论文标题
不及事的骰子锦标赛不是quasirandom
Intransitive dice tournament is not quasirandom
论文作者
论文摘要
我们定居于2016年由Conrey，Gabbard，Grant，Liu和Morrison提出的关于不及物骰子的猜想，以及2017年的Polymath。我们考虑使用$ n $ faces的广义骰子，并且我们说，如果随机的$ a $比随机的fac a a $ a $ a $ a $ a $ a $ beats $ b $比$ b $ b的随机数字更高。我们研究了从$ [0,1] $上的统一分布中抽出的面孔的随机骰子，并以等于$ n/2 $的面孔为条件。考虑到三个这样的随机骰子的“节拍”关系，polymath表明它们之间的八场可能的比赛在渐近上同样可能。特别是，三个骰子形成了一个不及物周期，概率收敛到$ 1/4 $。在本文中，我们证明，对于四个随机骰子而言，并非所有比赛都可能同样可能，并且瞬时比赛的可能性严格高于$ 3/8 $。
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 $3/8$.
