    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:
  E. Akin, Rock, paper, scissors, etc - topics in the theory of regular tournaments, preprint, arXiv: 1806.11241v1, (2018), v4(2020). Google Scholar  E. Akin, Intransitive dice - mimicking an arbitrary tournament, preprint, arXiv: 1901.09477v2, (2019). Google Scholar  J. K. Blitzstein and J. Hwang, Introduction to Probability, CRC Press, Boca Raton, 2014. Google Scholar  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  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  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  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:
  E. Akin, Rock, paper, scissors, etc - topics in the theory of regular tournaments, preprint, arXiv: 1806.11241v1, (2018), v4(2020). Google Scholar  E. Akin, Intransitive dice - mimicking an arbitrary tournament, preprint, arXiv: 1901.09477v2, (2019). Google Scholar  J. K. Blitzstein and J. Hwang, Introduction to Probability, CRC Press, Boca Raton, 2014. Google Scholar  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  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  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  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
  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  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  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  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  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  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  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