SIAM Journal on Discrete Mathematics, Volume 34, Issue 1, Page 399-414, January 2020.
The game of best choice (also known as the secretary problem) is a model for sequential decision making with a long history and many variations. The classical setup assumes that the sequence of candidate rankings is uniformly distributed. Given a statistic on the symmetric group, one can instead weight each permutation according to an exponential function in the statistic. We play the game of best choice on the Ewens and Mallows distributions that are obtained in this way from the number of left-to-right maxima and number of inversions in the permutation, respectively. For each of these, we give the optimal strategy and probability of winning. Moreover, we introduce a general class of permutation statistics that always produces games of best choice whose optimal strategies are positional, which simplifies their analysis considerably.

中文翻译：

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最佳选择加权游戏

SIAM离散数学杂志，第34卷，第1期，第399-414页，2020年1月。
最佳选择的博弈（也称为秘书问题）是具有悠久历史和众多变化的顺序决策模型。经典设置假定候选等级的序列是均匀分布的。给定对称组的统计信息，可以根据统计信息中的指数函数加权每个排列。我们在Ewens和Mallows分布上进行最佳选择的游戏，这种分布是通过这种方式分别从左到右的最大值和排列中的反转数获得的。对于每种方法，我们都给出了最佳策略和获胜的可能性。此外，我们介绍了一般的排列统计类别，该类别总是生成最佳选择的游戏，其最佳策略为位置游戏，这大大简化了它们的分析。