We introduce the \textit{prophet inequality with uncertain acceptance} model, in which a decision maker sequentially observes a sequence of independent options, each characterized by a value $x_i$ and an acceptance probability $p_i$, both sampled from a known joint distribution. At time $i$, the decision maker observes the value $x_i$ and must irrevocably and immediately decide whether to attempt to select it or to continue to the next time step. If the option is selected, the process terminates with probability $p_i$ and the decision maker obtains $x_i$; otherwise, she continues searching. In this setting, two natural benchmarks arise: the \textit{value-aware decision-maker}, who knows all value realizations in advance but not the acceptance outcomes, and the \textit{full-knowledge prophet}, who knows all realizations beforehand and can choose the best option among those that will be accepted. We characterize the worst-case competitive ratios between our defined agents and show that all these values equal $1/2$. In addition, we provide sufficient conditions under which the value-aware decision-maker surpasses the $1/2$-barrier against the more informed prophet. This demonstrates the (crucial) interest for the decision maker to improve her knowledge over the values rather than over the acceptances, and is obtained via a more general result that reduces the value-aware decision-maker's problem to a classical prophet inequality with scaled Bernoulli distributions, followed by a sequence of transformations that further reduce the problem to a unique optimization problem.

翻译：我们提出“不确定接受率的先知不等式”模型，在该模型中，决策者按顺序观察一系列独立选项，每个选项由一个值$x_i$和一个接受概率$p_i$表征，两者均从已知的联合分布中采样。在时间$i$，决策者观察到值$x_i$，必须立即做出不可撤销的决定：是尝试选择该选项，还是继续到下一个时间步。如果选项被选中，过程以概率$p_i$终止，决策者获得$x_i$；否则，她继续搜索。在此设定中，出现了两个自然的基准：一是“知情价值决策者”，她事先知道所有值的实现但不知道接受结果；二是“全知先知”，她事先知道所有实现，并能从那些将被接受的选项中选择最优者。我们刻画了所定义智能体之间最坏情况下的竞争比，并证明所有这些值都等于$1/2$。此外，我们提供了充分条件，使得知情价值决策者在面对信息更全面的先知时能够超越$1/2$的屏障。这表明决策者提升其对值的知识比提升对接受率的知识具有（关键的）益处，此结论通过一个更一般的结果获得，该结果将知情价值决策者的问题简化为一个经典先知不等式与缩放伯努利分布相结合的问题，随后通过一系列变换进一步将问题简化为一个唯一的优化问题。