论文标题
关于循环和非传播概率
On cyclic and nontransitive probabilities
论文作者
论文摘要
以经典的非过渡性悖论的激励，我们称为$ n $ -tuple $（x_1，\ dots，x_n）\ in [0,1]^n $ \ textit {cyclia {cyclia {cyclia}如果存在独立随机变量$ $ p（u_ {i+1}> u_i）= x_i $ for $ i = 1，\ dots，n-1 $和$ p（u_1> u_n）= x_n $。如果它是循环的，我们将元组$（x_1，\ dots，x_n）$ \ textit {nortransitive}称为$ \ textit {nontransitive}，此外还满足所有$ i $的$ x_i> 1/2 $。 令$ p_n $（resp。〜$ p_n^*$）表示随机选择的$ n $ -tuple $（x_1，\ dots，x_n）\ in [0,1]^n $是环状的（scrip。〜〜非发行）。我们确定$ p_3 $和$ p_3^*$，而对于$ n \ ge4 $，我们给出了$ p_n $的上限和下限，这表明$ p_n $收敛到$ 1 $ as $ n \ to \ infty $。我们还确定了循环三重中最小，中和最大元素的分布。
Motivated by classical nontransitivity paradoxes, we call an $n$-tuple $(x_1,\dots,x_n) \in[0,1]^n$ \textit{cyclic} if there exist independent random variables $U_1,\dots, U_n$ with $P(U_i=U_j)=0$ for $i\not=j$ such that $P(U_{i+1}>U_i)=x_i$ for $i=1,\dots,n-1$ and $P(U_1>U_n)=x_n$. We call the tuple $(x_1,\dots,x_n)$ \textit{nontransitive} if it is cyclic and in addition satisfies $x_i>1/2$ for all $i$. Let $p_n$ (resp.~$p_n^*$) denote the probability that a randomly chosen $n$-tuple $(x_1,\dots,x_n)\in[0,1]^n$ is cyclic (resp.~nontransitive). We determine $p_3$ and $p_3^*$ exactly, while for $n\ge4$ we give upper and lower bounds for $p_n$ that show that $p_n$ converges to $1$ as $n\to\infty$. We also determine the distribution of the smallest, middle, and largest elements in a cyclic triple.
