The Double-Edged Sword of Information: Revealed versus Hidden Lotteries in School Choice

In school choice, a lottery number is often used by the matching mechanism to break ties when there are more students who prefer the same school than the number of seats available. There has been growing theoretical and empirical interest in understanding the impact of revealing the lottery number to students. In practice, in recent years, the NYC Public Schools started revealing the lottery number to students to improve transparency. Theoretical findings from prior literature also suggest that revealing the lottery number strictly improves the number of matches under the deferred acceptance algorithm. However, these theoretical results are based on the over-simplifying assumption that all students share the same preference ranking for schools. In this work, we relax this assumption and allow students to have heterogeneous preference rankings. Under a game-theoretic model where student strategies form a Bayesian Nash equilibrium, we characterize scenarios where revealing the lottery number can either improve or worsen the matching outcome, measured by two metrics of match rate and social welfare. We further consider revealing partial information about the lottery, and demonstrate non-monotonic effects in the amount of information available to students. These results together illustrate complex tradeoffs induced by the lottery revealing policy.