Optimal Guarantees for Online Selection Over Time

Sebastian Perez-Salazar and Victor Verdugo

Papers from arXiv.org

Abstract: Prophet inequalities are a cornerstone in optimal stopping and online decision-making. Traditionally, they involve the sequential observation of $n$ non-negative independent random variables and face irrevocable accept-or-reject choices. The goal is to provide policies that provide a good approximation ratio against the optimal offline solution that can access all the values upfront -- the so-called prophet value. In the prophet inequality over time problem (POT), the decision-maker can commit to an accepted value for $\tau$ units of time, during which no new values can be accepted. This creates a trade-off between the duration of commitment and the opportunity to capture potentially higher future values. In this work, we provide best possible worst-case approximation ratios in the IID setting of POT for single-threshold algorithms and the optimal dynamic programming policy. We show a single-threshold algorithm that achieves an approximation ratio of $(1+e^{-2})/2\approx 0.567$, and we prove that no single-threshold algorithm can surpass this guarantee. With our techniques, we can analyze simple algorithms using $k$ thresholds and show that with $k=3$ it is possible to get an approximation ratio larger than $\approx 0.602$. Then, for each $n$, we prove it is possible to compute the tight worst-case approximation ratio of the optimal dynamic programming policy for instances with $n$ values by solving a convex optimization program. A limit analysis of the first-order optimality conditions yields a nonlinear differential equation showing that the optimal dynamic programming policy's asymptotic worst-case approximation ratio is $\approx 0.618$. Finally, we extend the discussion to adversarial settings and show an optimal worst-case approximation ratio of $\approx 0.162$ when the values are streamed in random order.

Date: 2024-08
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://arxiv.org/pdf/2408.11224 Latest version (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:2408.11224

Access Statistics for this paper

More papers in Papers from arXiv.org
Bibliographic data for series maintained by arXiv administrators ().