Selection problems with costly information, dating back to Weitzman's Pandora's Box problem, have received much attention recently. We study the general model of Costly Information Combinatorial Selection (CICS) that was recently introduced by Chawla et al. [2024] and Bowers et al. [2025]. In this problem, a decision maker needs to select a feasible subset of stochastic variables, and can only learn information about their values through a series of costly steps, modeled by a Markov decision process. The algorithmic objective is to maximize the total value of the selection minus the cost of information acquisition. However, determining the optimal algorithm is known to be a computationally challenging problem. To address this challenge, previous approaches have turned to approximation algorithms by considering a restricted class of committing policies that simplify the decision-making aspects of the problem and allow for efficient optimization. This motivates the question of bounding the commitment gap, measuring the worst case ratio in the performance of the optimal committing policy and the overall optimal. In this work, we obtain improved bounds on the commitment gap of CICS through a reduction to a simpler problem of Bayesian Combinatorial Selection where information is free. By establishing a close relationship between these problems, we are able to relate the commitment gap of CICS to ex ante free-order prophet inequalities. As a consequence, we obtain improved approximation results for CICS, including the well-studied variant of Pandora's Box with Optional Inspection under matroid feasibility constraints.

翻译：具有代价高昂信息的选择问题，可追溯至Weitzman的潘多拉盒子问题，近来备受关注。我们研究由Chawla等人[2024]和Bowers等人[2025]近期提出的代价信息组合选择（CICS）通用模型。在此问题中，决策者需选择一个随机变量的可行子集，且仅能通过一系列以马尔可夫决策过程建模的代价步骤来获取其值的信息。算法目标在于最大化所选总价值减去信息获取成本。然而，确定最优算法已知是计算困难问题。为应对此挑战，先前研究转向近似算法，通过考虑一类受限的承诺策略来简化问题的决策层面并实现高效优化。这引出了界定承诺差距的问题——即衡量最优承诺策略与全局最优策略在最坏情况下的性能比值。本工作中，我们通过将CICS归约至信息免费的贝叶斯组合选择这一更简单问题，获得了对CICS承诺差距的改进界。通过建立这些问题间的紧密联系，我们得以将CICS的承诺差距与事前自由顺序先知不等式相关联。由此，我们获得了CICS的改进近似结果，包括在拟阵可行性约束下被广泛研究的带可选检查的潘多拉盒子问题变体。