Choice correspondences are crucial in decision-making, especially when faced with indifferences or ties. While tie-breaking can transform a choice correspondence into a choice function, it often introduces inefficiencies. This paper introduces a novel notion of path-independence (PI) for choice correspondences, extending the existing concept of PI for choice functions. Intuitively, a choice correspondence is PI if any consistent tie-breaking produces a PI choice function. This new notion yields several important properties. First, PI choice correspondences are rationalizabile, meaning they can be represented as the maximization of a utility function. This extends a core feature of PI in choice functions. Second, we demonstrate that the set of choices selected by a PI choice correspondence for any subset forms a generalized matroid. This property reveals that PI choice correspondences exhibit a nice structural property. Third, we establish that choice correspondences rationalized by ordinally concave functions inherently satisfy the PI condition. This aligns with recent findings that a choice function satisfies PI if and only if it can be rationalized by an ordinally concave function.

The existence of competitive equilibria has been extensively studied in many economic models. In the literature of matching theory, attention has been paid to many-to-one matching models. These models consider matching between two types of agents, such as firms and workers, sellers and buyers, hospitals and doctors, or schools and students. In a many-to-one matching model with continuous transfers and quasilinear utilities, the gross substitutes condition (GS), which excludes the complementarity among workers in a firm's valuation, guarantees the existence of competitive equilibria. While GS is known to be a necessary condition in the maximal domain sense, competitive equilibria can exist even if GS is violated. Recent developments in this field have revealed that various conditions other than GS can guarantee the existence of competitive equilibria. For example, under the gross substitutes and complements condition (GSC), competitive equilibria exist even when certain types of complement. The concept of unimodular demand types includes both GS and GSC as special cases.