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Asymptotic Rounding Laws for Optimal Thresholds in Secretary Problems
Abstract
We study second-order asymptotics for optimal thresholds in secretary-type problems. Under a unimodality assumption and a local affine expansion of the discrete payoff increments, we establish a general principle showing that sufficiently accurate rational approximations to the leading threshold proportion yield exact optimal integer thresholds. Continued-fraction convergents provide an immediate application. We also derive an asymptotic rounding law and apply the results to several variants of the secretary problem with fixed and random horizons.
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BibTeX
@misc{galan2026asymptotic,
title = {{Asymptotic Rounding Laws for Optimal Thresholds in Secretary Problems}},
author = {Raúl Sánchez Galán},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.24552},
eprint = {2608.24552},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24552},
}
RIS
TY - UNPB
TI - Asymptotic Rounding Laws for Optimal Thresholds in Secretary Problems
AU - Raúl Sánchez Galán
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.24552
UR - https://arxiv.org/abs/2608.24552
AB - We study second-order asymptotics for optimal thresholds in secretary-type problems. Under a unimodality assumption and a local affine expansion of the discrete payoff increments, we establish a general principle showing that sufficiently accurate rational approximations to the leading threshold proportion yield exact optimal integer thresholds. Continued-fraction convergents provide an immediate application. We also derive an asymptotic rounding law and apply the results to several variants of the secretary problem with fixed and random horizons.
ER -