Equivalence of Pairwise Stability, Set-Wise Stability and the Core under Dynamic Reserves Choice Functions

Abstract

In this paper I show that the pairwise and set-wise stable allocations in many-to-one matching problems with contracts are equivalent under the choice procedures of schools that are constructed by dynamic reserves. I also show that the set of set-wise stable allocations is equal to the core. Hence, three seemingly different solution concepts coincide under the family of dynamic reserves choice functions for schools that fails the unilateral substitutability and the law of aggregate demand. Since many significant choice procedures developed in practical matching markets fall in this family of choice procedures, the equivalence provide us a way to express set-wise stability notion by the simpler pairwise stability for these real-life problems. It also makes it easier to define stability normatively for specific government goals.