A secretary for Messrs. Luce and Mallows

TL;DR

This paper extends the analysis of the secretary problem to cases where items arrive according to Luce and Mallows distributions, including new cases where the Mallows parameter exceeds one, and examines asymptotic strategies.

Contribution

It analyzes the secretary problem under Luce and Mallows distributions for all parameter ranges, including cases not previously studied, and compares these with known distributions.

Findings

Probabilities for the secretary problem coincide for Luce and Mallows distributions under certain conditions.

Asymptotic optimal strategies and limiting probabilities are derived for these distributions.

The analysis includes Luce distributions with various weight classes, such as Sukhatme weights.

Abstract

We analyze the secretary problem in the case that the $n$ ranked items arrive not in uniformly random order but rather according to a certain type of Luce distribution or according to a Mallows distribution on the set $S_n$ of permutations of $[n]$. The secretary problem for the Mallows distributions with parameter $q\in(0,1)$ was analyzed in a previous paper; in this paper the case $q>1$ is also analyzed. The Luce distribution with the class $\{q_j\}_{j=1}^\infty$ of weights is related in a certain sense to the Mallows distribution with parameter $q$, but is more difficult to analyze. It turns out that for every $n$ and every strategy, the probabilities for the secretary problem when the smallest number is considered of highest rank for the Luce distribution with this class of weights coincides with those for the corresponding Mallows distribution. We analyze the asymptotic optimal strategy and corresponding limiting probability for the above cases, as well as for the Luce distributions with other classes of weights, such as the Sukhatme weights.