Given n random variables $X_1, \ldots , X_n$ taken from known distributions, a gambler observes their realizations in this order, and needs to select one of them, immediately after it is being observed, so that its value is as high as possible. The classical prophet inequality shows a strategy that guarantees a value at least half (in expectation) of that an omniscience prophet that picks the maximum, and this ratio is tight. Esfandiari, Hajiaghayi, Liaghat, and Monemizadeh introduced a variant of the prophet inequality, the prophet secretary problem in [1]. The difference being that that the realizations arrive at a random permutation order, and not an adversarial order. Esfandiari et al. gave a simple $1-1/e \approx 0.632$ competitive algorithm for the problem. This was later improved in a surprising result by Azar, Chiplunkar and Kaplan [2] into a $1-1/e + 1/400 \approx 0.634$ competitive algorithm. In a subsequent result, Correa, Saona, and Ziliotto [3] took a systematic approach, introducing blind strategies, and gave an improved $0.669$ competitive algorithm. Since then, there has been no improvements on the lower bounds. Meanwhile, current upper bounds show that no algorithm can achieve a competitive ratio better than $0.7235$ [4]. In this paper, we give a $0.6724$-competitive algorithm for the prophet secretary problem. The algorithm follows blind strategies introduced by [3] but has a technical difference. We do this by re-interpretting the blind strategies, framing them as Poissonization strategies. We break the non-iid random variables into iid shards and argue about the competitive ratio in terms of events on shards. This gives significantly simpler and direct proofs, in addition to a tighter analysis on the competitive ratio. The analysis might be of independent interest for similar problems such as the prophet inequality with order-selection

翻译：给定来自已知分布的 $n$ 个随机变量 $X_1, \ldots, X_n$，一个博弈者按其顺序观察它们的实现值，并需要在观察到每个值后立即做出选择，以使得所选值尽可能大。经典先知不等式给出了一种策略，其保证期望值至少达到全知先知（选择最大值）的一半，且该比率是紧的。Esfandiari、Hajiaghayi、Liaghat 和 Monemizadeh 在文献 [1] 中引入了先知不等式的一个变体——先知秘书问题。区别在于实现值以随机排列顺序而非对抗性顺序到达。Esfandiari 等人为该问题提出了一个简单的 $1-1/e \approx 0.632$ 竞争比算法。随后，Azar、Chiplunkar 和 Kaplan 在文献 [2] 中给出了一个令人惊讶的结果，将其改进为 $1-1/e + 1/400 \approx 0.634$ 竞争比算法。在后续工作中，Correa、Saona 和 Ziliotto 在文献 [3] 中采用系统方法，引入了盲策略，并提出了一个改进的 $0.669$ 竞争比算法。此后，下界方面再无改进。与此同时，当前上界表明没有算法能实现优于 $0.7235$ 的竞争比 [4]。在本文中，我们为先知秘书问题提出了一个 $0.6724$ 竞争比算法。该算法遵循文献 [3] 中引入的盲策略，但存在技术差异。我们通过重新解释盲策略，将其构建为泊松化策略来实现这一点。我们将非独立同分布的随机变量分解为独立同分布的分片，并基于分片上的事件论证竞争比。这带来了显著更简单且直接的证明，同时竞争比分析也更加紧致。该分析可能对顺序选择先知不等式等类似问题具有独立参考价值。