Stable matchings, choice functions, and linear orders

Abstract

We consider a model of stable edge sets (``matchings'') in a bipartite graph $G=(V,E)$ in which the preferences for vertices of one side (``firms'') are given via choice functions subject to standard axioms of consistency, substitutability and cardinal monotonicity, whereas the preferences for the vertices of the other side (``workers'') via linear orders. For such a model, we present a combinatorial description of the structure of rotations and develop an algorithm to construct the poset of rotations, in time $O(|E|^2)$ (including oracle calls). As consequences, one can obtain a ``compact'' affine representation of stable matchings and efficiently solve some related problems. Keywords: bipartite graph, choice function, linear preferences, stable matching, affine representation, sequential choice