An Asymptotic Analysis on Generalized Secretary Problem

TL;DR

This paper provides an asymptotic analysis of the secretary problem, challenging the traditional 37% rule by proposing that the optimal cutoff is often proportional to the square root of the total candidates, especially when focusing on top candidates.

Contribution

It introduces a new asymptotic perspective on the secretary problem, showing that the optimal stopping rule should scale with the square root of the number of candidates, and discusses implications for different objectives.

Findings

Optimal cutoff is often O(√n), not 37%

In some objectives, Θ(n) skips are necessary

Challenging traditional secretary problem strategies

Abstract

As a famous result, the ``37\% Law'' for Secretary Problem has widely influenced peoples' perception on online decision strategies about choice. However, using this strategy, too many attractive candidates may be rejected in the first 37\%, and in practice people also tend to stop earlier\cite{Bearden_early}. In this paper, we argued that in most cases, the best-only optimization does not obtain an optimal outcome, while the optimal cutoff should be $O(\sqrt{n})$. And we also showed that in some strict objective that only cares several best candidates, $\Theta(n)$ skips are still needed.