Optimal Stopping Methodology for the Secretary Problem with Random Queries

TL;DR

This paper introduces a probabilistic optimal stopping approach for the secretary problem, incorporating external expert queries that may be unreliable, and develops strategies to optimize decision-making under these conditions.

Contribution

It extends the classical secretary problem by modeling expert advice as probabilistic and potentially faulty, and applies optimal stopping theory to develop new decision mechanisms.

Findings

Optimal querying strategies under expert unreliability

Decision mechanisms for faulty expert advice

Enhanced success probability in candidate selection

Abstract

Candidates arrive sequentially for an interview process which results in them being ranked relative to their predecessors. Based on the ranks available at each time, one must develop a decision mechanism that selects or dismisses the current candidate in an effort to maximize the chance to select the best. This classical version of the ``Secretary problem'' has been studied in depth using mostly combinatorial approaches, along with numerous other variants. In this work we consider a particular new version where during reviewing one is allowed to query an external expert to improve the probability of making the correct decision. Unlike existing formulations, we consider experts that are not necessarily infallible and may provide suggestions that can be faulty. For the solution of our problem we adopt a probabilistic methodology and view the querying times as consecutive stopping times which we optimize with the help of optimal stopping theory. For each querying time we must also design a mechanism to decide whether we should terminate the search at the querying time or not. This decision is straightforward under the usual assumption of infallible experts but, when experts are faulty, it has a far more intricate structure.