This paper examines sequential two-stage admission systems involving public and private schools, introducing a concept called "straightforwardness" to define a sequential notion of truthfulness. Unlike one-stage systems, sequential systems create a trade-off between the existence of a straightforward equilibrium and non-wastefulness. The paper identifies the unique set of rules for two-stage systems that ensure the existence of a straightforward equilibrium and minimize waste. It analyzes existing admission systems in Turkey and Sweden within this general framework. In many countries, both private and public schools are part of the education system. In OECD countries, 18% of 15-year-olds are admitted to privately managed schools. School seats are often assigned in two or more stages. The main difference between one-stage and multi-stage systems is that the latter includes rules for determining available schools, participating students, and assignment mechanisms in each stage. Examples include New York City and Boston, where admissions to exam schools and regular public schools are conducted separately and sequentially. In Turkey, the order of admissions to public and private schools and whether students can apply to both without commitment has changed. In Sweden, a centralized public school assignment follows private school admissions. Almost all Swedish municipalities have either parallel or sequential assignments. The work of Abdulkadiroğlu and Sönmez (2003) initiated research on centralized school choice, focusing on single-stage systems. However, sequential systems are more common. The paper's framework considers multi-stage systems with any combination of centralized and decentralized admissions, which is typical in real-life student admissions. When students apply through sequential admissions, properties from static systems may be compromised. This is because students make multiple decisions over time, leading to spillovers between schools. A key question is the trade-off between welfare and incentive issues. The paper addresses how a multi-stage system should be organized in terms of timing, participation, and other factors.This paper examines sequential two-stage admission systems involving public and private schools, introducing a concept called "straightforwardness" to define a sequential notion of truthfulness. Unlike one-stage systems, sequential systems create a trade-off between the existence of a straightforward equilibrium and non-wastefulness. The paper identifies the unique set of rules for two-stage systems that ensure the existence of a straightforward equilibrium and minimize waste. It analyzes existing admission systems in Turkey and Sweden within this general framework. In many countries, both private and public schools are part of the education system. In OECD countries, 18% of 15-year-olds are admitted to privately managed schools. School seats are often assigned in two or more stages. The main difference between one-stage and multi-stage systems is that the latter includes rules for determining available schools, participating students, and assignment mechanisms in each stage. Examples include New York City and Boston, where admissions to exam schools and regular public schools are conducted separately and sequentially. In Turkey, the order of admissions to public and private schools and whether students can apply to both without commitment has changed. In Sweden, a centralized public school assignment follows private school admissions. Almost all Swedish municipalities have either parallel or sequential assignments. The work of Abdulkadiroğlu and Sönmez (2003) initiated research on centralized school choice, focusing on single-stage systems. However, sequential systems are more common. The paper's framework considers multi-stage systems with any combination of centralized and decentralized admissions, which is typical in real-life student admissions. When students apply through sequential admissions, properties from static systems may be compromised. This is because students make multiple decisions over time, leading to spillovers between schools. A key question is the trade-off between welfare and incentive issues. The paper addresses how a multi-stage system should be organized in terms of timing, participation, and other factors.