Prophet inequalities for rewards maximization are fundamental to optimal stopping theory with extensive applications to mechanism design and online optimization. We study the \emph{cost minimization} counterpart of the classical prophet inequality: a decision maker is facing a sequence of costs $X_1, X_2, \dots, X_n$ drawn from known distributions in an online manner and \emph{must} ``stop'' at some point and take the last cost seen. The goal is to compete with a ``prophet'' who can see the realizations of all $X_i$'s upfront and always select the minimum, obtaining a cost of $\mathbb{E}[\min_i X_i]$. If the $X_i$'s are not identically distributed, no strategy can achieve a bounded approximation, even for random arrival order and $n = 2$. This leads us to consider the case where the $X_i$'s are independent and identically distributed (I.I.D.). For the I.I.D. case, we show that if the distribution satisfies a mild condition, the optimal stopping strategy achieves a (distribution-dependent) constant-factor approximation to the prophet's cost. Moreover, for MHR distributions, this constant is at most $2$. All our results are tight. We also demonstrate an example distribution that does not satisfy the condition and for which the competitive ratio of any algorithm is infinite. Turning our attention to single-threshold strategies, we design a threshold that achieves a $O\left(polylog{n}\right)$-factor approximation, where the exponent in the logarithmic factor is a distribution-dependent constant, and we show a matching lower bound. Finally, we note that our results can be used to design approximately optimal posted price-style mechanisms for procurement auctions which may be of independent interest. Our techniques utilize the \emph{hazard rate} of the distribution in a novel way, allowing for a fine-grained analysis which could find further applications in prophet inequalities.

翻译：奖励最大化的先知不等式是最优停止理论的基础，在机制设计和在线优化中有广泛应用。我们研究了经典先知不等式的对应问题——成本最小化：决策者面临一系列按在线方式从已知分布中抽取的成本$X_1, X_2, \dots, X_n$，必须选择某一“停止”时刻并接受最后观测到的成本。目标是与一位“先知”竞争——该先知能预先获知所有$X_i$的实现值，并始终选择最小值，从而获得成本$\mathbb{E}[\min_i X_i]$。若$X_i$非同分布，即使随机到达顺序且$n=2$，也无策略能实现有界近似。这促使我们考虑$X_i$独立同分布（I.I.D.）的情形。对于I.I.D.情形，我们证明若分布满足温和条件，最优停止策略可实现与先知成本相差常数因子（依赖分布）的近似。特别地，对于MHR分布，该常数不超过2。所有结果均为紧界。我们还构造了一个不满足该条件的示例分布，使得任意算法的竞争比无穷大。针对单阈值策略，我们设计了一个阈值，可实现$O\left(polylog{n}\right)$因子近似（对数因子的指数为分布相关常数），并证明了匹配的下界。最后，我们的结果可用于设计采购拍卖中近似最优的标价机制，这或许具有独立意义。我们的技术方法新颖地利用了分布的\textit{风险率}，实现了精细分析，可望在先知不等式中获得进一步应用。