Expanding one slot changes everyone's allocation.
Bengtsson and Engström (arXiv:2603.21917) prove a “cascade identity” for two-stage least squares in capacity-constrained settings. When slots are rationed — university admissions, medical residencies, public housing — adding one slot to program k doesn't just affect the marginal entrant. It triggers a chain of reallocations. The student who gets in displaces from their second-choice program, freeing a slot there, which pulls someone from their third-choice, and so on. The cascade propagates through the entire allocation system.
The identity states that the 2SLS coefficient equals the total societal effect of expanding treatment k by one slot, including all cascading reallocations. This requires only instrument relevance and that the instrument operates on the same margin as the policy — the same margin condition. No monotonicity assumption about individual treatment effects. No exclusion restriction beyond the standard one.
This reinterprets what 2SLS estimates in these settings. The conventional reading is “the effect of treatment on the treated.” The cascade identity reads it as “the system-level effect of one additional slot” — a fundamentally different policy parameter. The individual treatment effect and the system effect coincide only when there are no cascades, which means no capacity constraints, which means the rationing structure doesn't matter.
The application uses lottery-based Swedish university admissions with charitable giving as outcome. But the theoretical contribution is general: any system where fixed supply constrains allocation — from organ transplants to school choice — has cascading reallocation effects, and 2SLS captures the total cascade, not just the direct effect. The parameter you thought you were estimating is not the parameter you have.