Lottery Design for School Choice

This paper studies outcomes of the deferred acceptance algorithm in large random matching markets where priorities are generated either by a single lottery or by independent lotteries. In contrast to prior work, my model permits students to submit lists of varying lengths and schools to vary in their popularity and number of seats. In a limiting regime where the number of students and schools grow while the length of student lists and number of seats at each school remain bounded, I provide exact expressions for the number of students who list l schools and match to one of their top k choices, for each k ≤ l. These expressions provide three main insights. First, there is a persistent tradeoff between using a single lottery and independent lotteries. For students who submit short lists, the rank distribution under a single lottery stochastically dominates the corresponding distribution under independent lotteries. However, the students who submit the longest lists are always more likely to match when schools use independent lotteries. Second, I compare the total number of matches in the two lottery systems, and find that the shape of the list length distribution plays a key role. If this distribution has an increasing hazard rate, then independent lotteries match more students. If it has a decreasing hazard rate, the comparison reverses. To my knowledge, this is the first analytical result comparing the size of stable matchings under different priority rules. Finally, I study the fraction of assigned students who receive their first choice. Under independent lotteries, this fraction may be arbitrarily small, even if schools are equally popular. Under a single lottery, we provide a tight lower bound on this fraction which depends on the ratio r of the popularity of the most to least popular school. When each school has a single seat, the fraction of assigned students who receive their first choice is at least √r=(1 + r). This guarantee increases to 2=(1 + √r) as the number of seats at each school increases.