Constrained Optimization Approaches to Solving Real World School Choice Problems

Matching markets have been the subject of intensive mechanism design efforts over the past 15 years with school choice being a frequent application, which has led Boston and other large cities employ centralized matching procedures to allocate students to public schools. One common recommendation of market designers is to build a school assignment mechanism in two steps. First, given the school system’s public policy goals, choose a priority structure that gives certain students higher priorities at certain schools than other students. For example, to encourage a student to attend her neighborhood school, the Boston Public Schools (BPS) until recently gave higher priority for seats at a school to students that live close to that school. Second, given the priority structure chosen, solicit student preferences over the schools and run a version of the Gale-Shapley algorithm to generate the school assignment. There is much to recommend this advice. First, the outcome of the Gale-Shapley algorithm is incentive compatible, so the students have no motive to try and manipulate the assignment system. Second, the outcome is stable, which reflects a notion of fairness that requires that an agent with a high priority for a good be allocated that good before a lower priority agent is provided the good. However, it is far from obvious that encoding policy goals into the priority structure and then running the Gale-Shapley algorithm is the most effective method of implementing goals such as encouraging neighborhood schools. In addition, it is not clear what the trade-offs are between goals such as student welfare and encouraging neighborhood schools. I argue in this paper that matching problems in general, and school choice problems in particular, can be posed as computationally tractable constrained optimization problems. The constrained optimization approach has the advantage of allowing the market designer to explicitly describe the policy goals in the objective function and encode required properties such as stability and incentive compatibility in the constraints. Using my approach I can compute the global optimum for various desiderata (e.g., the student welfare maximizing school assignment) or assess trade-offs between the goals (e.g., encouraging neighborhood schools and school diversity). The optimization approach also allows me to easily assess the benefits of weakening the constraints of the problem by, for example, adding capacity to a popular school. Treating a school choice problem as a constrained optimization problem is not a novel theoretical idea. However, I combine this idea with the more recent approach of modeling large school choice problems as a match between a continuum of seats at a finite set of schools and a continuum of each of a finite set of student types. Since each student has a negligible effect on the aggregate outcome, I can write the incentive compatibility conditions in a computationally tractable form. Although the resulting optimization problem