The Cutoff Structure of Top Trading Cycles in School Choice∗

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 assignment by specifying priorities. However, the common combinatorial description of TTC does little to explain the relationship between student priorities and their eventual assignment. This creates difficulties in transparently communicating TTC to parents and in guiding policy choices of school boards. We show that the TTC assignment can be described by n2 admission thresholds, where n is the number of schools. These thresholds can be observed after mechanism is run, and can serve as non-personalized prices that allow parents to verify their assignment. In a continuum model these thresholds can be computed directly from the distribution of preferences and priorities, providing a framework that can be used to evaluate policy choices. We provide closed form solutions for the assignment under a family of distributions, and derive comparative statics. As an application of the model we solve for the welfare maximizing investment in school quality, and find that a more egalitarian investment can be more efficient because it promotes more efficient sorting by students. ∗This paper was previously circulated as “The Simple Structure of Top Trading Cycles in School Choice”. A supplementary online appendix is available at http://www.columbia.edu/ ̃iyl2104/Leshno,Lo-TTC%20(online%20appendix).pdf. We thank (in random order) Cyndia Yu, John Tsitsiklis, Yeon-Koo Che, Larry Samuelson, Eduardo Azevedo, Nikhil Agarwal, Jay Sethuraman, Chiara Margaria, Itai Ashlagi, Laura Doval, Parag Pathak, Jonah Rockoff, Yusuke Narita, Fuhito Kojima, Alvin Roth and seminar participants in Berkeley, Columbia, Stanford, Yale, University of Toronto, Duke, NYU, Matching in Practice workshop, and Cireq Montreal Microeconomics meeting for their helpful comments. †Columbia University, email: [email protected] and [email protected]