# Stable Secretaries

## TL;DR

This paper introduces a new variant of the secretary problem involving multiple positions and secretaries, focusing on stable matchings and analyzing both random and adversarial arrival orders.

## Contribution

It formulates a novel online matching problem with stability considerations and provides bounds for different arrival scenarios.

## Key findings

- Upper and lower bounds for random arrivals
- Upper and lower bounds for adversarial arrivals
- Analysis of stability via blocking pairs

## Abstract

We define and study a new variant of the secretary problem. Whereas in the classic setting multiple secretaries compete for a single position, we study the case where the secretaries arrive one at a time and are assigned, in an on-line fashion, to one of multiple positions. Secretaries are ranked according to talent, as in the original formulation, and in addition positions are ranked according to attractiveness. To evaluate an online matching mechanism, we use the notion of blocking pairs from stable matching theory: our goal is to maximize the number of positions (or secretaries) that do not take part in a blocking pair. This is compared with a stable matching in which no blocking pair exists. We consider the case where secretaries arrive randomly, as well as that of an adversarial arrival order, and provide corresponding upper and lower bounds.