The trade-off between minimal instability and larger improvements over deferred acceptance

The celebrated Efficiency-Adjusted Deferred Acceptance mechanism (EADA) improves the efficiency of the DA algorithm via consented priority violations. Notwithstanding its many merits, we show that EADA can improve only two students when an alternative mechanism that Pareto-dominates DA could benefit all but one student. This shortfall in the number of students improved is not exclusive of EADA but extends to all setwise minimally unstable mechanisms, i.e. those that generate a set of blocking pairs that is never a strict superset of that of another mechanism. The incompatibility between number of students improved and minimal instability disappears when blocking pairs are compared cardinally rather than by set inclusion. In some problems, EADA can be doubly dominated: improving fewer students while generating more blocking pairs.