Sequential school choice with public and private schools

We investigate sequential two-stage admission systems with public and private schools. A sequential notion of truthfulness, called straightforwardness, is introduced. Contrary to one-stage systems, sequentiality leads to a trade-off between the existence of a straightforward equilibrium and non-wastefulness. We identify the unique set of rules for two-stage systems that guarantees the existence of a straightforward equilibrium and reduces waste. Existing admission systems in Türkiye and Sweden are analyzed within our general framework.