Stable Matchings with Choice Correspondences Under Acyclicity

We study the existence of stable matchings in markets where agents are described by choice correspondences rather than preference relations. For many-to-many matching markets, we introduce a condition called Settled Set Persistence (SSP) and show that substitutability together with SSP guarantees the existence of a CY-stable matching. The proof is constructive and yields a dynamic, one-at-a-time offer algorithm. We also show that path independence and SSP are logically independent, neither implies the other, so SSP identifies a different sufficient condition for stability. For one-to-one matching markets, we introduce a replacement-based notion of stability and a binary acyclicity condition on pairwise choices, and show that this condition guarantees the existence of stable matchings. Since path independence implies binary acyclicity but not conversely, our results identify weaker behavioral conditions for stability.