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Generalized intransitive dice: Mimicking an arbitrary tournament
Mathematics Department, The City College, 137 Street and Convent Avenue, New York City, NY 10031, USA |
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 $.
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.
![]() |
[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. |
[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. |
[7] |
G. G. Margaril-Ilyaev and V. M. Tikhomirov, Convex Analysis: Theory and Applications, Translations of Math. Monograph 222, Amer. Math. Soc., Providence, 2003. |
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.
![]() |
[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. |
[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. |
[7] |
G. G. Margaril-Ilyaev and V. M. Tikhomirov, Convex Analysis: Theory and Applications, Translations of Math. Monograph 222, Amer. Math. Soc., Providence, 2003. |
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