Commitment Gap via Correlation Gap

2025-08-27

AI Summary

This paper studies the Combinatorial Information Acquisition with Costs (CICS) problem, focusing on characterizing the *commitment gap*—the performance loss of commitment strategies relative to globally optimal adaptive policies. To tackle this challenge, we introduce a novel analytical framework grounded in *free-order prophet inequalities*, establishing for the first time a rigorous theoretical connection between costly information acquisition and costless Bayesian selection. We model information acquisition as a Markov Decision Process and integrate *ex ante* prophet inequalities with approximation algorithm design. Under matroid constraints—including the Pandora’s Box problem—we achieve significantly improved approximation ratios, breaking prior hardness barriers. Our core contribution is a tight characterization tool for the commitment gap, enabling precise quantification of adaptivity loss under information costs. This advances the theoretical foundations of information-cost-aware combinatorial optimization and extends the frontier of sequential decision-making under uncertainty with costly observations.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Game Theory and Economic Paradigms: Imperfect Information
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Studying costly information combinatorial selection optimization problem
• Bounding commitment gap between committing and optimal policies
• Improving approximation algorithms via free-order prophet inequalities

Innovation

• Reduction to Bayesian Combinatorial Selection
• Relating commitment gap to prophet inequalities
• Improved approximation for Pandora's Box