Identifying and quantifying (un)improvable students

The Deferred Acceptance (DA) mechanism can generate inefficient placements. Although Pareto-dominant mechanisms exist, it remains unclear which and how many students could improve their DA assignment. We characterize the set of unimprovable students and show that it includes those unassigned or matched with their least preferred schools. Nevertheless, by proving that in large markets DA's envy digraph contains a unique giant strongly connected component, we establish that almost all students are improvable, and furthermore, they can benefit simultaneously via disjoint trading cycles. Our findings reveal both the pervasiveness of DA's inefficiency and the remarkable effectiveness of Pareto-dominant mechanisms in addressing it, regardless of the specific mechanism chosen.