# American Institute of Mathematical Sciences

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, 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

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
 [1] Nicola Pace, Angelo Sonnino. On the existence of PD-sets: Algorithms arising from automorphism groups of codes. Advances in Mathematics of Communications, 2021, 15 (2) : 267-277. doi: 10.3934/amc.2020065 [2] Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, 2020, 3 (4) : 263-277. doi: 10.3934/mfc.2020010 [3] Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455 [4] Jean-Paul Chehab. Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021002 [5] Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 [6] Niklas Kolbe, Nikolaos Sfakianakis, Christian Stinner, Christina Surulescu, Jonas Lenz. Modeling multiple taxis: Tumor invasion with phenotypic heterogeneity, haptotaxis, and unilateral interspecies repellence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 443-481. doi: 10.3934/dcdsb.2020284 [7] Huijuan Song, Bei Hu, Zejia Wang. Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 667-691. doi: 10.3934/dcdsb.2020084

Impact Factor:

Article outline