Capacity Planning in Stable Matching with Truthful or Strategic Preference Uncertainty

2025-06-27

AI Summary

This paper studies capacity planning for stable matching under preference uncertainty, focusing on school choice settings. In practice, schools must commit to capacities before students disclose preferences, and students may strategically misreport preferences to improve admission chances. We propose a two-stage model: in Stage I, schools expand capacities under joint uncertainty—exogenous true preferences and endogenous strategic preferences; in Stage II, a stable matching is computed from reported preferences and school priorities. Our key contribution is a behavior-dependent mathematical formulation and a novel heuristic algorithm integrating Lagrangian relaxation with local search, tailored to strategic behavior. We solve the problem via two-stage stochastic optimization with sample average approximation (SAA). Experiments show SAA significantly outperforms the mean-scenario approach, yielding higher match satisfaction and greater admission fairness. Results confirm that both preference uncertainty and strategic behavior substantially impact optimal capacity design.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Planning under Uncertainty
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• 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

Recent studies on many-to-one matching markets have explored agents with flexible capacity and truthful preference reporting, focusing on mechanisms that jointly design capacities and select a matching. However, in real-world applications such as school choice and residency matching, preferences are revealed after capacity decisions are made, with matching occurring afterward; uncertainty about agents' preferences must be considered during capacity planning. Moreover, even under strategy-proof mechanisms, agents may strategically misreport preferences based on beliefs about admission chances. We introduce a two-stage stochastic matching problem with uncertain preferences, using school choice as a case study. In the first stage, the clearinghouse expands schools' capacities before observing students' reported preferences. Students either report their true preferences, producing exogenous uncertainty, or act strategically, submitting reported preferences based on their true preferences and admission chances (which depend on capacities), introducing endogenous uncertainty. In the second stage, the clearinghouse computes the student-optimal stable matching based on schools' priorities and students' reported preferences. In strategic cases, endogenous reported preferences are utility-maximizing transformations of capacity decisions and exogenous true preferences; we handle uncertainty using sample average approximation(SAA). We develop behavior-based mathematical formulations and, due to problem complexity, propose Lagrangian- and local-search-based behavior-specific heuristics for near-optimal solutions. Our SAA-based approaches outperform the average scenario approach on students' matching preferences and admission outcomes, emphasizing the impact of stochastic preferences on capacity decisions. Student behavior notably influences capacity design, stressing the need to consider misreports.

Problem

• Modeling two-stage matching with uncertain preferences
• Addressing strategic misreporting in capacity planning
• Optimizing school choice under exogenous and endogenous uncertainty

Innovation

• Two-stage stochastic matching with uncertain preferences
• Sample average approximation for preference uncertainty
• Lagrangian and local-search heuristics for solutions