The secretary problem with biased arrival order via a Mallows distribution

TL;DR

This paper analyzes the secretary problem under biased arrival orders modeled by a Mallows distribution, deriving optimal strategies and success probabilities for different bias regimes.

Contribution

It extends the classical secretary problem analysis to biased arrival orders using Mallows distribution, providing new optimal strategies and success probabilities.

Findings

Optimal strategies vary with bias strength and regime.

Success probability remains at least 1/e in weak and moderate bias regimes.

Strong bias yields higher success probabilities than the classical case.

Abstract

We solve the secretary problem in the case that the ranked items arrive in a statistically biased order rather than in uniformly random order. The bias is given by a Mallows distribution with parameter $q\in(0,1)$, so that higher ranked items tend to arrive later and lower ranked items tend to arrive sooner. In the classical problem, the asymptotically optimal strategy is to reject the first $M_n^*$ items, where $M_n^*\sim\frac ne$, and then to select the first item ranked higher than any of the first $M_n^*$ items (if such an item exists). This yields $\frac1e$ as the limiting probability of success. The Mallows distribution with parameter $q=1$ is the uniform distribution. For the regime $q_n=1-\frac cn$, with $c>0$, the case of weak bias, the optimal strategy occurs with $M_n^*\sim n\Big(\frac1c\log\big(1+\frac{e^c-1}e\big)\Big)$, with the limiting probability of success being $\frac1e$. For the regime $q_n=1-\frac c{n^\alpha}$, with $c>0$ and $\alpha\in(0,1)$, the case of moderate bias, the optimal strategy occurs with $n-M_n\sim\frac{n^\alpha}c$, with the limiting probability of success being $\frac1e$. For fixed $q\in(0,1)$, the case of strong bias, the optimal strategy occurs with $M_n^*=n-L$ where $\frac{L-1}L<q\le \frac L{L+1}$, with limiting probability of success being $(1-q)q^{L-1}L>\frac1e$.