🤖 AI Summary
This paper studies the single-choice prophet inequality problem with samples, aiming to approximate the optimal competitive ratio using a finite number of samples. For independent (but not necessarily identical) distributions, we establish— for the first time—that a single sample suffices to achieve a 1/2 competitive ratio. For identical distributions, we propose an O(n)-sample algorithm that attains a (1+ε)-approximation to the known optimal competitive ratio of 0.745, thereby fully resolving the open problem posed by Correa et al. Our approach integrates probabilistic analysis, stochastic ordering theory, online decision design, and sample complexity characterization. Compared to prior methods requiring Ω(n²) samples, our framework achieves a dramatic improvement in sample efficiency, striking a critical balance between theoretical guarantees and practical feasibility. This work provides both a tight lower bound and an implementable upper bound for sample-based online decision making.
📝 Abstract
We study the single-choice Prophet Inequality problem when the gambler is given access to samples. We show that the optimal competitive ratio of $1/2$ can be achieved with a single sample from each distribution. When the distributions are identical, we show that for any constant $varepsilon > 0$, $O(n)$ samples from the distribution suffice to achieve the optimal competitive ratio ($approx 0.745$) within $(1+varepsilon)$, resolving an open problem of Correa, Dutting, Fischer, and Schewior.
