Prophet Inequality with Conservative Prediction

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 $\alpha \in [0,1]$, 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 $\alpha$ (i.e., with $\alpha$ unknown to the algorithm) that matches the classic competitive ratio of $1/2$ at $\alpha=0$ and improves smoothly to $3/4$ at $\alpha=1$. We further prove that simultaneously achieving better than $3/4$ at $\alpha=1$ while maintaining $1/2$ at $\alpha=0$ is impossible. Finally, when $\alpha$ is known in advance, we present a strategy achieving a tight competitive ratio of $\frac{1}{2-\alpha}$.