# The Secretary Recommendation Problem

## TL;DR

This paper studies a two-agent secretary problem with signaling, analyzing how information transmission affects the sender's performance and proposing mechanisms that optimize persuasion under various conditions.

## Contribution

It introduces a novel two-agent framework with signaling in the secretary problem and characterizes optimal persuasive mechanisms under different scenarios.

## Key findings

- Quantifies performance loss due to online arrival.
- Provides mechanisms that recover a constant fraction of the sender's utility.
- Explores conditions leading to standard and divergent secretary problem behaviors.

## Abstract

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.