Measuring the Welfare Gains from Cardinal-Preference Mechanisms in School Choice

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.