Stable Matchings with Choice Correspondences Under Acyclicity

We study the existence of stable matchings when agents have choice correspondences instead of preference relations. We extend the framework of \cite{chambers2017choice} by weakening the Path Independence assumption. For many-to-many markets, we show that stable matchings exist when choice correspondences satisfy Substitutability and a new General Acyclicity condition. We provide a constructive proof using a Grow or Discard Algorithm that iteratively expands or eliminates contracts until a strongly maximal Individually Rational set is reached. We provide an algorithm to obtain stable matchings in which rejected contracts are not permanently discarded, distinguishing our approach significantly from standard DAA-type algorithms. For one-to-one markets, we show that Path Independence alone does not guarantee stability. We introduce a replacement-based notion of stability and provide an algorithm that constructs stable matchings when choice correspondences satisfy Binary Acyclicity.