Incentivizing Hidden Types in Secretary Problem

TL;DR

This paper analyzes a game-theoretic model of the secretary problem where applicants can incur costs to reveal their types, and the optimal hiring strategy is characterized with a focus on the impact of these costs on success probability.

Contribution

It introduces a full learning equilibrium framework for the secretary problem with costly type revelation, proving its existence, uniqueness, and optimality, and analyzes how costs affect success probability.

Findings

Equilibrium acceptance probabilities depend on a threshold and cost.

Success probability decreases polynomially with the number of applicants due to costs.

The model extends classical secretary problem results to include strategic type revelation with costs.

Abstract

We study a game between $N$ job applicants who incur a cost $c$ (relative to the job value) to reveal their type during interviews and an administrator who seeks to maximize the probability of hiring the best. We define a full learning equilibrium and prove its existence, uniqueness, and optimality. In equilibrium, the administrator accepts the current best applicant $n$ with probability $c$ if $n<n^*$ and with probability 1 if $n\ge n^*$ for a threshold $n^*$ independent of $c$. In contrast to the case without cost, where the success probability converges to $1/\mathrm{e}\approx 0.37$ as $N$ tends to infinity, with cost the success probability decays like $N^{-c}$.