Winning Probabilities of Balanced and Nontransitive n-tuples of Dice

TL;DR

This paper characterizes the range of winning probabilities for balanced, nontransitive n-tuples of dice, showing that any rational probability within a specific interval can be achieved, thus fully solving a longstanding problem.

Contribution

It proves that all rational winning probabilities within the established bounds are attainable for balanced, nontransitive dice, completing the characterization of their probability spectrum.

Findings

Winning probability less than  for n=3

Extension of bounds to n for all n

Existence of balanced, nontransitive dice for all rational probabilities in the interval

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 $\pi_n := 1-\frac{1}{4\cos^2\left( \frac{\pi}{n+2} \right)}$. In this paper, we will show that for $n \geq 3$ and every rational $p \in \left( \frac{1}{2}, \pi_n \right]$, 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.