Possible probability and irreducibility of balanced non-transitive dice

A Direct Construction of Non-Transitive Dice Sets

In this paper, we give a direct construction for a set of dice realizing any given tournament $T$. The construction for a tournament with $n$ vertices requires a number of sides on the order of $n$, which appears to be the best general construction to date. Our construction relies only on a standard theorem from graph theory.

随机骰子三元组具有传递性的概率

The probability that a random triple of dice is transitive

针对随机n面骰子的传递性问题，本文证明在自然模型中，三个随机骰子A击败B且B击败C时，A击败C的概率约为1/2，验证了Conrey等人的猜想。

The Probability of Intransitivity in Dice and Close Elections

We study the phenomenon of intransitivity in models of dice and voting. First, we follow a recent thread of research for $n$-sided dice with pairwise ordering induced by the probability, relative to $1/2$, that a throw from one die is higher than the other. We build on a recent result of Polymath showing that three dice with i.i.d. faces drawn from the uniform distribution on $\{1,\ldots,n\}$ and conditioned on the average of faces equal to $(n+1)/2$ are intransitive with asymptotic probability $1/4$. We show that if dice faces are drawn from a non-uniform continuous mean zero distribution con…