Longer Lists Yield Better Matchings

Many centralized mechanisms for two-sided matching markets that enjoy strong theoretical properties assume that the planner solicits full information on the preferences of each participating agent. In particular, they expect that participants compile and communicate their complete preference lists over agents from the other side of the market. However, real-world markets are often very large and agents cannot always be expected to even produce a ranking of all options on the other side. It is therefore important to understand the impact of incomplete or truncated lists on the quality of the resultant matching. In this paper, we focus on the Serial Dictatorship mechanism in a model where each agent of the proposing side (students) has a random preference list of length $d$, sampled independently and uniformly at random from $n$ schools, each of which has one seat. Our main result shows that if the students primarily care about being matched to any school of their list (as opposed to ending up unmatched), then all students in position $i\leq n$ will prefer markets with longer lists, when $n$ is large enough. Schools on the other hand will always prefer longer lists in our model. We moreover investigate the impact of $d$ on the rank of the school that a student gets matched to. Our main result suggests that markets that are well-approximated by our hypothesis and where the demand of schools does not exceed supply should be designed with preference lists as long as reasonable, since longer lists would favor all agents.