In the classic prophet inequality, a well-known problem in optimal stopping theory, samples from independent random variables (possibly differently distributed) arrive online. A gambler who knows the distributions, but cannot see the future, must decide at each point in time whether to stop and pick the current sample or to continue and lose that sample forever. The goal of the gambler is to maximize the expected value of what she picks and the performance measure is the worst case ratio between the expected value the gambler gets and what a prophet that sees all the realizations in advance gets. In the late seventies, Krengel and Sucheston (Bull Am Math Soc 83(4):745–747, 1977), established that this worst case ratio is 0.5. A particularly interesting variant is the so-called prophet secretary problem, in which the only difference is that the samples arrive in a uniformly random order. For this variant several algorithms are known to achieve a constant of $$1-1/e \approx 0.632$$ and very recently this barrier was slightly improved by Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018). In this paper we introduce a new type of multi-threshold strategy, called blind strategy. Such a strategy sets a nonincreasing sequence of thresholds that depends only on the distribution of the maximum of the random variables, and the gambler stops the first time a sample surpasses the threshold of the stage. Our main result shows that these strategies can achieve a constant of 0.669 for the prophet secretary problem, improving upon the best known result of Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018), and even that of Beyhaghi et al. (Improved approximations for posted price and second price mechanisms. CoRR arXiv:1807.03435 , 2018) that works in the case in which the gambler can select the order of the samples. The crux of the result is a very precise analysis of the underlying stopping time distribution for the gambler’s strategy that is inspired by the theory of Schur-convex functions. We further prove that our family of blind strategies cannot lead to a constant better than 0.675. Finally we prove that no algorithm for the gambler can achieve a constant better than $$\sqrt{3}-1 \approx 0.732$$ , which also improves upon a recent result of Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018). This implies that the upper bound on what the gambler can get in the prophet secretary problem is strictly lower than what she can get in the i.i.d. case. This constitutes the first separation between the prophet secretary problem and the i.i.d. prophet inequality.

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先知秘书通过盲目的策略

在经典的先知不等式（最优停止理论中的一个众所周知的问题）中，来自独立随机变量（可能分布不同）的样本在线到达。一个知道分布但看不到未来的赌徒必须在每个时间点决定是停止并选择当前样本还是继续并永远失去该样本。赌徒的目标是最大化她选择的期望值，而绩效衡量标准是赌徒获得的期望值与提前看到所有实现的先知获得的期望值之间的最坏情况比率。在 70 年代后期，克伦格尔和苏切斯顿（Bull Am Math Soc 83(4):745–747, 1977）确定这个最坏情况的比率是 0.5。一个特别有趣的变体是所谓的先知秘书问题，其中唯一的区别是样本以均匀随机的顺序到达。对于这种变体，已知有几种算法可以实现 $$1-1/e \approx 0.632$$ 的常数，并且最近 Azar 等人略微改进了这一障碍。（在：ACM 经济学和计算会议论文集，EC，2018 年）。在本文中，我们介绍了一种新型的多阈值策略，称为盲策略。这种策略设置了一个不递增的阈值序列，该序列仅取决于随机变量最大值的分布，赌徒在样本第一次超过阶段阈值时停止。我们的主要结果表明，对于先知秘书问题，这些策略可以达到 0.669 的常数，改进了 Azar 等人最著名的结果。（在：ACM 经济学和计算会议论文集，EC，2018 年），甚至 Beyhaghi 等人的论文集。（改进了公布价格和第二价格机制的近似值。CoRR arXiv:1807.03435 , 2018）在赌徒可以选择样本顺序的情况下有效。结果的关键是对赌徒策略的潜在停止时间分布进行了非常精确的分析，其灵感来自 Schur-凸函数理论。我们进一步证明，我们的盲策略系列不能导致优于 0.675 的常数。最后，我们证明了赌徒的算法没有比 $$\sqrt{3}-1 \approx 0.732$$ 更好的常数，这也改进了 Azar 等人最近的结果。（在：ACM 经济学和计算会议论文集，EC，2018 年）。这意味着赌徒在先知秘书问题中可以获得的上限严格低于她在 iid 情况下可以获得的上限。这构成了先知秘书问题和 iid 先知不平等之间的第一次分离。