Prophet inequalities compare online stopping strategies against an omniscient "prophet" using distributional knowledge. In this work, we augment this model with a conservative prediction of the maximum realized value. We quantify the quality of this prediction using a parameter $α\in [0,1]$, ranging from inaccurate to perfect. Our goal is to improve performance when predictions are accurate (consistency) while maintaining theoretical guarantees when they are not (robustness). We propose a threshold-based strategy oblivious to $α$ (i.e., with $α$ unknown to the algorithm) that matches the classic competitive ratio of $1/2$ at $α=0$ and improves smoothly to $3/4$ at $α=1$. We further prove that simultaneously achieving better than $3/4$ at $α=1$ while maintaining $1/2$ at $α=0$ is impossible. Finally, when $α$ is known in advance, we present a strategy achieving a tight competitive ratio of $\frac{1}{2-α}$.

翻译：先知不等式利用分布知识比较在线停止策略与全知“先知”的性能。在本工作中，我们通过引入对已实现最大值的保守预测来扩展该模型。我们使用参数$α\in [0,1]$量化该预测的质量，其范围从完全不准确到完全准确。我们的目标是在预测准确时提升性能（一致性），同时在预测不准确时保持理论保证（鲁棒性）。我们提出了一种不依赖于$α$的阈值策略（即算法未知$α$），该策略在$α=0$时达到经典的$1/2$竞争比，并随着$α$增大平滑提升至$α=1$时的$3/4$。我们进一步证明，若要在$α=1$时获得优于$3/4$的竞争比，同时保持在$α=0$时的$1/2$竞争比，是不可能的。最后，当$α$可提前获知时，我们提出一种策略能达到紧致的$\frac{1}{2-α}$竞争比。