  We study a multi-stage admission system, known as the Tiered Deferred
Acceptance mechanism, designed to benefit some schools over others. The current
US public school and Chinese college admission systems are two examples. In
this system, schools are partitioned into tiers, and the Deferred Acceptance
algorithm is applied within each tier. Once assigned, students cannot apply to
schools in subsequent tiers. This mechanism is not strategyproof. Therefore, we
study the Nash equilibria of the induced preference revelation game. We show
that Nash equilibrium outcomes are nested in the sense that merging tiers
preserves all equilibrium outcomes. We also identify within-tier acyclicity as
a necessary and sufficient condition for the mechanism to implement stable
matchings in equilibrium. Our findings suggest that transitioning from the
Deferred Acceptance mechanism to the Tiered Deferred Acceptance mechanism may
not improve student quality at top-tier schools as intended.
