mineola, NY, 2015.  Google Scholar

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G. Sabidussi, The composition of graphs, Duke Math. J., 26 (1959), 693-696.  doi: 10.1215/S0012-7094-59-02667-5.  Google Scholar

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G. Sabidussi, The lexicographic product of graphs, Duke Math. J., 28 (1961), 573-578.  doi: 10.1215/S0012-7094-61-02857-5.  Google Scholar

show all references

References:
[1]

E. Akin, Rock, paper, scissors, etc. - topics in the theory of regular tournaments, arXiv: 1806.11241v1, (2018), v4(2020). Google Scholar

[2]

E. Akin, Generalized intransitive dice: Mimicking an arbitrary tournament, J. of Dynamics and Games, 8 (2021), 1-20.  doi: 10.3934/jdg.2020030.  Google Scholar

[3]

B. Alspach, On point-symmetric tournaments, Canad. Math. Bull., 13 (1970), 317-323.  doi: 10.4153/CMB-1970-061-7.  Google Scholar

[4]

B. Conrey, J. Gabbard, K. Grant, A. Liu and K. E. Morrison, Intransitive Dice, Math. Mag., 89 (2016), 133-143.  doi: 10.4169/math.mag.89.2.133.  Google Scholar

[5]

M. Goldberg and J. W. Moon, On the composition of two tournaments, Duke Math. J., 37 (1970), 323-332.  doi: 10.1215/S0012-7094-70-03742-7.  Google Scholar

[6]

F. Harary and L. Moser, The theory of round robin tournaments, Amer. Math. Monthly, 73 (1966), 231-246.  doi: 10.1080/00029890.1966.11970749.  Google Scholar

[7]

J. W. Moon, Topics on Tournaments, Holt, Rinehart and Winston, New York, NY, 1968, Reprinted, Dover Publications, Mineola, NY, 2015.  Google Scholar

[8]

G. Sabidussi, The composition of graphs, Duke Math. J., 26 (1959), 693-696.  doi: 10.1215/S0012-7094-59-02667-5.  Google Scholar

[9]

G. Sabidussi, The lexicographic product of graphs, Duke Math. J., 28 (1961), 573-578.  doi: 10.1215/S0012-7094-61-02857-5.  Google Scholar

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