非传递骰子锦标赛不是拟随机的
Intransitive dice tournament is not quasirandom
AI总结：
本文证明四个随机骰子的非传递锦标赛并非拟随机，传递锦标赛概率严格高于3/8，解决了相关猜想。
摘要：
我们解决了Conrey、Gabbard、Grant、Liu和Morrison在2016年以及Polymath在2017年提出的关于非传递骰子的一个猜想版本。我们考虑具有$n$个面的广义骰子，并称骰子$A$击败$B$，如果从$A$中随机取一个面，其显示的数字大于从$B$中随机取一个面的数字的概率更高。我们研究随机骰子，其面独立同分布地从$[0,1]$上的均匀分布中抽取，并满足所有面的和等于$n/2$的条件。对于三个这样的随机骰子之间的“击败”关系，Polymath证明了八种可能的锦标赛中每一种在渐近意义下出现的概率相等。特别地，三个骰子形成非传递循环的概率收敛于$1/4$。在本文中，我们证明对于四个随机骰子，并非所有锦标赛都等可能，且传递锦标赛的概率严格高于$3/8$。
英文摘要：
We settle a version of the conjecture about intransitive dice posed by Conrey, Gabbard, Grant, Liu and Morrison in 2016 and Polymath in 2017. We consider generalized dice with $n$ faces and we say that a die $A$ beats $B$ if a random face of $A$ is more likely to show a higher number than a random face of $B$. We study random dice with faces drawn iid from the uniform distribution on $[0,1]$ and conditioned on the sum of the faces equal to $n/2$. Considering the "beats" relation for three such random dice, Polymath showed that each of eight possible tournaments between them is asymptotically equally likely. In particular, three dice form an intransitive cycle with probability converging to $1/4$. In this paper we prove that for four random dice not all tournaments are equally likely and the probability of a transitive tournament is strictly higher than $3/8$.
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