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Optimal Stopping Methodology for the Secretary Problem with Random Queries

George V. Moustakides, Xujun Liu, Olgica Milenkovic

TL;DR

This paper studies a randomized secretary problem in which the decision maker can query an external, potentially faulty, expert a bounded number of times. The authors cast the problem as a triple stochastic optimization over the querying times, the final stopping time, and the stopping rule at each query, and solve it with optimal stopping theory to obtain a fully explicit threshold policy. They introduce deterministic sequences $A_t^k$ and $U_t^k$ that recursively define the optimal thresholds $r_k$ and $s_k(m)$, yielding an optimal strategy where $T_k$ is the first time after a computed threshold with rank $z_t=1$ and stopping at $T_k$ depends on whether $T_k$ exceeds a per-response threshold. A numerical example highlights how the probabilistic expert model improves success probabilities when the expert is informative, while showing saturation effects when the expert is imperfect; the classical secretary problem is recovered as a baseline when $M=2$ and the expert provides no information. These results provide a principled, non-asymptotic upper bound and a constructive policy for secretary-like problems with unreliable external guidance, with direct implications for sequential hiring and decision-making under uncertainty.

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.

Optimal Stopping Methodology for the Secretary Problem with Random Queries

TL;DR

This paper studies a randomized secretary problem in which the decision maker can query an external, potentially faulty, expert a bounded number of times. The authors cast the problem as a triple stochastic optimization over the querying times, the final stopping time, and the stopping rule at each query, and solve it with optimal stopping theory to obtain a fully explicit threshold policy. They introduce deterministic sequences and that recursively define the optimal thresholds and , yielding an optimal strategy where is the first time after a computed threshold with rank and stopping at depends on whether exceeds a per-response threshold. A numerical example highlights how the probabilistic expert model improves success probabilities when the expert is informative, while showing saturation effects when the expert is imperfect; the classical secretary problem is recovered as a baseline when and the expert provides no information. These results provide a principled, non-asymptotic upper bound and a constructive policy for secretary-like problems with unreliable external guidance, with direct implications for sequential hiring and decision-making under uncertainty.

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.

Paper Structure

This paper contains 6 sections, 4 theorems, 60 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Optimizing the Success Probability
  4. Simplified Form of the Optimal Components
  5. Numerical Example
  6. Acknowledgements

Key Result

Lemma 1

For $n\geq t\geq t_1>0$ we have where $\mathbbm{I}_A$ denotes the indicator function of the event $A$ and where for $b\geq a$ we define $\mathbbm{1}_a^b=\prod_{\ell=a}^b\mathbbm{I}_{\{z_\ell>1\}}$, while for $b<a$ we let $\mathbbm{1}_a^b=1$.

Figures (1)

  • Figure 1: Success probability as a function of the number of queries $K$ when $n=100$ objects and $M=2$ responses with symmetric probabilities $\mathsf{p}(1)=1-\mathsf{p}(2)=1-\mathsf{q}(1)=\mathsf{q}(2)=\mathsf{p}$ for $\mathsf{p}=0.5,0.6,0.7,$$0.8,0.9,0.95,0.98,1$.

Theorems & Definitions (11)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Lemma 3
  • Remark 4
  • Remark 5
  • Remark 6
  • ...and 1 more