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.

翻译：潘多拉魔盒问题（Weitzman 1979）是经济理论中的一个核心模型，描述了行为主体（潘多拉）搜索最佳选择（盒子）的过程。本文研究了该问题的一个重要泛化形式：行为主体既可以付费完全打开盒子以获知其确切价值，也可以以更低成本部分打开盒子。这引入了信息获取与成本效率之间的新权衡。我们建立了一个难度结论，并运用随机优化领域的多种技术对该模型进行了全面分析，包括：（1）识别最优策略的结构性质，从而揭示最优决策的洞见；（2）推导问题松弛及可证明的近似最优解；（3）刻画特殊但非平凡情形下的最优策略；（4）开展广泛的数值研究，比较多种策略的性能，并进一步提供关于最优策略的洞见。自始至终，我们证明扩展潘多拉魔盒最优解决方案的直观阈值基策略能够有效指导搜索决策。