In a typical school choice application, the students have strict preferences over the schools while the schools have coarse priorities over the students based on their distance and their enrolled siblings. The outcome of a centralized admission mechanism is then usually obtained by the Deferred Acceptance (DA) algorithm with random tie-breaking. Therefore, every possible outcome of this mechanism is a stable solution for the coarse priorities that will arise with certain probability. This implies a probabilistic assignment, where the admission probability for each student-school pair is specified. In this paper, we propose a new efficiency-improving stable `smart lottery' mechanism. We aim to improve the probabilistic assignment ex-ante in a stochastic dominance sense, while ensuring that the improved random matching is still ex-post stable, meaning that it can be decomposed into stable matchings regarding the original coarse priorities. Therefore, this smart lottery mechanism can provide a clear Pareto-improvement in expectation for any cardinal utilities compared to the standard DA with lottery solution, without sacrificing the stability of the final outcome. We show that although the underlying computational problem is NP-hard, we can solve the problem by using advanced optimization techniques such as integer programming with column generation. We conduct computational experiments on generated and real instances. Our results show that the welfare gains by our mechanism are substantially larger than the expected gains by standard methods that realize efficiency improvements after ties have already been broken.

翻译：在典型的学校选择应用中，学生对于学校具有严格的偏好排序，而学校则根据学生的居住距离及在校兄弟姐妹情况对学生形成粗略的优先级排序。集中录取机制的结果通常通过延迟接受算法结合随机破平方式获得。因此，该机制所有可能的结果都是针对特定概率下出现的粗略优先级所形成的稳定解。这定义了一种概率分配机制，其中每个学生-学校对的录取概率得以明确。本文提出一种新的效率改进型稳定“智能抽签”机制。我们的目标是在随机占优意义上改进事前概率分配，同时确保改进后的随机匹配仍满足事后稳定性——即该匹配可分解为关于原始粗略优先级的稳定匹配。因此，与采用随机破平的标准延迟接受算法相比，该智能抽签机制能在不牺牲最终结果稳定性的前提下，为任何基数效用函数提供明确的期望帕累托改进。我们证明，尽管底层计算问题属于NP难问题，但可通过列生成整数规划等先进优化技术进行求解。我们在生成实例和真实实例上进行了计算实验，结果表明：相较于传统方法在破平后实现效率改进的预期收益，本机制带来的福利增益显著更大。