The "Thirty-seven Percent Rule" and the Secretary Problem with Relative Ranks

TL;DR

This paper analyzes a variant of the secretary problem focusing on minimizing the expected rank of the selected item using a stopping rule based on relative ranks, achieving better results than classical methods.

Contribution

It introduces an optimal stopping rule with parameters that improve expected rank minimization, extending classical secretary problem results to relative ranks and generalizations.

Findings

Optimal stopping point at k ~ n/e

Expected rank can be less than any positive power of n

Generalization to rank within the lowest d

Abstract

We revisit the problem of selecting an item from $n$ choices that appear before us in random sequential order so as to minimize the expected rank of the item selected. In particular, we examine the stopping rule where we reject the first $k$ items and then select the first subsequent item that ranks lower than the $l$-th lowest-ranked item among the first $k$. We prove that the optimal rule has $k \sim n/{\mathrm e}$, as in the classical secretary problem where our sole objective is to select the item of lowest rank; however, with the optimally chosen $l$, here we can get the expected rank of the item selected to be less than any positive power of $n$ (as $n$ approaches infinity). We also introduce a common generalization where our goal is to minimize the expected rank of the item selected, but this rank must be within the lowest $d$.