Student-optimal Stable Mechanism Research Articles

Constrained school choice: an experimental QRE analysis

The theoretical literature on public school choice proposes centralized mechanisms that assign children to schools on the basis of parents’ preferences and the priorities children have for different schools. The related experimental literature analyzes in detail how various mechanisms fare in terms of welfare and stability of the resulting matchings, yet often provides only aggregate statistics of the individual behavior that leads to these outcomes (i.e., the degree to which subjects tell the truth in the induced simultaneous move game). In this paper, we show that the quantal response equilibrium (QRE) adequately describes individual behavior and the resulting matching in three constrained problems for which the immediate acceptance mechanism and the student-optimal stable mechanism coincide. Specifically, the comparative statics of the logit-QRE with risk-neutral and expected-payoff-maximizing agents capture the directional changes of subject behavior and the prevalence of the different stable matchings when cardinal payoffs (i.e., relative preference intensities) are modified in the experiment.

On the equivalence of two competing affirmative actions in school choice

This note analyzes the outcome equivalence conditions of two popular affirmative action policies, majority quota and minority reserve, under the student optimal stable mechanism. These two affirmative actions generate an identical matching outcome, if the market either is effectively competitive or contains a sufficiently large number of schools.

Improving schools through school choice: An experimental study of deferred acceptance

In the context of school choice, we experimentally study the student-optimal stable mechanism where subjects take the role of students and schools are passive. Specifically, we study if a school can be better off when it unambiguously improves in the students’ true preferences and its (theoretic) student-optimal stable match remains the same or gets worse. Using first-order stochastic dominance to evaluate the schools’ distributions over their actual matches, we find that schools’ welfare almost always changes in the same direction as the change of the student-optimal stable matching, i.e., incentives to improve school quality are nearly idle.

Consistent Pareto improvement over the student-optimal stable mechanism

We show that there is no consistent Pareto improvement over any stable mechanism. We introduce the following weakly consistent Pareto improvement (over the student-optimal stable matching) requirement: whenever a set of students, each of whom is assigned to a school that is underdemanded at the student-optimal stable matching, is removed with their assigned seats, then the assignments of the remaining students do not change. Our main result is that the efficiency-adjusted deferred acceptance mechanism (Kesten in Q J Econ 125(3):1297–1348, 2010) is the unique weakly consistent Pareto improvement over the student-optimal stable mechanism.

Welfare and incentives in partitioned school choice markets

This paper examines the effects of partition structure of schools on students' welfare and on incentives students face under the iterative student optimal stable mechanism (I-SOSM), introduced by Manjunath and Turhan (2016), in divided school enrollment systems. I find that when school partition gets coarser students' welfare weakly increases under the I-SOSM for any number of iterations. I also show that under coarser school partitions the I-SOSM becomes weakly less manipulable for students (when iterated sufficiently many times to reach a stable assignment) according to the “as strongly manipulable as” criteria defined by Pathak and Sönmez (2013). These results suggest that when full integration is not possible keeping school partition as coarse as possible benefits students with respect to their welfare and incentives they face if stability is a concern for policymakers.

Welfare and Incentives in Partitioned School Choice Markets

Divided enrollment systems cause school assignments to be unfair and wasteful. Iterative version of the student-optimal stable mechanism (I-SOSM), proposed by Manjunath and Turhan (2016), achieves individually rational and fair assignments in such partitioned school choice markets for any number of iteration. It also reaches a non-wasteful matching when iterated sufficiently many times. This paper examines the effects of partition structure of schools on students' welfare and incentives they face under I-SOSM. I find that when school partition gets coarser students' welfare weakly increases under I-SOSM for any number of iteration. I also show that under coarser school partitions I-SOSM becomes weakly less manipulable for students according to a criteria defined by Pathak and Sonmez (2013). These results suggest that when full integration is not possible keeping school partition as coarse as possible benefits students with respect to their welfare and incentives they face if stability is a concern for policymakers.

On the welfare effects of affirmative actions in school choice

Recent evidence, from both academia and practice, indicates that implementing affirmative action policies in school choice problems may induce substantial welfare losses on the intended beneficiaries. This paper addresses the following two questions: what are the causes of such perverse consequences, and when we can effectively implement affirmative action policies without unsatisfactory outcomes. Using the minority reserve policy in the student optimal stable mechanism as an example, I show that two acyclicity conditions, type-specific acyclicity and strongly type-specific acyclicity, are crucial for the effective implementations of minority reserve policies. I further illustrate how restrictive these two acyclicity conditions are, and the intrinsic difficulty of embedding diversity goals into stable matching mechanisms. Under some regularity conditions, I demonstrate that the minority reserve policy is very unlikely to cause welfare losses on any minority students when the number of schools is sufficiently large.

On Effective Affirmative Action in School Choice

Recent evidence, from both academia and practice, indicates that implementing affirmative action policies in school choice problems may induce substantial welfare losses on the intended beneficiaries. This paper addresses the following two questions: what are the causes of such perverse consequences, and when we can effectively implement affirmative action policies without unsatisfactory outcomes. Using the minority reserve policy in the student optimal stable mechanism as an example, I show that two acyclicity conditions, type-specific acyclicity and strongly type-specific acyclicity, are crucial for the effective implementations of minority reserve policies. I further illustrate how restrictive these two acyclicity conditions are, and the intrinsic difficulty of embedding diversity goals into stable matching mechanisms. Under some regularity conditions, I demonstrate that the minority reserve policy is very unlikely to cause welfare losses on any minority students when the number of schools is sufficiently large.

Consistent Pareto Improvement over the Student Optimal Stable Mechanism

We show that there is no consistent Pareto improvement over any stable mechanism. To overcome this impossibility, we introduce the following weak consistency requirement: Whenever a set of students, each of whom is assigned to a school that is under-demanded at the student-optimal stable matching, are removed with their assigned seats, then the assignments of the remaining students should not change. We show that EADA (Kesten, 2010) is the unique mechanism that is weakly consistent and Pareto improves over the student-optimal stable mechanism.

Level-K Reasoning in School Choice

Assuming students have heterogeneous sophistication and use the level-k reasoning, we compare the Boston mechanism (BM) and the student-optimal stable mechanism (SOSM). We show that for any distribution of students’ sophistication, BM is never less efficient than SOSM and sometimes is more efficient. We also show that more sophisticated students do not necessarily game BM better. Our results are different from those obtained from the Nash equilibrium analysis, and provide new insights on school choice.

Coalitions and cliques in the school choice problem

The school choice mechanism design problem focuses on assignment mechanisms matching students to public schools in a given school district. The well-known Gale Shapley Student Optimal Stable Matching Mechanism (SOSM) is the most efficient stable mechanism proposed so far as a solution to this problem. However its inefficiency is well-documented, and recently the Efficiency Adjusted Deferred Acceptance Mechanism (EADAM) was proposed as a remedy for this weakness. In this note we describe two related adjustments to SOSM with the intention to address the same inefficiency issue. In one we create possibly artificial coalitions among students where some students modify their preference profiles in order to improve the outcome for some other students. Our second approach involves trading cliques among students where those involved improve their assignments by waiving some of their priorities. The coalition method yields the EADAM outcome among other Pareto dominations of the SOSM outcome, while the clique method yields all possible Pareto optimal Pareto dominations of SOSM. The clique method furthermore incorporates a natural solution to the problem of breaking possible ties within preference and priority profiles. We discuss the practical implications and limitations of our approach in the final section of the article.

When is the Boston mechanism strategy-proof?

Kumano (2013) is the first to investigate the Boston school choice mechanism (BOSM) under restricted priority domains. This paper strengthens and extends his result and shows that the BOSM is strategy-proof, if and only if it is fair, if and only if it is equivalent to the student-optimal stable mechanism (SOSM), and if and only if the number of total seats at any two schools exceeds the number of students.

Course Allocation via Stable Matching

The allocation of students to courses is a wide-spread and repeated task in higher education, often accomplished by a simple first-come first-served (FCFS) procedure. FCFS is neither stable nor strategy-proof, however. The Nobel Prize in Economic Sciences was awarded to Al Roth and Lloyd Shapley for their work on the theory of stable allocations. This theory was influential in many areas, but found surprisingly little application in course allocation as of yet. In this paper, different approaches for course allocation with a focus on appropriate stable matching mechanisms are surveyed. Two such mechanisms are discussed in more detail, the Gale-Shapley student optimal stable mechanism (SOSM) and the efficiency adjusted deferred acceptance mechanism (EADAM). EADAM can be seen as a fundamental recent contribution which recovers efficiency losses from SOSM at the expense of strategy-proofness. In addition to these two important mechanisms, a survey of recent extensions with respect to the assignment of schedules of courses rather than individual courses is provided. The survey of the theoretical literature is complemented with results of a field experiment, which help understand the benefits of stable matching mechanisms in course allocation applications.

Fictitious students creation incentives in school choice problems

We identify a new channel through which schools can potentially manipulate the well-known student and school-optimal stable mechanisms. We introduce two fictitious students creation manipulation notions where one of them is stronger. While the student and school-optimal stable mechanisms turn out to be weakly fictitious student-proof under acyclic (Ergin in Econometrica 88:485–494, 2002) and essentially homogeneous (Kojima in Games Econ Behav 82:1–14, 2013) priority structures, respectively, they still lack strong fictitious student-proofness. We then compare the mechanisms in terms of their vulnerability to manipulations in the sense of Pathak and Sonmez (Am Econ Rev 103(1):80–106, 2013) and find out that the student-optimal stable mechanism is more manipulable than the school-optimal one. Lastly, in the large market setting of Kojima and Pathak (Am Econ Rev 99(3):608–627, 2009), the student-optimal stable mechanism becomes weakly fictitious student-proof as the market is getting large.

Equilibria under deferred acceptance: Dropping strategies, filled positions, and welfare

We study many-to-one matching markets where hospitals have responsive preferences over students. We study the game induced by the student-optimal stable matching mechanism. We assume that students play their weakly dominant strategy of truth-telling.Roth and Sotomayor (1990) showed that equilibrium outcomes can be unstable. We prove that any stable matching is obtained in some equilibrium. We also show that the exhaustive class of dropping strategies does not necessarily generate the full set of equilibrium outcomes. Finally, we find that the ‘rural hospital theorem’ cannot be extended to the set of equilibrium outcomes and that welfare levels are in general unrelated to the set of stable matchings. Two important consequences are that, contrary to one-to-one matching markets, (a) filled positions depend on the equilibrium that is reached and (b) welfare levels are not bounded by the optimal stable matchings (with respect to the true preferences).

Equilibria Under Deferred Acceptance: Dropping Strategies, Filled Positions, and Welfare (Equilibrios bajo el mecanismo de aceptaci´on diferida: Estrategias de eliminaci´on, posiciones ocupadas y bienestar)

This paper studies many-to-one matching markets where each student is assigned to a hospital. Each hospital has possibly multiple positions and responsive preferences. We study the game induced by the student-optimal stable matching mechanism. We assume that students play their weakly dominant strategy of truth-telling.Roth and Sotomayor (1990) showed that there can be unstable equilibrium outcomes. We prove that any stable matching can be obtained in some equilibrium. We also show that the exhaustive class of dropping strategies does not necessarily generate the full set of equilibrium outcomes. Finally, we find that the so-called 'rural hospital theorem' cannot be extended to the set of equilibrium outcomes and that welfare levels are in general unrelated to the set of stable matchings. Two important consequences are that, contrary to one-to-one matching markets, (a) filled positions depend on the particular equilibrium that is reached and (b) welfare levels are not bounded by the student and hospital-optimal stable matchings (with respect to the true preferences).

Local and global consistency properties for student placement

In the context of resource allocation on the basis of priorities, Ergin (2002) identifies a necessary and sufficient condition on the priority structure such that the student-optimal stable mechanism satisfies a consistency principle. Ergin (2002) formulates consistency as a local property based on a fixed population of agents and fixed resources—we refer to this condition as local consistency and to his condition on the priority structure as local acyclicity. A related but stronger necessary and sufficient condition on the priority structure such that the student-optimal stable mechanism satisfies a more standard global consistency property is unit acyclicity.We provide necessary and sufficient conditions for the student-optimal stable mechanism to satisfy converse consistency principles. First, we identify a necessary and sufficient condition (local shift-freeness) on the priority structure such that the student-optimal stable mechanism satisfies local converse consistency. Interestingly, local acyclicity implies local shift-freeness and hence the student-optimal stable mechanism more frequently satisfies local converse consistency than local consistency. Second, in order for the student-optimal stable mechanism to be globally conversely consistent, one again has to impose unit acyclicity on the priority structure. Hence, unit acyclicity is a necessary and sufficient condition on the priority structure for the student-optimal stable mechanism to satisfy global consistency or global converse consistency.

Group robust stability in matching markets

We introduce the notion of group robust stability which requires robustness against a combined manipulation, first misreporting preferences and then rematching, by any group of students in the school choice type of matching markets. Our first result shows that there is no group robustly stable mechanism even under acyclic priority structures. Next, we define a weak version of group robust stability, called weak group robust stability. Our main theorem, then, proves that there is a weakly group robustly stable mechanism if and only if the priority structure of schools is acyclic, and in that case, it coincides with the student-optimal stable mechanism.

Robust stability in matching markets

In a matching problem between students and schools, a mechanism is said to be robustly stable if it is stable, strategy-proof, and immune to a combined manipulation, where a student first misreports her preferences and then blocks the matching that is produced by the mechanism. We find that even when school priorities are publicly known and only students can behave strategically, there is a priority structure for which no robustly stable mechanism exists. Our main result shows that there exists a robustly stable mechanism if and only if the priority structure of schools is acyclic (Ergin 2002), and in that case, the student-optimal stable mechanism is the unique robustly stable mechanism.

Fictitious Student Creation Incentives in School Choice Problems

We address the question of whether schools can manipulate the student-optimal stable mechanism by creating fictitious students in school choice problems. To this end, we introduce two different manipulation concepts, where one of them is stronger. Our first result demonstrates that the student-optimal stable mechanism is not strongly fictitious student-proof even under acyclic priority structures (Ergin (2002)). Then, for the weak one, our main theorem shows that the student-optimal stable mechanism is weakly fictitious student-proof if and only if the priority structure of schools is acyclic.