What Pareto-efficiency adjustments cannot fix

The Deferred Acceptance (DA) algorithm is stable and strategy-proof, but can produce matchings that are Pareto-inefficient for students, and thus several alternatives have been proposed to correct this inefficiency that only involve consented priority violations. However, we show that these approaches cannot correct DA's suboptimal rank distribution, because this shortcoming can arise even in cases where DA is Pareto-efficient. We also examine student segregation in settings with tiered priority structures. We prove that the demographic composition of every school is perfectly preserved under any Pareto-efficient rule that dominates DA, and consequently fully segregated schools under DA maintain their extreme homogeneity.