The Pandora’s Box Problem with Sequential Inspections

The Pandora’s box problem is a core model in economic theory that captures an agent’s (Pandora’s) search for the best alternative (box). We study an important generalization of the problem in which the agent can either fully open boxes for a certain fee to reveal their exact values or partially open them at a reduced cost. This introduces a new trade-off between information acquisition and cost efficiency. We establish a hardness result and employ an array of techniques in stochastic optimization to provide a comprehensive analysis of this model. This includes (i) the identification of structural properties of the optimal policy that provide insights about optimal decisions, (ii) the derivation of problem relaxations and provably near-optimal solutions, (iii) the characterization of the optimal policy in special yet nontrivial cases, and (iv) an extensive numerical study that compares the performance of various policies and provides additional insights about the optimal policy. Throughout, we show that intuitive threshold-based policies that extend the Pandora’s box optimal solution can effectively guide search decisions.

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2024.0733.