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July  2021, 8(3): 187-202. doi: 10.3934/jdg.2021005

## Generalized intransitive dice II: Partition constructions

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

Received  September 2020 Published  July 2021 Early access  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 $N = 3^{n-2}$.

Citation: Ethan Akin, Julia Saccamano. Generalized intransitive dice II: Partition constructions. Journal of Dynamics and Games, 2021, 8 (3) : 187-202. 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). [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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##### References:
 [1] E. Akin, Rock, paper, scissors, etc. - topics in the theory of regular tournaments, arXiv: 1806.11241v1, (2018), v4(2020). [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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