Generalized intransitive dice II: Partition constructions

Ethan Akin; Julia Saccamano

doi:10.3934/jdg.2021005

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2021, Volume 8, Issue 3: 187-202. Doi: 10.3934/jdg.2021005

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Generalized intransitive dice II: Partition constructions

Ethan Akin and
Julia Saccamano

Mathematics Department, The City College, 137 Street and Convent Avenue, New York City, NY 10031, USA

Received: August 31, 2020

Early access: March 2021

Published: July 2021

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Abstract

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} $.

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Mathematics Subject Classification: 05C20, 05C25, 05C62, 05C70.

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References

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