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.

翻译：近期关于多对一匹配市场的研究主要关注具有灵活容量和真实偏好报告的智能体，重点探讨联合设计容量并选择匹配的机制。然而，在诸如学校选择和住院医师匹配等实际应用中，偏好是在容量决策确定后才被揭示，匹配随后进行；因此在容量规划阶段必须考虑智能体偏好的不确定性。此外，即使在策略证明机制下，智能体也可能基于对录取机会的信念而策略性地误报偏好。本文以学校选择为案例研究，引入了一个具有偏好不确定性的两阶段随机匹配问题。在第一阶段，匹配清算所在观察到学生报告的偏好之前扩展学校的容量。学生要么报告其真实偏好（产生外生不确定性），要么采取策略性行为，根据其真实偏好和录取机会（后者取决于容量）提交报告的偏好，从而引入内生不确定性。在第二阶段，清算所基于学校的优先级顺序和学生报告的偏好计算学生最优稳定匹配。在策略性情况下，内生的报告偏好是容量决策和外生真实偏好的效用最大化变换；我们采用样本平均近似（SAA）方法处理不确定性。针对问题的复杂性，我们开发了基于行为的数学表述，并提出了基于拉格朗日松弛和局部搜索的、针对特定行为的启发式算法以寻求近似最优解。我们的基于SAA的方法在学生匹配偏好和录取结果方面优于平均情景方法，凸显了随机偏好对容量决策的影响。学生行为显著影响容量设计，强调了考虑误报的必要性。