The hiring problem with rank-based strategies

TL;DR

This paper analyzes rank-based hiring strategies, providing limit theorems that reveal a dichotomy in their long-term behavior, either dominated by early events or exhibiting mixing and normality.

Contribution

It extends previous results by establishing general limit theorems for rank-based strategies, revealing a dichotomy in their asymptotic behavior.

Findings

Strategies are either dominated by early events with almost sure convergence.

Some strategies exhibit mixing behavior with asymptotic normality.

The results unify and extend prior analyses of specific strategies.

Abstract

The hiring problem is studied for general strategies based only on the relative ranking of the candidates; this includes some well known strategies studied before such as hiring above the median. We give general limit theorems for the number of hired candidates and some other properties, extending previous results. The results exhibit a dichotomy between two classes of rank-based strategies: either the asymptotics of the process are determined by the early events, with a.s. convergence of suitably normalized random variables, or there is a mixing behaviour without long-term memory and with asymptotic normality.