  Many schools in large urban districts have more applicants than seats.
Centralized school assignment algorithms ration seats at over-subscribed
schools using randomly assigned lottery numbers, non-lottery tie-breakers like
test scores, or both. The New York City public high school match illustrates
the latter, using test scores and other criteria to rank applicants at
``screened'' schools, combined with lottery tie-breaking at unscreened
``lottery'' schools. We show how to identify causal effects of school
attendance in such settings. Our approach generalizes regression discontinuity
methods to allow for multiple treatments and multiple running variables, some
of which are randomly assigned. The key to this generalization is a local
propensity score that quantifies the school assignment probabilities induced by
lottery and non-lottery tie-breakers. The local propensity score is applied in
an empirical assessment of the predictive value of New York City's school
report cards. Schools that receive a high grade indeed improve SAT math scores
and increase graduation rates, though by much less than OLS estimates suggest.
Selection bias in OLS estimates is egregious for screened schools.
