带有不确定接受的先知不等式
Prophet Inequalities with Delayed and Uncertain Acceptance
AI总结：
研究了带有不确定接受的先知不等式模型，分析了价值意识决策者与全知先知的竞争比，证明两者均为1/2，并探讨了价值意识决策者在特定条件下超越1/2阈值的条件。
摘要：
我们引入了带有不确定接受的先知不等式模型，在此模型中，决策者依次观察一系列独立选项，每个选项由价值x_i和接受概率p_i定义，两者均从已知的联合分布中采样。在时间i，决策者观察到x_i并必须立即且不可撤销地决定是否尝试选择该选项或继续到下一步。如果选择该选项，过程以概率p_i终止，决策者获得x_i；否则，继续搜索。在此设定中，两种自然的基准出现：价值意识决策者知道所有价值实现但不知道接受结果，全知先知知道所有实现并可选择最佳接受选项。我们刻画了所定义代理之间的最坏竞争比，并证明所有这些值均为1/2。此外，我们提供了充分条件，使得价值意识决策者在更知情的先知面前超越1/2阈值。这展示了决策者改进价值知识而非接受结果的重要性，并通过更一般性的结果将价值意识决策者的问题转化为经典先知不等式问题，随后通过一系列变换将其简化为一个独特的优化问题。
英文摘要：
We introduce the *prophet inequality with delayed and uncertain acceptance* model, a variant of the classical prophet inequality in which a decision-maker sequentially evaluates options whose acceptance is uncertain and revealed only after a random delay. At each time step, the decision-maker observes the realized value of the arriving option and must irrevocably decide whether to select it or continue searching. Once an option is selected, the process is suspended for a delay drawn from a known distribution with expectation $d$, during which no other option can be considered. When the delay expires, the selection succeeds with a known probability. If it succeeds, the decision-maker receives the realized value and the process terminates; otherwise, the search resumes. In addition to the online decision-maker, we consider two stronger benchmarks: the value-aware decision-maker, who knows all value realizations in advance but not the acceptance outcomes, and the prophet, who knows both the values and the acceptance outcomes. We show that, in the worst case, both decision-makers achieve a $1/(d + 2)$ fraction of the prophet's value, while the decision-maker achieves half of the value-aware decision-maker's value. We then turn to the no-delay case, where we establish an equivalence between our model and the classical prophet inequality, and show that the value-aware decision-maker beats the $1/2$ barrier against the prophet whenever all acceptance probabilities are bounded away from $0$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
