  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.
