Lottery Design for School Choice.

This article analyzes the outcomes of the deferred acceptance algorithm in large random school choice markets, focusing on how two tie-breaking lottery procedures—independent lotteries at each school versus a single district-wide lottery—affect match results. The model allows students to submit lists of varying lengths and schools to differ in popularity and capacity, providing exact asymptotic expressions for the fraction of students matched to one of their top choices under each lottery system. Key findings reveal a tradeoff: single lotteries favor students with short lists and yield more first-choice matches, while independent lotteries benefit students with longer lists and often produce a higher total number of matches, especially when the distribution of list lengths has an increasing hazard rate. Additionally, the paper quantifies how the fraction of assigned students receiving their first choice depends on the relative popularity of schools and the number of seats available, offering policy-relevant insights into lottery design and its implications for equity and efficiency in school assignment systems.