中文摘要
在公立学校选择中，具有严格偏好的学生被分配到学校。各学校被赋予针对学生的优先序。为纳入不同应用场景中的约束条件，优先序通常被建模为学生集合上的选择函数。已有观点认为，匹配最理想的评价标准是稳定性，即不应存在任何阻断对：不应有学生相比自己被分配到的学校更偏好另一所学校，同时满足以下两种情形之一：该学生的优先序高于某位已被该校录取的学生，或该校尚有空余名额。我们提出一种阻断概念，额外要求将该学生分配到其更偏好的学校是可行的。随后，我们为匹配集合定义如下稳定性标准：一个匹配集合是合法的，当且仅当集合外的每个匹配都被集合内的某个匹配阻断，且集合内任意两个匹配均不阻断对方。我们证明，在优先序满足非常基本的条件时，总是存在唯一的合法匹配集合，且该集合与稳定匹配集合具有共同的结构性质：（i）它是一个格；（ii）它满足乡村医院定理。学生最优合法匹配是有效率的，并为稳定性与效率之间的冲突提供了一种解决方案。
Abstract
Abstract In public school choice, students with strict preferences are assigned to schools. Schools are endowed with priorities over students. Incorporating constraints from different applications, priorities are often modelled as choice functions over sets of students. It has been argued that the most desirable criterion for an assignment is stability; there should not exist any blocking pair: no student shall prefer some school to her assigned school and have higher priority than some student who got into that school or the school has an empty seat. We propose a blocking notion where in addition it must be possible to assign the student to her preferred school. We then define the following stability criterion for a set of assignments: a set of assignments is legal if and only if any assignment outside the set is blocked with some assignment in the set and no two assignments inside the set block each other. We show that under very basic conditions on priorities, there always exists a unique legal set of assignments, and that this set has a structure common to the set of stable assignments: (i) it is a lattice and (ii) it satisfies the rural hospitals theorem. The student-optimal legal assignment is efficient and provides a solution for the conflict between stability and efficiency.