In a prophet inequality problem, $n$ independent random variables are presented to a gambler one by one. The gambler decides when to stop the sequence and obtains the most recent value as reward. We evaluate a stopping rule by the worst-case ratio between its expected reward and the expectation of the maximum variable. In the classic setting, the order is fixed, and the optimal ratio is known to be 1/2. Three variants of this problem have been extensively studied: the prophet-secretary model, where variables arrive in uniformly random order; the free-order model, where the gambler chooses the arrival order; and the i.i.d. model, where the distributions are all the same, rendering the arrival order irrelevant. Most of the literature assumes that distributions are known to the gambler. Recent work has considered the question of what is achievable when the gambler has access only to a few samples per distribution. Surprisingly, in the fixed-order case, a single sample from each distribution is enough to approximate the optimal ratio, but this is not the case in any of the three variants. We provide a unified proof that for all three variants of the problem, a constant number of samples (independent of n) for each distribution is good enough to approximate the optimal ratios. Prior to our work, this was known to be the case only in the i.i.d. variant. We complement our result showing that our algorithms can be implemented in polynomial time. A key ingredient in our proof is an existential result based on a minimax argument, which states that there must exist an algorithm that attains the optimal ratio and does not rely on the knowledge of the upper tail of the distributions. A second key ingredient is a refined sample-based version of a decomposition of the instance into "small" and "large" variables, first introduced by Liu et al. [EC'21].

翻译：在先知不等式问题中，$n$个独立随机变量依次呈现给一个赌徒。赌徒决定何时停止序列，并获得最近的值作为奖励。我们通过期望奖励与最大变量期望值之间的最坏情况比率来评估停止规则。在经典设定中，顺序是固定的，已知最优比率为1/2。该问题的三种变体已被广泛研究：先知-秘书模型（变量以均匀随机顺序到达）、自由顺序模型（赌徒选择到达顺序）以及独立同分布模型（所有分布相同，到达顺序无关）。大多数文献假设赌徒已知分布。近期研究探讨了当赌徒仅能访问每个分布的少量样本时能实现什么。令人惊讶的是，在固定顺序情况下，每个分布的一个样本就足以逼近最优比率，但这在三种变体中都并非如此。我们提供了一个统一证明：对于该问题的所有三种变体，每个分布的常数个样本（与n无关）足以逼近最优比率。在我们工作之前，已知这仅在独立同分布变体中成立。我们补充了结果，表明我们的算法可以在多项式时间内实现。证明中的一个关键成分是基于极小极大论证的存在性结果，该结果表明必然存在一种达到最优比率且不需要依赖分布上尾信息的算法。第二个关键成分是实例分解为“小”和“大”变量的精细化基于样本版本，该分解最初由Liu等人[EC'21]提出。