The Pandora's Box Problem with Sequential Inspections

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.

Figures (10)

• Figure 1: Illustration of the dynamics of a single box.
• Figure 2: Illustration of the optimal policy for the single closed box case. Here, we present the case where $\boldsymbol{\sigma_i^{F/P} \geq 0, \sigma_i^P \geq 0}$.
• Figure 3: A visualization of how the opening thresholds of the prototypical boxes are distributed.
• Figure EC.1: Illustration of thresholds in Example \ref{['example:many_boxes']}
• Figure EC.2: A graphical illustration of policy $\pi$ and policy $\pi'$ for Case 2, in the Proof of Lemma \ref{['lemma:just_strong_partial']}.

Theorems & Definitions (35)

• Example 1
• Example 2
• Definition 1: Thresholds
• Theorem 1
• Theorem 2
• Definition 2
• Theorem 3
• Corollary 1
• Theorem 4
• Theorem 5