Competitive equilibria in school assignment

doi:10.1016/j.geb.2017.10.003

Games and Economic Behavior

Volume 108, March 2018, Pages 269-274

https://doi.org/10.1016/j.geb.2017.10.003 Get rights and content

Abstract

Top Trading Cycles was originally developed as an elegant method for finding a competitive equilibrium of Shapley and Scarf's housing market. We extend the definition of a competitive equilibrium to the school assignment problem and show that there remains a profound relationship between Top Trading Cycles and a competitive equilibrium. Specifically, in every competitive equilibrium with weakly decreasing prices, the equilibrium assignment is unique and exactly corresponds to the Top Trading Cycles assignment. This provides a new way of interpreting the worth of a student's priority at a given school. It also provides a new way of explaining Top Trading Cycles to students and a school board.

Introduction

A hallmark of Lloyd Shapley's work is to introduce a seemingly simple model and to provide an elegant solution to the problem. A great part of his lasting legacy is that many years later these problems and solutions have been the basis for modern market design. A perfect example of this is school assignment. One of the seminal papers in market design is Abdulkadiroğlu and Sönmez (2003) which introduced school assignment as a market design problem.¹ One can see the influence of Gale and Shapley (1962) and Shapley and Scarf (1974) throughout that paper. While Abdulkadiroğlu and Sönmez's model is described as a variation of Gale and Shapley's College Admissions model, it can also be viewed as a generalization of the Shapley and Scarf (1974) house exchange model.² Moreover, the two solutions Abdulkadiroğlu and Sönmez propose, the Deferred Acceptance algorithm and the Top Trading Cycles algorithm (hereafter TTC), are both natural modifications of the algorithms introduced in Gale and Shapley (1962) and Shapley and Scarf (1974), respectively.

While Shapley and Scarf (1974) is primarily remembered for the introduction of TTC, this was not the paper's original purpose. Debreu and Scarf (1963) established the connection between the core and a competitive equilibrium under standard regularity conditions such as convexity of preferences, transferable utility, and perfect divisibility of the objects. Shapley and Scarf's stated purpose is to demonstrate that the same relationship between the core and a system of competitive prices can exist in a model where none of these assumptions hold. In their housing exchange model, there are n people and n houses. Each person owns one house and has only ordinal preferences over the other houses. Shapley and Scarf demonstrate that in this model, where none of the standard regularity conditions hold, a core assignment always exists and corresponds in a natural way to the outcome from a system of competitive equilibrium prices.

Many papers have studied the properties of TTC when applied to the school assignment problem. However, to the best of our knowledge, no paper has considered whether or not a competitive equilibrium exists in this more general model and if so, what its properties are. Stated differently, in the housing model it is not clear a priori the relative value of each house. However, this is revealed by solving for the competitive equilibrium. In the school assignment problem, no student owns a seat at a school. Instead, a student has a priority at each school. These priorities represent a claim on the school, and therefore result in some degree of “ownership” over the school. However, it is not clear a priori the value of a priority at a particular school.

In this paper we return to the classic concept of a competitive equilibrium in order to better understand the value of a particular priority at a particular school in the school assignment problem. Our approach is to assign a value to each priority. We allow each student to sell her most valuable priority and buy the best priority that she is able to afford (where the “best” priority is the priority that gains her admittance to the best possible school). We define an equilibrium to occur when the prices clear the market. Specifically, for each school a with capacity for q students, q priorities are sold, q priorities are purchased, and these priorities exactly coincide.

We demonstrate that a competitive equilibrium always exists for the school assignment problem (Theorem 1).³ This result will not be surprising to readers familiar with TTC. Just as in the housing market, we show that TTC can be used to find a competitive equilibrium of the school assignment problem. However, our main result is surprising. We show that all competitive equilibria where prices are weakly decreasing induce the same assignment: the assignment made by TTC (Theorem 2).

Interestingly, a similar result is true for the original Shapley and Scarf housing model. Roth and Postlewaite (1977) demonstrate that the unique assignment in the core is the assignment made by TTC. As any competitive assignment is in the core, this establishes that a student purchases the same house in any competitive equilibrium. However, the school assignment problem is significantly more complicated than the housing market problem. While there exists only one “reasonable” assignment for a housing problem, there exist several for the school assignment problem.⁴ Therefore, it is surprising that monotonic equilibria have such a consistent structure. We demonstrate that alternative assignments can result from a competitive equilibrium when the worth of priorities are not monotonic (Example 1).

Section snippets

Model

We consider a finite set of students $I = {i, j, k, \dots}$ and a finite set of schools $S = {a, b, c, \dots}$ . Each student $i \in I$ has a complete, irreflexive, and transitive preference relation $P_{i}$ over $S \cup {\emptyset}$ where ∅ denotes the option of being unassigned. Here, $a P_{i} b$ indicates that student i strictly prefers school a to school b. Given $P_{i}$ , we define the symmetric extension $R_{i}$ by $a R_{i} b$ if and only if $a P_{i} b$ or $a = b$ . A school a is acceptable for student i if $a P_{i} \emptyset$ .

The capacity of each school $a \in S$ is given by $q_{a}$ . Let $q_{\emptyset} = | I |$ .