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Generalized intransitive dice II: Partition constructions
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} $. 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} $.
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. |
| [3] |
B. Alspach,
On point-symmetric tournaments, Canad. Math. Bull., 13 (1970), 317-323.
doi: 10.4153/CMB-1970-061-7. |
| [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. |
| [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. |
| [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. |
| [7] |
J. W. Moon, Topics on Tournaments, Holt, Rinehart and Winston, New York, NY, 1968, Reprinted, Dover Publications, Mineola, NY, 2015. |
| [8] |
G. Sabidussi,
The composition of graphs, Duke Math. J., 26 (1959), 693-696.
doi: 10.1215/S0012-7094-59-02667-5. |
| [9] |
G. Sabidussi,
The lexicographic product of graphs, Duke Math. J., 28 (1961), 573-578.
doi: 10.1215/S0012-7094-61-02857-5. |
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. |
| [3] |
B. Alspach,
On point-symmetric tournaments, Canad. Math. Bull., 13 (1970), 317-323.
doi: 10.4153/CMB-1970-061-7. |
| [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. |
| [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. |
| [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. |
| [7] |
J. W. Moon, Topics on Tournaments, Holt, Rinehart and Winston, New York, NY, 1968, Reprinted, Dover Publications, Mineola, NY, 2015. |
| [8] |
G. Sabidussi,
The composition of graphs, Duke Math. J., 26 (1959), 693-696.
doi: 10.1215/S0012-7094-59-02667-5. |
| [9] |
G. Sabidussi,
The lexicographic product of graphs, Duke Math. J., 28 (1961), 573-578.
doi: 10.1215/S0012-7094-61-02857-5. |
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