Paths to stability for college admissions with budget constraints

We study two‐sided matching problem considered in Abizada (Theor Econ 11(2), 735–756, 2016), where one side (colleges) can make monetary transfers (offer stipends) to the other (students) subject to budget constraints. Colleges have strict preferences over sets of students and value money only to the extent that it allows them to enroll better or additional students. A student can attend at most one college and receive a stipend from it. Each student has preferences over college‐stipend bundles. Although in the presence of budget constraints, the conditions that are essential for most of the results on stability in the literature fail, Abizada (Theor Econ 11(2), 735–756, 2016) shows that for this model a pairwise stable allocation always exists. In this paper, we show that starting from an arbitrary allocation, there is a sequence of allocations, each allocation being obtained from the previous one by ‘satisfying’ a blocking pair, such that the final allocation is pairwise stable.