Efficient iBF: Balanced Integration of Fragmented Matching Markets for Welfare Improvement

Matching markets often suffer from fragmentation, which leads to inefficiency. We model a fragmented market in a school-choice context and offer a practically relevant method for integration. Specifically, each student and school belong to a region, and we allow for inter-regional transfer of students with "balancedness" constraint: a matching is said to be balanced if, for each region, the outflow of students from that region to other regions is equal to the inflow of students from the latter to the former. Using a directed bipartite graph defined on students and schools, we characterize the set of Pareto efficient matchings among those that are individually rational, balanced and fair (efficient iBF). We also provide a class of polynomial-time algorithms to compute such matchings. When each region favors local students in their priority, the outcome of an algorithm from this class weakly improves student welfare upon the outcome where each region independently uses the deferred acceptance mechanism. Various real-life examples of fragmentation are discussed, and we illustrate how our method would address the issue.