Many school districts apply the student-proposing deferred acceptance algorithm after ties among students are broken exogenously. We compare two common tie-breaking rules: one in which all schools use a single common lottery, and one in which every school uses a separate independent lottery. We identify the balance between supply and demand as the determining factor in this comparison.

First we analyze a two-sided matching model with random preferences in over-demanded and under-demanded markets. In a market with a surplus of seats, a common lottery is less equitable, and there are efficiency tradeoffs between the two tie-breaking rules. However, a common lottery is always preferable when there is a shortage of seats in the sense of stochastic dominance of rank distribution.
The theory suggests that popular schools should use a common lottery to resolve ties.
We run numerical experiments with New York City school choice data after partitioning the market into popular and non-popular schools. The experiments support our findings.