Efficient and (or) fair allocations under market-clearing constraints

Matching markets often encounter admissibility issues due to social concerns and regulations that must be respected. A key situation that has not been thoroughly analyzed in the literature involves the market-clearing requirement, which ensures balance in allocations across multiple matching markets, similar to supply-and-demand dynamics. To address these admissibility issues, we introduce the concept of an admissible set for such problems. We propose two solutions. The first solution is the "fairness-guaranteed stable solution." We identify a requirement on admissible sets that is necessary and sufficient for the non-emptiness of this solution. This requirement ensures that an allocation where no agent is assigned any resources is admissible. We then conduct welfare analysis and comparative statics of this solution. The second solution is called "efficiency-guaranteed stability," which focuses on maximizing efficiency within the constraints of the admissible set. We show that only specific admissible sets allow this solution to be non-empty.