Match Stability with a Costly and Flexible Number of Positions

One of the primary objectives of two-sided matching systems is to facilitate the pairing of two groups of agents in a manner that eliminates any incentive for pair deviation. Such challenges are quite prevalent and can have significant and long-lasting ramifications for participants, including students applying to colleges. While much of the existing research in this field addresses the problem using fixed quotas, real-world applications, like college admissions, demonstrate that this is not always applicable. We introduce the concept of slot stability, recognizing the potential motivation for organizations to modify their quotas after the matching process. We propose two algorithms designed to create stable and slot stable matches by employing flexible, endogenous quotas to address this issue. Additionally, we demonstrate that our algorithm aligns with the concerns raised by colleges implementing waitlist systems, effectively mitigating behaviors that can lead to unstable outcomes.