-
Previous Article
Permanence in polymatrix replicators
- JDG Home
- This Issue
- Next Article
Generalized intransitive dice: Mimicking an arbitrary tournament
Mathematics Department, The City College, 137 Street and Convent Avenue, New York City, NY 10031, USA |
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 $.
References:
| [1] |
E. Akin, Rock, paper, scissors, etc - topics in the theory of regular tournaments, preprint, arXiv: 1806.11241v1, (2018), v4(2020). |
| [2] |
E. Akin, Intransitive dice - mimicking an arbitrary tournament, preprint, arXiv: 1901.09477v2, (2019). |
| [3] |
J. K. Blitzstein and J. Hwang, Introduction to Probability, CRC Press, Boca Raton, 2014.
|
| [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. |
| [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. |
| [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. |
| [7] |
G. G. Margaril-Ilyaev and V. M. Tikhomirov, Convex Analysis: Theory and Applications, Translations of Math. Monograph 222, Amer. Math. Soc., Providence, 2003. |
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). |
| [2] |
E. Akin, Intransitive dice - mimicking an arbitrary tournament, preprint, arXiv: 1901.09477v2, (2019). |
| [3] |
J. K. Blitzstein and J. Hwang, Introduction to Probability, CRC Press, Boca Raton, 2014.
|
| [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. |
| [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. |
| [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. |
| [7] |
G. G. Margaril-Ilyaev and V. M. Tikhomirov, Convex Analysis: Theory and Applications, Translations of Math. Monograph 222, Amer. Math. Soc., Providence, 2003. |
| [1] |
Ethan Akin, Julia Saccamano. Generalized intransitive dice II: Partition constructions. Journal of Dynamics and Games, 2021, 8 (3) : 187-202. doi: 10.3934/jdg.2021005 |
| [2] |
Jan J. Dijkstra and Jan van Mill. Homeomorphism groups of manifolds and Erdos space. Electronic Research Announcements, 2004, 10: 29-38. |
| [3] |
Christian Licht, Thibaut Weller. Approximation of semi-groups in the sense of Trotter and asymptotic mathematical modeling in physics of continuous media. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1709-1741. doi: 10.3934/dcdss.2019114 |
| [4] |
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 and Continuous Dynamical Systems - B, 2019, 24 (10) : 5569-5596. doi: 10.3934/dcdsb.2019072 |
| [5] |
Ludovic Rifford. Ricci curvatures in Carnot groups. Mathematical Control and Related Fields, 2013, 3 (4) : 467-487. doi: 10.3934/mcrf.2013.3.467 |
| [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 and 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] |
Robert McOwen, Peter Topalov. Groups of asymptotic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6331-6377. doi: 10.3934/dcds.2016075 |
| [10] |
Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725 |
| [11] |
Hans Ulrich Besche, Bettina Eick and E. A. O'Brien. The groups of order at most 2000. Electronic Research Announcements, 2001, 7: 1-4. |
| [12] |
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 |
| [13] |
Marc Peigné. On some exotic Schottky groups. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 559-579. doi: 10.3934/dcds.2011.31.559 |
| [14] |
Paul Skerritt, Cornelia Vizman. Dual pairs for matrix groups. Journal of Geometric Mechanics, 2019, 11 (2) : 255-275. doi: 10.3934/jgm.2019014 |
| [15] |
Uri Bader, Alex Furman. Boundaries, Weyl groups, and Superrigidity. Electronic Research Announcements, 2012, 19: 41-48. doi: 10.3934/era.2012.19.41 |
| [16] |
Javier Pérez Álvarez. Invariant structures on Lie groups. Journal of Geometric Mechanics, 2020, 12 (2) : 141-148. doi: 10.3934/jgm.2020007 |
| [17] |
Martin Kassabov. Symmetric groups and expanders. Electronic Research Announcements, 2005, 11: 47-56. |
| [18] |
André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351 |
| [19] |
Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517 |
| [20] |
Elon Lindenstrauss. Pointwise theorems for amenable groups. Electronic Research Announcements, 1999, 5: 82-90. |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]