Centralized student-school matching systems, adopted in the United States, Chile, the Netherlands, and the United Kingdom, assign students to schools based on their submitted preference rankings and schools' priority policies. Yet welfare in these systems is typically measured through a narrow lens: a few aggregate statistics published by education authorities, such as the share of students unmatched or assigned to a top-ranked choice. A single aggregate cannot reveal how outcomes vary across students, or why. Welfare depends on three interacting forces: the rules policymakers set (algorithm, tie-breaking, list length limits); the choices students make (which and how many schools to rank); and chance (lottery draws and students' residential location). A rule that reduces unmatched rates may simultaneously lower top-ranked assignment rates for specific groups, and evaluating such trade-offs in advance is challenging because individual-level preference data are rarely available. We present Trace (Trade-offs in Rules, Agency, Chance, and Equity), an open-source framework that synthesizes individual-level preference profiles from aggregate data using mixtures of Mallows models and evaluates alternative rules across multiple welfare metrics, disaggregating outcomes by choice and chance. We apply Trace to two real-world systems: NYC high school matching, where we study minimum list length requirements, and Chilean school assignment, where we compare the current multiple tie-breaking rule with a counterfactual single tie-breaking rule. Both reveal substantial, heterogeneous welfare trade-offs, underscoring the value of simulation-based evaluation before implementing school choice reforms.

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