Secretary Problem Research Articles

A note on the satisficing policy of the secretary problem

Wu and Li (2022) presented a satisficing policy for the secretary problem (SP). Moreover, Wu and Li (2022) derived asymptotic formulas of the maximum probability of choosing one of the k satisfactory applicants with special cases k = 2 and 3, respectively, in the satisficing policy. “Based on the work of Wu and Li (2022), this paper derives the asymptotic formulas for the probability of selecting one of the k satisfactory applicants and provides the equation for the approximate optimal cutoff value to achieve the asymptotic maximum probability in the general case of k satisfactory applicants under the satisficing policy.” Moreover, a series of computer simulations are conducted to test the theoretical results for k = 4 and 5.

Robust Online Selection with Uncertain Offer Acceptance

Online advertising has motivated interest in online selection problems. Displaying ads to the right users benefits both the platform (e.g., via pay-per-click) and the advertisers (by increasing their reach). In practice, not all users click on displayed ads, while the platform’s algorithm may miss the users most disposed to do so. This mismatch decreases the platform’s revenue and the advertiser’s chances to reach the right customers. With this motivation, we propose a secretary problem where a candidate may or may not accept an offer according to a known probability p. Because we do not know the top candidate willing to accept an offer, the goal is to maximize a robust objective defined as the minimum over integers k of the probability of choosing one of the top k candidates, given that one of these candidates will accept an offer. Using Markov decision process theory, we derive a linear program for this max-min objective whose solution encodes an optimal policy. The derivation may be of independent interest, as it is generalizable and can be used to obtain linear programs for many online selection models. We further relax this linear program into an infinite counterpart, which we use to provide bounds for the objective and closed-form policies. For [Formula: see text], an optimal policy is a simple threshold rule that observes the first [Formula: see text] fraction of candidates and subsequently makes offers to the best candidate observed so far. Funding: Financial support from the U.S. National Science Foundation [Grants CCF-2106444, CCF-1910423, and CMMI 1552479] is gratefully acknowledged.

The Secretary Problem with Independent Sampling

The secretary problem is probably the most well-studied optimal stopping problem with many applications in economics and management. In the secretary problem, a decision maker faces an unknown sequence of values, revealed successively, and has to make irrevocable take-it-or-leave-it decisions. Her goal is to select the maximum value in the sequence. Although in the classic secretary problem the values of upcoming elements are entirely unknown, in many realistic situations, the decision maker has access to some information, for example, from past data. In this paper, we take a sampling approach and assume that before starting the sequence, each element is sampled independently with probability p. We study both the adversarial and the random arrival models. Our main result is to obtain the best possible algorithms for both settings and all values of p. As p grows to one, the obtained guarantees converge to the optimal guarantees in the full information case. Notably, we establish that the best possible algorithm in the adversarial order setting is a fixed threshold algorithm. In the random order setting, we characterize the best possible algorithm by a sequence of thresholds, dictating at which point in time we should accept a value. Surprisingly, this sequence is independent of p. We complement our theoretical results with numerical experiments on data of people playing the secretary problem repeatedly. Our results help explain some behavioral issues they raised and indicate that people play a strategy similar to our optimal algorithms from the start onwards, albeit slightly suboptimally. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was partially funded by the Agencia Nacional de Investigación y Desarrollo Chile [Grants FB210005 (Centro de Modelamiento Matamático), AIM23_0004 (Millenium Institute for Research in Market Imperfections and Public Policy), AFB230002 (Instituto Sistemas Complejos de Ingeniería), ACT210005, and Fondo Nacional de Desarrollo Científico y Tecnológico 1220054], the Alexander von Humboldt Foundation with funds from the German Federal Ministry of Education and Research, and the German Research Foundation within the Research Training Group Advanced Optimization in a Networked Economy [Grant Graduiertenkolleg 2201]. The views expressed in this paper do not necessarily reflect those of the European Central Bank or the Eurosystem. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2021.01580 .

Competitive Analysis of Algorithms for an Online Distribution Problem

We study an online distribution problem in which a producer has to send a load from an origin to a destination. At each time period before the deadline, they ask for transportation price quotes and have to decide to either accept or not accept the minimum offered price. If this price is not accepted, they have to pay a penalty cost, which may be the cost to ask for new quotes, the penalty cost for a late delivery, or the inventory cost to store the load for a certain duration. The aim is to minimize the sum of the transportation and the penalty costs. This problem has interesting real-world applications, given that transportation quotes can be obtained from professional websites nowadays. We show that the classical online algorithm used to solve the well-known Secretary problem is not able to provide, on average, effective solutions to our problem, given the trade-off between the transportation and the penalty costs. Therefore, we design two classes of online algorithms. The first class is based on a given time of acceptance, while the second is based on a given threshold price. We formally prove the competitive ratio of each algorithm, i.e., the worst-case performance of the online algorithm with respect to the optimal solution of the offline problem, in which all transportation prices are known at the beginning, rather than being revealed over time. The computational results show the algorithms’ performance on average and in the worst-case scenario when the transportation prices are generated on the basis of given probability distributions.

Neurophysiological insights into sequential decision-making: exploring the secretary problem through ERPs and TBR dynamics.

Decision-making under uncertainty, a cornerstone of human cognition, is encapsulated by the "secretary problem" in optimal stopping theory. Our study examines this decision-making challenge, where participants are required to sequentially evaluate and make irreversible choices under conditions that simulate cognitive overload. We probed neurophysiological responses by engaging 27 students in a secretary problem simulation while undergoing EEG monitoring, focusing on Event-Related Potentials (ERPs) P200 and P400, and Theta to Beta Ratio (TBR) dynamics.Results revealed a nuanced pattern: the P200 component's amplitude declined from the initial to the middle offers, suggesting a diminishing attention span as participants grew accustomed to the task. This attenuation reversed at the final offer, indicating a heightened cognitive processing as the task concluded. In contrast, the P400 component's amplitude peaked at the middle offer, hinting at increased cognitive evaluation, and tapered off at the final decision. Additionally, TBR dynamics illustrated a fluctuation in attentional control and emotional regulation throughout the decision-making sequence, enhancing our understanding of the cognitive strategies employed.The research elucidates the dynamic interplay of cognitive processes in high-stakes environments, with neurophysiological markers fluctuating significantly as participants navigated sequential choices. By correlating these fluctuations with decision-making behavior, we provide insights into the evolving strategies from heightened alertness to strategic evaluation. Our findings offer insights that could inform the use of neurophysiological data in the development of decision-making frameworks, potentially contributing to the practical application of cognitive research in real-life contexts.

Self-Interested Coalitional Crowdsensing for Multi-Agent Interactive Environment Monitoring.

As a promising paradigm, mobile crowdsensing (MCS) takes advantage of sensing abilities and cooperates with multi-agent reinforcement learning technologies to provide services for users in large sensing areas, such as smart transportation, environment monitoring, etc. In most cases, strategy training for multi-agent reinforcement learning requires substantial interaction with the sensing environment, which results in unaffordable costs. Thus, environment reconstruction via extraction of the causal effect model from past data is an effective way to smoothly accomplish environment monitoring. However, the sensing environment is often so complex that the observable and unobservable data collected are sparse and heterogeneous, affecting the accuracy of the reconstruction. In this paper, we focus on developing a robust multi-agent environment monitoring framework, called self-interested coalitional crowdsensing for multi-agent interactive environment monitoring (SCC-MIE), including environment reconstruction and worker selection. In SCC-MIE, we start from a multi-agent generative adversarial imitation learning framework to introduce a new self-interested coalitional learning strategy, which forges cooperation between a reconstructor and a discriminator to learn the sensing environment together with the hidden confounder while providing interpretability on the results of environment monitoring. Based on this, we utilize the secretary problem to select suitable workers to collect data for accurate environment monitoring in a real-time manner. It is shown that SCC-MIE realizes a significant performance improvement in environment monitoring compared to the existing models.

Decision-making analysis for a new variant of the classical secretary problem

Decision-making analysis for a new variant of the classical secretary problem

Optimal Stopping Methodology for the Secretary Problem with Random Queries – ADDENDUM

Optimal Stopping Methodology for the Secretary Problem with Random Queries – ADDENDUM

Secretary Problems with Biased Evaluations Using Partial Ordinal Information

The k-secretary problem deals with online selection of at most k numerically scored applicants, where selection decisions are immediate upon their arrival and irrevocable, with the goal of maximizing total score. There is, however, wide prevalence of bias in evaluations of applicants from different demographic groups (e.g., gender, age, race), and the assumption of an algorithm observing their true score is unreasonable in practice. In this work, we propose the poset secretary problem, where selection decisions must be made by observing a partial order over the applicants. This partial order is constructed to account for uncertainty and biases in applicant scores. We assume that each applicant has a fixed score, which is not visible to the algorithm and is consistent with the partial order. Using a random partitioning technique from the matroid secretary literature, we provide order-optimal competitive algorithms for the poset secretary problem and provide matching lower bounds. Further, we develop the theory of thresholding in posets to provide a tight, adaptive-thresholding algorithm under regimes where k grows quickly enough, thus matching the adaptiveness to k shown for the classical k-secretary problem. We then study a special case, in which applicants belong to g disjoint demographic groups and the bias is group specific. We provide competitive algorithms for adversarial and stochastic variants of this special case, including a framework, Gap, for parallelizing any vanilla k-secretary algorithm to the group setting. Finally, we perform a case study on real-world data to demonstrate the responsiveness of our algorithms to data and their impact on selection rates. This paper was accepted by Hamid Nazerzadeh, data science. Funding: This work was supported by the National Science Foundation [Grant 2112533]. S. Gupta’s research was supported in part by NSF CAREER Award 2239824. Supplemental Material: The data files are available at https://doi.org/10.1287/mnsc.2023.4926 .

Optimal stopping methodology for the secretary problem with random queries

AbstractCandidates arrive sequentially for an interview process which results in them being ranked relative to their predecessors. Based on the ranks available at each time, a decision mechanism must be developed that selects or dismisses the current candidate in an effort to maximize the chance of selecting the best. This classical version of the ‘secretary problem’ has been studied in depth, mostly using combinatorial approaches, along with numerous other variants. We consider a particular new version where, during reviewing, it is possible 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 or not we should terminate the search at the querying time. This decision is straightforward under the usual assumption of infallible experts, but when experts are faulty it has a far more intricate structure.

Query-based selection of optimal candidates under the Mallows model

We study the secretary problem in which rank-ordered lists are generated by the Mallows model and the goal is to identify the highest ranked candidate through a sequential interview process which does not allow rejected candidates to be revisited. The main difference between our formulation and existing models is that, during the selection process, we are given a fixed number of opportunities to query an infallible expert whether the current candidate is the highest-ranked or not. If the response is positive, the selection process terminates, otherwise, the search continues until a new potentially optimal candidate is identified. Our optimal interview strategy, as well as the expected number of candidates interviewed and expected number of queries used can be determined through the evaluation of well-defined recurrence relations. Specifically, if we are allowed to query s−1 times and to make a final selection without querying (thus, making s selections in total) then the optimum scheme is characterized by s thresholds that depend on the parameter θ of the Mallows distribution but are independent on the maximum number of queries.

The Secretary Problem with Predictions

The value maximization version of the secretary problem is the problem of hiring a candidate with the largest value from a randomly ordered sequence of candidates. In this work, we consider a setting where predictions of candidate values are provided in advance. We propose an algorithm that achieves a nearly optimal value if the predictions are accurate and results in a constant-factor competitive ratio otherwise. We also show that the worst-case competitive ratio of an algorithm cannot be higher than some constant [Formula: see text], which is the best possible competitive ratio when we ignore predictions, if the algorithm performs nearly optimally when the predictions are accurate. Additionally, for the multiple-choice secretary problem, we propose an algorithm with a similar theoretical guarantee. We empirically illustrate that if the predictions are accurate, the proposed algorithms perform well; meanwhile, if the predictions are inaccurate, performance is comparable to existing algorithms that do not use predictions.

Secretaries with Advice

The secretary problem is probably the purest model of decision making under uncertainty. In this paper, we ask which advice we can give the algorithm to improve its success probability. We propose a general model that unifies a broad range of problems: from the classic secretary problem with no advice to the variant where the quality of a secretary is drawn from a known distribution and the algorithm learns each candidate’s quality on arrival, more modern versions of advice in the form of samples, and a machine-learning inspired model where a classifier gives us a noisy signal about whether the current secretary is the best on the market. Our main technique is a factor-revealing linear program (LP) that captures all of these problems. We use this LP formulation to gain structural insight into the optimal policy. Using tools from linear programming, we present a tight analysis of optimal algorithms for secretaries with samples, optimal algorithms when secretaries’ qualities are drawn from a known distribution, and optimal algorithms for a new noisy binary advice model.

On Hiring Secretaries with Stochastic Departures

The paper studies generalization of the secretary problem, where decisions do not have to be made immediately upon applicants’ arrivals. After arriving, each applicant stays in the system for some (random) amount of time and then leaves, whereupon the algorithm has to decide irrevocably whether to select this applicant or not. The arrival and waiting times are drawn from known distributions, and the decision maker’s goal is to maximize the probability of selecting the best applicant overall. The paper characterizes the optimal policy for this setting, showing that when deciding whether to select an applicant, it suffices to know only the time and the number of applicants that have arrived so far. Furthermore, the policy is monotone nondecreasing in the number of applicants seen so far, and, under certain natural conditions, monotone nonincreasing in time. Furthermore, when the number of applicants is large, a single threshold policy is almost optimal.

Secretary and online matching problems with machine learned advice

The classic analysis of online algorithms, due to its worst-case nature, can be quite pessimistic when the input instance at hand is far from worst-case. In contrast, machine learning approaches shine in exploiting patterns in past inputs in order to predict the future. However, such predictions, although usually accurate, can be arbitrarily poor. Inspired by a recent line of work, we augment three well-known online settings with machine learned predictions about the future, and develop algorithms that take these predictions into account. In particular, we study the following online selection problems: (i) the classic secretary problem, (ii) online bipartite matching and (iii) the graphic matroid secretary problem. Our algorithms still come with a worst-case performance guarantee in the case that predictions are subpar while obtaining an improved competitive ratio (over the best-known classic online algorithm for each problem) when the predictions are sufficiently accurate. For each algorithm, we establish a trade-off between the competitive ratios obtained in the two respective cases.

Sample-Driven Optimal Stopping: From the Secretary Problem to the i.i.d. Prophet Inequality

We take a unifying approach to single selection optimal stopping problems with random arrival order and independent sampling of items. In the problem we consider, a decision maker (DM) initially gets to sample each of N items independently with probability p, and can observe the relative rankings of these sampled items. Then, the DM faces the remaining items in an online fashion, observing the relative rankings of all revealed items. While scanning the sequence the DM makes irrevocable stop/continue decisions and her reward for stopping the sequence facing the item with rank i is Yi. The goal of the DM is to maximize her reward. We start by studying the case in which the values Yi are known to the DM, and then move to the case in which these values are adversarial. For the former case we are able to recover several classic results in the area, thus giving a unifying framework for single selection optimal stopping. For the latter, we pin down the optimal algorithm, obtaining the optimal competitive ratios for all values of p. Funding: This work was partially supported by The Center for Mathematical Modeling at the University of Chile (ANID FB210005), Grant Anillo Information and Computation in Market Design (ANID ACT210005), FONDECYT 1220054 and 1181180, and a Meta Research PhD Fellowship.

Secretary problem and two almost the same consecutive applicants

We present a new variant of the secretary problem. Let $A$ be a totally ordered set of $n$ \emph{applicants}. Given $P\subseteq A$ and $x\in A$, let $rr(P,x)=\vert\{z\in P \mid z\leq x\}\vert\mbox{ }$ be the \emph{relative rank of} $x$ \emph{with regard to} $P$, and let $rr_n(x)=rr(A,x)$. Let $x_1,x_2,\dots,x_n\in A$ be a random sequence of distinct applicants. The aim is to select $1<j\leq n$ such that $rr_n(x_{j-1})-rr_n(x_j)\in\{-1,1\}$. Let $\alpha$ be a real constant with $0<\alpha<1$. Suppose the following stopping rule $\tau_n(\alpha)$: reject first $\alpha n$ applicants and then select the first $x_j$ such that $rr(P_j,x_{j-1})-rr(P_j,x_j)\in\{-1,1\}$, where $P_j=\{x_i\mid 1\leq i\leq j\}$. Let $p_{n,\tau}(\alpha)$ be the probability that $\tau_n(\alpha)$ selects $x_j$ such that $rr_n(x_{j-1})-rr_n(x_j)\in\{-1,1\}$. We show that \[\lim_{n\rightarrow\infty}p_{n,\tau}(\alpha)\leq \lim_{n\rightarrow\infty}p_{n,\tau}\left(\frac{1}{2}\right)=\frac{1}{2}\mbox{.}\]

The secretary problem with non-uniform arrivals via a left-to-right minimum exponentially tilted distribution

The secretary problem with non-uniform arrivals via a left-to-right minimum exponentially tilted distribution

Decision-Making Analysis for a new variant of the classical Secretary Problem

Decision-Making Analysis for a new variant of the classical Secretary Problem

Tight Guarantees for Static Threshold Policies in the Prophet Secretary Problem

In the prophet secretary problem, n values are drawn independently from known distributions and presented in a uniformly random order. A decision maker must accept or reject each value when it is presented and may accept at most k values in total. The objective is to maximize the expected sum of accepted values. We analyze the performance of static threshold policies, which accept the first k values exceeding a fixed threshold (or all such values, if fewer than k exist). We show that an appropriate threshold guarantees [Formula: see text] times the value of the offline optimal solution. Note that [Formula: see text], and by Stirling’s approximation, [Formula: see text]. This represents the best-known guarantee for the prophet secretary problem for all k &gt; 1 and is tight for all k for the class of static threshold policies. We provide two simple methods for setting the threshold. Our first method sets a threshold such that [Formula: see text] values are accepted in expectation, and offers an optimal guarantee for all k. Our second sets a threshold such that the expected number of values exceeding the threshold is equal to k. This approach gives an optimal guarantee if k &gt; 4 but gives suboptimal guarantees for [Formula: see text]. Our proofs use a new result for optimizing sums of independent Bernoulli random variables, which extends a result of Hoeffding from 1956 and could be of independent interest. Supplemental Material: The online appendices are available at https://doi.org/10.1287/opre.2022.2419 .