NBER Working Papers · 2026 · Hülya Eraslan、Jeremy T. Fox、YingHua He
中文摘要
序数机制使用排序，但不使用偏好强度。基数机制还会征询学生对学校的效用，因而能够利用学生对分配彩票的偏好。我们将概率序列机制与四种基数机制进行比较；这些机制在依次更强的约束下最大化同一福利目标，分别为：仅容量约束、ε-无嫉妒、无嫉妒，以及福利最大化的基数偏好伪市场。由于可行集相互嵌套，随着约束从仅有容量约束收紧至近似无嫉妒、精确无嫉妒，以及共同价格下的等额预算，最大福利弱下降（非严格下降）。我们就大市场中的如实报告确立了正面和负面结果。论文中定理的全部44个组成部分均已在Lean 4证明辅助器中完成机器验证。我们开发了计算西雅图898名学生和11所学校的基数偏好伪市场均衡的方法。利用根据西雅图高中择校数据进行集合识别的基数偏好，我们估计，在精确容量约束下，仅容量约束机制相较于概率序列机制将平均福利提高0.053，这相当于把5.3个百分点的分配概率从平均学生的最差学校转移到最好学校。精确无嫉妒保留了这一增益的79%；所找到的最佳伪市场均衡带来的平均增益为0.002。这些估计结果既表明了基数信息的价值，也揭示了伪市场公平性约束的福利成本。
Abstract
Ordinal mechanisms use rankings but not preference intensities. Cardinal mechanisms also elicit school utilities, allowing them to use students’ preferences over assignment lotteries. We compare probabilistic serial with four cardinal mechanisms that maximize the same welfare objective under successively stronger restrictions: capacity-only, ε-envy-free, envy-free, and a welfare-maximizing cardinal-preference pseudomarket. Because the feasible sets are nested, maximum welfare weakly falls as the restrictions tighten from capacity alone to approximate no-envy, exact no-envy, and equal budgets with common prices. We establish positive and negative results on large-market truthtelling. All 44 theorem parts in the paper are machine-checked in the Lean 4 proof assistant. We develop methods for computing cardinal-preference pseudomarket equilibria for Seattle’s 898 students and 11 schools. Using set-identified cardinal preferences from Seattle high-school choice data, we estimate that, under exact capacity, capacity-only raises mean welfare over probabilistic serial by 0.053, equivalent to shifting 5.3 percentage points of assignment probability from the average student’s worst school to the top school. Exact envy-freeness retains 79% of this gain; best-found pseudomarket equilibria yield a mean gain of 0.002. The estimates show both the value of cardinal information and the welfare cost of pseudomarket fairness restrictions.