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Using school choice as a motivating example, we introduce a stylized model of a many-to-one matching market where the clearinghouse aims to implement contingent priorities, i.e., priorities that depend on the current assignment, to prioritize students with siblings and match them together. We provide a series of guidelines and introduce two natural approaches to implement them: (i) absolute, whereby a prioritized student can displace any student without siblings assigned to the school, and (ii) partial, whereby prioritized students can only displace students that have a less favorable lottery than their priority provider. We study several properties of the corresponding mechanisms, including the existence of a stable assignment under contingent priorities, the complexity of deciding whether there exists one, and its incentive properties. Furthermore, we introduce a soft version of these priorities to guarantee existence, and we provide mathematical programming formulations to find such stable matching or certify that one does not exist. Finally, using data from the Chilean school choice system, we show that our framework can significantly increase the number of students assigned to their top preference and the number of siblings assigned together relative to current practice.

Motivated by online dating apps, we study how to select subset of profiles to show to each user in each period in a two-sided matching platform. Users on each side observe the profiles set by the platform and decide which of them to like. A match occurs if and only if two users mutually like each other, potentially in different periods. The goal of the platform is to maximize the total expected number of matches. We study different platform designs, varying (i) how users interact with each other, i.e., whether one or both sides of the market can initiate an interaction, and (ii) the timing of matches, i.e., whether the platform allows non-sequential matches in addition to sequential ones. We focus on the case with two periods and study the performance of different approaches. First, we show that algorithms that exploit the submodularity of the problem and properties of its feasible region can achieve constant factor approximation guarantees that depend on the platform design, ranging from $1-1/e$ to $1/3$. Finally, we show that the Dating Heuristic (DH) (Rios et al., 2023), which is commonly used and achieves good performance in practice, provides an approximation guarantee of $1-1/e$ for all platform designs. We show theoretically and empirically that the performance of the DH is robust to the platform design. Our simulation results -- using real data from our industry partner -- also show that platforms using a one-directional design should initiate interactions with the side that leads to the smallest expected backlog per profile displayed, balancing size and selectivity. Moreover, we find that a one-directional design can lead to at least half of the matches obtained with a two-directional design. Finally, our results show that avoiding non-sequential matches has no sizable effect, regardless of the platform design.

Motivated by the shortage of seats that the Chilean school choice system is facing, we introduce the problem of jointly increasing school capacities and finding a student-optimal assignment in the expanded market. Due to the theoretical and practical complexity of the problem, we provide a comprehensive set of tools to solve the problem, including different mathematical programming formulations, a cutting plane algorithm, and two heuristics that allow obtaining near-optimal solutions quickly. On the theoretical side, we show the correctness of our formulations, different properties of the objective and feasible region that facilitate computation, and also several properties of the underlying mechanism to find a student-optimal matching under capacity expansions. On the computational side, we use data from the Chilean school choice system to demonstrate the impact of our framework and derive insights that could help alleviate the problem. Our results show that each additional seat can benefit multiple students and that we can effectively target the assignment of previously unassigned students or improve the assignment of several students through improvement chains. Nevertheless, our results show that the marginal effect of each additional seat is decreasing and that simply adding seats is insufficient to ensure every student gets assigned to some school. Finally, we discuss several extensions of our framework, showcasing its flexibility to accommodate different needs.