  A classic trade-off that school districts face when deciding which matching
algorithm to use is that it is not possible to always respect both priorities
and preferences. The student-proposing deferred acceptance algorithm (DA)
respects priorities but can lead to inefficient allocations. We identify a new
condition on school choice markets under which DA is efficient. Our condition
generalizes earlier conditions by placing restrictions on how preferences and
priorities relate to one another only on the parts that are relevant for the
assignment. Whenever there is a unique allocation that respects priorities, our
condition captures all the environments for which DA is efficient. We show
through stylized examples and simulations that our condition significantly
expands the range of known environments for which DA is efficient. We also
discuss how our condition sheds light on existing empirical findings.
