Symmetric mechanisms for two-sided matching problems

We focus on the basic one-to-one two-sided matching model, where there are two disjoint sets of agents of equal size, and each agent in a set has preferences on the agents in the other set, modelled by linear orders. The goal is to find a matching that associates each agent in one set with one and only one agent in the other set based on the agents' preferences. A mechanism is a rule that associates a set of matchings to each preference profile. Stability, which refers to the capability to select only stable matchings, is an important property a mechanism should fulfill. Another crucial property, especially useful for applications, is resoluteness, which requires that the mechanism always selects a unique matching. The two versions of the deferred acceptance algorithm are examples of stable and resolute mechanisms. However, these mechanisms are severely unfair since they strongly favor one of the two sides of the market. In this paper, we introduce a property that mechanisms may meet which relates to fairness. Such property, called symmetry, is formulated in a way able to capture different levels of fairness within and across the two sets of agents and generalize existing notions. We prove several possibility and impossibility results, mainly involving the most general notion of symmetry, known as gender fairness: among others, a resolute and gender fair mechanism exists if and only if each side of the market consists of an odd number of agents; there exists no resolute, stable and gender fair mechanism.