  We investigate notions of group fairness in bipartite matching markets
involving agents and jobs, where agents are grouped based on sensitive
attributes. Employing a geometric approach, we characterize how many agents can
be matched in each group, showing that the set of feasible matchings forms a
(discrete) polymatroid. We show how we can define weakly-fair matchings
geometrically, for which poly-matroid properties imply that they are maximal.
Next, we focus on strong fairness notions (inspired by group-fairness metrics
in machine learning), where each group gets their exact same fraction of their
entitlement, and we explore the Price of Fairness (PoF), i.e., the loss in
optimality when imposing such fairness constraints. Importantly, we advocate
for the notion of opportunity fairness, where a group entitlement is the
maximum number of agents that can be matched without the presence of other
competing groups. We show that the opportunity PoF is bounded independently of
the number of agents and jobs, but may be linear in the number of groups.
Finally, we provide improved bounds with additional structural properties, or
with stochastic graphs.
