AI 中文总结
本文研究Steck提出的对称群概率分布，给出其基于排列统计量的新组合公式，定义带过滤步骤的最优选择博弈，证明最优胜率渐近为1/e且最优策略与经典模型不同。
AI 中文摘要
我们研究了由Steck在20世纪70年代初提出的对称群上的一种显著概率分布，该分布源于一种自然过程，分别对排列的数值和位置进行连续与离散选择的交织。Steck用矩阵行列式来表示他的分布，而我们则针对该分布，基于“自底向上最大值”（即逆排列的自左向右最大值）排列统计量，提出了新的组合公式。这些公式在恒等排列的情况下，可特例化为Pitman–Stanley在20世纪90年代末的一项结果。随后，我们利用Steck分布定义了一种最优选择博弈（秘书问题的变体），该博弈会随时间对候选池引入过滤过程。我们针对存在单次过滤步骤的情形求解了该模型。结果表明，在最优策略下赢得游戏的概率与经典模型（渐近为1/e）相近，但面试官必须采用不同的（非位置性）策略才能达到该概率。最优策略的决策取决于面试位置与过滤步骤后下一个自底向上最大值之间的关系。在其他结果中，我们证明，最优策略在拒绝与接受之间的转换位置，始终处于总候选数比例的1/e与1/e + (1 - 1/e)y之间，其中y为未过滤候选的比例。
英文摘要
We investigate a remarkable probability distribution on the symmetric group, due to Steck from the early 1970's, arising from a natural process that intertwines continuous and discrete selections for the values and positions, respectively, of a permutation. Steck used a matrix determinant to express his distribution, whereas we contribute new combinatorial formulas for it in terms of "bottom-to-top maxima" (that are simply the left-to-right maxima of the inverse) permutation statistics. These formulas specialize, in the case of the identity permutation, to a result of Pitman--Stanley from the late 1990's. We then use the Steck distribution to define a game of best choice (secretary problem variation) that incorporates a filtering process for the pool of candidates over time. We solve the model for the case where there is a single filtering step. It turns out that the probability of winning the game under optimal play is similar to the classical model ($1/e$, asymptotically), but that the interviewer must employ a different (non-positional) strategy in order to attain it. The optimal strategy depends on the relationship between interview position and the next bottom-to-top maximum value after the filtering step. Among other results, we prove that the optimal strategy always transitions from rejection to acceptance between positions $(1/e)$ and $(1/e) + (1 - 1/e)y$ as a proportion of the total candidates considered, where $y$ is the proportion of unfiltered candidates.
Comments35 pages
