The Pandora's Box Problem with Sequential Inspections

TL;DR

This paper generalizes the Pandora's box problem by allowing partial inspections at reduced costs, analyzing the resulting tradeoffs and proposing near-optimal policies through stochastic optimization techniques.

Contribution

It introduces a new model with partial inspections, establishes structural properties, relaxations, and characterizations of optimal policies, supported by numerical analysis.

Findings

Threshold-based policies extend Pandora's box solutions effectively.

Near-optimal solutions can be derived from problem relaxations.

Numerical studies compare policy performance and provide insights.

Abstract

The Pandora's box problem (Weitzman 1979) 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 where 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 tradeoff 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 (1) the identification of structural properties of the optimal policy that provide insights about optimal decisions; (2) the derivation of problem relaxations and provably near-optimal solutions; (3) the characterization of the optimal policy in special yet non-trivial cases; and (4) an extensive numerical study that compares the performance of various policies, and which 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.