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. ConreyJ. GabbardK. GrantA. 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. ConreyJ. GabbardK. GrantA. 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]

Jan J. Dijkstra and Jan van Mill. Homeomorphism groups of manifolds and Erdos space. Electronic Research Announcements, 2004, 10: 29-38.

[2]

Christian Licht, Thibaut Weller. Approximation of semi-groups in the sense of Trotter and asymptotic mathematical modeling in physics of continuous media. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1709-1741. doi: 10.3934/dcdss.2019114

[3]

Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5569-5596. doi: 10.3934/dcdsb.2019072

[4]

Ludovic Rifford. Ricci curvatures in Carnot groups. Mathematical Control & Related Fields, 2013, 3 (4) : 467-487. doi: 10.3934/mcrf.2013.3.467

[5]

Eduard Duryev, Charles Fougeron, Selim Ghazouani. Dilation surfaces and their Veech groups. Journal of Modern Dynamics, 2019, 14: 121-151. doi: 10.3934/jmd.2019005

[6]

Sergei V. Ivanov. On aspherical presentations of groups. Electronic Research Announcements, 1998, 4: 109-114.

[7]

Benjamin Weiss. Entropy and actions of sofic groups. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3375-3383. doi: 10.3934/dcdsb.2015.20.3375

[8]

Neal Koblitz, Alfred Menezes. Another look at generic groups. Advances in Mathematics of Communications, 2007, 1 (1) : 13-28. doi: 10.3934/amc.2007.1.13

[9]

Emmanuel Breuillard, Ben Green, Terence Tao. Linear approximate groups. Electronic Research Announcements, 2010, 17: 57-67. doi: 10.3934/era.2010.17.57

[10]

Robert McOwen, Peter Topalov. Groups of asymptotic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6331-6377. doi: 10.3934/dcds.2016075

[11]

Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725

[12]

Elon Lindenstrauss. Pointwise theorems for amenable groups. Electronic Research Announcements, 1999, 5: 82-90.

[13]

Hans Ulrich Besche, Bettina Eick and E. A. O'Brien. The groups of order at most 2000. Electronic Research Announcements, 2001, 7: 1-4.

[14]

Światosław R. Gal, Jarek Kędra. On distortion in groups of homeomorphisms. Journal of Modern Dynamics, 2011, 5 (3) : 609-622. doi: 10.3934/jmd.2011.5.609

[15]

Marc Peigné. On some exotic Schottky groups. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 559-579. doi: 10.3934/dcds.2011.31.559

[16]

Paul Skerritt, Cornelia Vizman. Dual pairs for matrix groups. Journal of Geometric Mechanics, 2019, 11 (2) : 255-275. doi: 10.3934/jgm.2019014

[17]

Javier Pérez Álvarez. Invariant structures on Lie groups. Journal of Geometric Mechanics, 2020, 12 (2) : 141-148. doi: 10.3934/jgm.2020007

[18]

Uri Bader, Alex Furman. Boundaries, Weyl groups, and Superrigidity. Electronic Research Announcements, 2012, 19: 41-48. doi: 10.3934/era.2012.19.41

[19]

Martin Kassabov. Symmetric groups and expanders. Electronic Research Announcements, 2005, 11: 47-56.

[20]

Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517

 Impact Factor: 

Article outline

[Back to Top]