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：我们引入了一种名为“不确定接受率下的先知不等式”模型。在该模型中，决策者依次观察一系列独立的选项，每个选项由从已知联合分布中抽取的一个价值 $x_i$ 和一个接受概率 $p_i$ 共同表征。在时刻 $i$，决策者观察到价值 $x_i$ 后，必须立即且不可撤销地决定是尝试选择该选项，还是继续进入下一个时间步。若选择该选项，过程将以概率 $p_i$ 终止，决策者获得价值 $x_i$；否则，她将继续搜索。在此设定下，出现了两个自然的基准：一是“价值感知型决策者”，她事先知晓所有价值的实现结果，但不知道接受结果；二是“全知先知”，她事先知晓所有实现结果，并能选择那些将被接受的选项中最佳的一个。我们刻画了所定义主体间的竞争比的最坏情况，并证明这些比值均等于 $1/2$。此外，我们给出了价值感知型决策者在与信息更充分的先知博弈中超越 $1/2$ 这一界限的充分条件。这表明决策者提升自身对价值（而非接受情况）的知识具有关键意义。该结果通过一个更通用的结论得出：该结论首先将价值感知型决策者的问题归结为具有缩放伯努利分布的经典先知不等式问题，随后通过一系列变换进一步将问题简化至一个唯一的优化问题。