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January  2021, 8(1): 1-20. doi: 10.3934/jdg.2020030

Generalized intransitive dice: Mimicking an arbitrary tournament

Ethan Akin ,

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

Received  July 2019 Published  September 2020

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} $. Examples are known of intransitive $ 6 $-sided dice, i.e. $ D_1 \to D_2 \to D_3 $ but $ D_3 \to D_1 $. A tournament of size $ n $ is a choice of direction $ i \to j $ for each edge of the complete graph on $ n $ vertices. We show that if $ R $ is tournament on the set $ [n] = \{ 1, \dots, n \} $, then for sufficiently large $ N $ there exist sets of independent $ N $-sided dice $ \{ D_1, \dots, D_n \} $ such that $ D_i \to D_j $ if and only if $ i \to j $ in $ R $.

Keywords: Intransitive dice, nontransitive dice, digraph, tournament, mimicking a tournament, modeling a tournament, homeomorphism groups.
Mathematics Subject Classification: Primary:05C20, 05C25, 05C62, 05C70.
Citation: Ethan Akin. Generalized intransitive dice: Mimicking an arbitrary tournament. Journal of Dynamics & Games, 2021, 8 (1) : 1-20. doi: 10.3934/jdg.2020030
References:
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E. Akin, Rock, paper, scissors, etc - topics in the theory of regular tournaments, preprint, arXiv: 1806.11241v1, (2018), v4(2020). Google Scholar

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E. Akin, Intransitive dice - mimicking an arbitrary tournament, preprint, arXiv: 1901.09477v2, (2019). Google Scholar

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

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show all references

References:
[1]

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

[2]

E. Akin, Intransitive dice - mimicking an arbitrary tournament, preprint, arXiv: 1901.09477v2, (2019). Google Scholar

[3] J. K. Blitzstein and J. Hwang, Introduction to Probability, CRC Press, Boca Raton, 2014.   Google Scholar
[4]

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

[5]

M. Finkelstein and E. O. Thorp, Nontransitive dice with equal means, in Optimal Play: Mathematical Studies of Games and Gambling (S. N. Ethier and W. R. Eadington, eds.), Reno: Institute for the Study of Gambling and Commercial Gaming, 2007. Google Scholar

[6]

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

[7]

G. G. Margaril-Ilyaev and V. M. Tikhomirov, Convex Analysis: Theory and Applications, Translations of Math. Monograph 222, Amer. Math. Soc., Providence, 2003.  Google Scholar

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