The Simple Structure of Top Trading Cycles in School Choice∗

Many cities determine the assignment of students to schools through a school choice mechanism. The prominent Top Trading Cycles (TTC) mechanism has attractive properties for school choice, as it is strategy-proof, Pareto efficient, and allows school boards to guide the allocation by specifying priorities. However the combinatorial description of TTC makes it non-transparent to parents and difficult for designers to analyze. We give a tractable characterization of the TTC mechanism for school choice: the TTC assignment can be simply described by n2 admission thresholds, where n is the number of schools, and these thresholds can be easily observed after running the mechanism. To calculate these thresholds, we define TTC through trade balance equations. In a continuum model these equations correspond to a differential equation whose solution can give closed form expressions for the admission thresholds. The model allows us to compute comparative statics, and evaluate welfare. As applications of the model we solve for optimal investment in school quality, explore the design of the priority structure, and provide comparisons between TTC and other school choice mechanisms that help explain empirical findings. To validate the continuum model we show that it gives a good approximation for strongly converging economies. Our analysis draws on an interesting connection between continuous trading procedures and continuous time Markov