# American Institute of Mathematical Sciences

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

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.  Google Scholar [3] B. Alspach, On point-symmetric tournaments, Canad. Math. Bull., 13 (1970), 317-323.  doi: 10.4153/CMB-1970-061-7.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] J. W. Moon, Topics on Tournaments, Holt, Rinehart and Winston, New York, NY, 1968, Reprinted, Dover Publications, Mineola, NY, 2015.  Google Scholar [8] G. Sabidussi, The composition of graphs, Duke Math. J., 26 (1959), 693-696.  doi: 10.1215/S0012-7094-59-02667-5.  Google Scholar [9] G. Sabidussi, The lexicographic product of graphs, Duke Math. J., 28 (1961), 573-578.  doi: 10.1215/S0012-7094-61-02857-5.  Google Scholar
 [1] Nadezhda Maltugueva, Nikolay Pogodaev. Modeling of crowds in regions with moving obstacles. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021066 [2] Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065 [3] Fuzhi Li, Dongmei Xu. Regular dynamics for stochastic Fitzhugh-Nagumo systems with additive noise on thin domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3517-3542. doi: 10.3934/dcdsb.2020244 [4] Baba Issa Camara, Houda Mokrani, Evans K. Afenya. Mathematical modeling of glioma therapy using oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 565-578. doi: 10.3934/mbe.2013.10.565 [5] Eduardo Casas, Christian Clason, Arnd Rösch. Preface special issue on system modeling and optimization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021008 [6] Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623 [7] Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393 [8] Brandy Rapatski, James Yorke. Modeling HIV outbreaks: The male to female prevalence ratio in the core population. Mathematical Biosciences & Engineering, 2009, 6 (1) : 135-143. doi: 10.3934/mbe.2009.6.135 [9] Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263 [10] Andrey Kovtanyuk, Alexander Chebotarev, Nikolai Botkin, Varvara Turova, Irina Sidorenko, Renée Lampe. Modeling the pressure distribution in a spatially averaged cerebral capillary network. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021016 [11] Qi Deng, Zhipeng Qiu, Ting Guo, Libin Rong. Modeling within-host viral dynamics: The role of CTL immune responses in the evolution of drug resistance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3543-3562. doi: 10.3934/dcdsb.2020245 [12] Wenbin Yang, Yujing Gao, Xiaojuan Wang. Diffusion modeling of tumor-CD4$^+$-cytokine interactions with treatments: asymptotic behavior and stationary patterns. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021090

Impact Factor: