Balanced Non-Transitive Dice II: Tournaments

2017/06/27 by Alex Schaefer, Schaefer, Alex · 2 citations
Mathematics · Physics and Astronomy · #05A99 #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.1706.08986

openalex publication_date 2017/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We further study sets of labeled dice in which the relation "is a better die than" is non-transitive. Focusing on sets with an additional symmetry we call "balance," we prove that sets of n such m-sided dice exist for all n,m ≥ 3. We then show how to construct a set of n dice such that the relation behaves according to the direction of the arrows of any tournament (complete directed graph) on n vertices.

Cited by

• Winning Probabilities of Balanced and Nontransitive n-tuples of Dice
• Sample Complexity of Identifying the Nonredundancy of Nontransitive Games in Dueling Bandits

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