The paper "Sequential school choice with public and private schools" by Tommy Andersson, Umut Dur, Sinan Ertemel, and Onur Kesten explores sequential two-stage admission systems involving both public and private schools. The authors introduce a new concept called "straightforwardness," which is a sequential notion of truthfulness. Unlike one-stage systems, sequentiality introduces 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 reduce waste. It analyzes existing admission systems in Türkiye and Sweden within this framework. The introduction highlights the prevalence of multi-stage admission systems in various countries, such as New York City, Boston, and the Greater Stockholm Region, and discusses the challenges these systems face, including the potential for spillovers and the need to balance welfare and incentive issues. The paper aims to provide insights into how multi-stage systems should be ideally organized in terms of timing, participation, and other factors.The paper "Sequential school choice with public and private schools" by Tommy Andersson, Umut Dur, Sinan Ertemel, and Onur Kesten explores sequential two-stage admission systems involving both public and private schools. The authors introduce a new concept called "straightforwardness," which is a sequential notion of truthfulness. Unlike one-stage systems, sequentiality introduces 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 reduce waste. It analyzes existing admission systems in Türkiye and Sweden within this framework. The introduction highlights the prevalence of multi-stage admission systems in various countries, such as New York City, Boston, and the Greater Stockholm Region, and discusses the challenges these systems face, including the potential for spillovers and the need to balance welfare and incentive issues. The paper aims to provide insights into how multi-stage systems should be ideally organized in terms of timing, participation, and other factors.