What Matters in School Choice Tie-breakings?: How Competition Guides Design

School districts that adopt the Deferred Acceptance (DA) mechanism to assign students to schools face the tradeoff between fairness and efficiency when selecting how to exogenously break ties among equivalent students. We analyze a model with random generated preferences for students and compare two tie-breaking rules: a single lottery (STB) and DA with a separate lottery for each school (MTB). We consider three different notions for this comparison: stochastic dominance of rank distributions, variance of students' ranks, and number of Pareto improving pairs (pairs of students whom would be better off by swapping their positions). We identify that the balance between supply and demand is the determinant factor in these comparisons. When there is a shortage of seats, the rank distribution under STB stochastically dominates the rank distribution under MTB, and also, has a smaller variance. In addition, MTB creates ``many'' Pareto improving pairs, while STB creates none. When there is a surplus of seats, we show that neither random assignment under these mechanisms stochastically dominates the other, variance of the rank distribution is lower under MTB, and MTB generates few Pareto improving pairs. Our findings imply that ``popular" schools should use a single common lottery to break ties. Finally, numerical experiments using NYC school choice data confirm our predictions.