# American Institute of Mathematical Sciences

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

## Generalized intransitive dice: Mimicking an arbitrary tournament

 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 $R$.

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:
 [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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##### 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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