The Secretary Recommendation Problem

In this paper we revisit the basic variant of the classical secretary problem. We propose a new approach in which we separate between an agent that evaluates the secretary performance and one that has to make the hiring decision. The evaluating agent (the sender) signals the quality of the candidate to the hiring agent (the receiver) who must make a decision. Whenever the two agents’ interests are not fully aligned, this induces an information transmission (signaling) challenge for the sender. We study the sender’s optimization problem subject to persuasiveness constraints of the receiver for several variants of the problem. Our results quantify the loss in performance for the sender due to online arrival. We provide optimal and near-optimal persuasive mechanisms that recover at least a constant fraction of a natural utility benchmark for the sender. The separation of evaluation and decision making can have a substantial impact on the approximation results. While in some scenarios, techniques and results closely mirror the conditions in the standard secretary problem, we also explore conditions that lead to very different characteristics.

💡 Research Summary

The paper introduces a novel twist on the classic secretary problem by separating the evaluation of candidates from the hiring decision. Two agents are modeled: a sender (the evaluator) who observes each candidate’s pair of values—ξ for the sender’s utility and ρ for the receiver’s utility—and a receiver (the hiring manager) who must decide whether to hire the candidate based on a signal sent by the sender. The sender commits to a signaling strategy in advance and can only send a simple “HIRE” or “NOT” recommendation each round. The receiver is required to be persuasive: following the recommendation must be optimal for her given her beliefs.

Two main environments are studied. In the basic scenario, the sender knows the entire set of (ξ, ρ) pairs before any arrivals. Here the authors compute the Pareto frontier of the candidate set and design a Pareto mechanism that selects a candidate on this frontier guaranteeing the receiver a prescribed expected utility μ_R while maximizing the sender’s expected utility. This mechanism is shown to be optimal for the sender (OPT).

In the secretary scenario, the values are adversarially chosen and unknown to the sender until each candidate arrives; only the uniform random order is common knowledge. This setting generalizes the classic secretary problem. The sender must now adaptively maintain the Pareto frontier of the candidates seen so far and issue recommendations that still guarantee the receiver μ_R. The authors prove that a simple online adaptation of the Pareto mechanism yields a (1/(3√3) − o(1))‑approximation of the sender’s optimal utility in the basic scenario. Thus, despite online arrival and information asymmetry, the loss is bounded by a constant factor, improving on the classic 1/e bound when interests are misaligned.

The paper further distinguishes four variants based on whether the receiver also learns the values of rejected candidates (disclosure) and whether utilities are cardinal (maximizing expected value) or ordinal (maximizing the probability of hiring the best candidate). For each of the sixteen possible combinations (basic/secretary × disclosure/no‑disclosure × cardinal/ordinal for each player) the authors either give an optimal mechanism or a constant‑factor approximation. In most cases the approximation factor remains a constant; only in the secretary‑plus‑disclosure case does the expected utility degrade by Θ(1/n).

Key technical contributions include: (1) framing the problem as an online Bayesian persuasion model with commitment; (2) leveraging the Pareto frontier to design direct, persuasive signaling rules; (3) proving that online adaptation of this frontier suffices for constant‑factor guarantees; and (4) providing a systematic analysis of disclosure effects on the receiver’s belief updates.

The results have practical relevance for settings where evaluation and hiring are delegated—such as HR departments, external consultants, or investment committees—showing that even with sequential arrivals and misaligned incentives, a well‑designed persuasive recommendation system can retain most of the sender’s potential utility. The work also opens a new research direction at the intersection of online selection problems and Bayesian persuasion.