Winning Probabilities of Balanced and Nontransitive n-tuples of Dice

TL;DR

The paper resolves the problem of realized winning probabilities for balanced and nontransitive $n$-tuples of dice by showing that for every $n\ge 3$ and every rational $p\in\left(\tfrac{1}{2},\pi_n\right]$ (with $\pi_n=1-\frac{1}{4\cos^2(\frac{\pi}{n+2})}$) there exists a BN $n$-tuple achieving $w=p$, thereby extending the $n=3$ classification to all $n\ge 3$. The construction relies on a combinatorial framework of central words, enabling precise control of pairwise win counts through $N_\sigma(A_i<A_{i+1})$ and the corresponding probabilities. Central to the argument are two lemmas: mainLem1, which generalizes a prior result to $n\ge 4$ by inductively building central words with prescribed counts, and RationalExistenceLem, which guarantees the existence of rational parameter choices meeting a system of inequalities that encode the balanced/nontransitive constraints. Together with the known upper bound $w(A_1,\dots,A_n)\le\pi_n$ established by Bogdanov and Komisarski, the authors obtain a complete characterization of attainable winning probabilities for BN $n$-tuples, thereby fully answering the posed problem. The work also situates these results within the broader literature on nontransitive dice and suggests avenues for extending the side-count problem to more general $(m_1,\dots,m_n)$.

Abstract

For a positive integer $n$, an $n$-tuple of dice $(A_1,A_2,\dots,A_n)$ is called balanced if $P(A_1<A_2) = P(A_2<A_3) = \cdots = P(A_n<A_1)$ and nontransitive if $P(A_1<A_2), P(A_2<A_3), \dots, P(A_n<A_1)$ are each greater than $\frac{1}{2}$. For a balanced and nontransitive $n$-tuple of dice $(A_1,A_2,\dots,A_n)$, we define the winning probability $w(A_1,A_2,\dots,A_n) := P(A_1 < A_2)$. The works of Trybula and Kim et al. together show that for a balanced and nontransitve triple of dice $(A_1,A_2,A_3)$, the least upper bound on the winning probability is $\frac{-1+\sqrt{5}}{2}$. Kim et al. then asked what the least upper bound on the winning probability was for the $n \geq 4$ cases. Bogdanov and Komisarski independently have shown that for $n\geq 3$ and a balanced and nontransitive $n$-tuple of dice $(A_1,A_2,\dots,A_n)$, the winning probability is less than $π_n := 1-\frac{1}{4\cos^2\left( \fracπ{n+2} \right)}$. In this paper, we will show that for $n \geq 3$ and every rational $p \in \left( \frac{1}{2}, π_n \right]$, there exists a balanced and nontransitive $n$-tuple of dice with winning probability $p$. Paired with Bogdanov and Komisarski's results, this fully answers the problem posed by Kim et al. and establishes a complete characterization of the winning probabilities for nontransitive and balanced $n$-tuples of dice.

Figures (4)

• Figure 1: Efron Dice; $P(A_1 < A_2) = P(A_2 < A_3) = P(A_3 < A_4) = P(A_4 < A_1) = \frac{2}{3}$
• Figure 2: Modified Efron Dice
• Figure 3: A triple of BN 6-sided dice with winning probability $\frac{5}{9}$
• Figure 4: A triple of BN dice with winning probability $\frac{3}{5}$

Theorems & Definitions (19)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.4
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3