Deferred Acceptance Mechanism Research Articles

Strategyproof and fair matching mechanism for union of symmetric m-convex constraints.

We identify a new class of distributional constraints defined as a union of symmetric M-convex sets, which can represent a wide range of real-life constraints in two-sided matching settings. Since M-convexity is not closed under union, a union of symmetric M-convex sets does not belong to this well-behaved class of constraints. Consequently, devising a fair and strategyproof mechanism to handle this new class is challenging. We present a novel mechanism for it called Quota Reduction Deferred Acceptance (QRDA), which repeatedly applies the standard Deferred Acceptance mechanism by sequentially reducing artificially introduced maximum quotas. We show that QRDA is fair and strategyproof when handling a union of symmetric M-convex sets, which extends previous results obtained for a subclass of the union of symmetric M-convex sets: ratio constraints. QRDA always yields a weakly better matching for students than a baseline mechanism called Artificial Cap Deferred Acceptance (ACDA). We also experimentally show that QRDA outperforms ACDA in terms of nonwastefulness.

Dominated choices under deferred acceptance mechanism: The effect of admission selectivity

College applicants often make dominated choices even when a strategically simple mechanism such as deferred acceptance is in place. We study Hungarian college admissions, where deferred acceptance is used, and still many college applicants make revealed dominated choices: they forgo the free opportunity to receive a tuition waiver. Using two empirical strategies, we show that when admission with a tuition waiver becomes more selective, applicants make more revealed dominated choices. Our results suggest that dominated choices respond to economic incentives.

School Choice with Consent: an Experiment

Abstract Public school choice often yields student assignments that are neither fair nor efficient. The efficiency-adjusted deferred acceptance mechanism allows students to consent to waive priorities that have no effect on their assignments. A burgeoning recent literature places the efficiency-adjusted deferred acceptance mechanism at the centre of the trade-off between efficiency and fairness in school choice. Meanwhile, the Flemish Ministry of Education has taken the first steps to implement this algorithm in Belgium. We provide the first experimental evidence on the performance of the efficiency-adjusted deferred acceptance mechanism against the celebrated deferred acceptance mechanism. We find that both efficiency and truth-telling rates are higher under the efficiency-adjusted deferred acceptance mechanism than under the deferred acceptance mechanism, even though the efficiency-adjusted deferred acceptance mechanism is not strategy proof. When the priority waiver is enforced, efficiency further increases, while truth-telling rates decrease relative to variants of the efficiency-adjusted deferred acceptance mechanism where students can dodge the waiver. Our results challenge the importance of strategy proofness as a prerequisite for truth telling and portend a new trade-off between efficiency and vulnerability to preference manipulation.

Reallocation with priorities

We consider a reallocation problem with priorities where each agent is initially endowed with a house and is willing to exchange it but each house has a priority ordering over the agents of the market. In this setting, it is well known that there is no individually rational and stable mechanism. As a result, the literature has introduced a modified stability notion called μ0-stability. In contrast to college admission problems, in which priorities are present but there is no initial endowment, we show that the ownership-adjusted Deferred Acceptance mechanism identified in the literature is not the only individually rational, strategy-proof and μ0-stable mechanism. By introducing a new axiom called the independence of irrelevant agents and using the standard axiom of unanimity, we show that the ownership-adjusted Deferred Acceptance mechanism is the unique mechanism that is individually rational, strategy-proof, μ0-stable, unanimous and independent of irrelevant agents.

Minimal-Access Rights in School Choice and the Deferred Acceptance Mechanism

A classical school choice problem consists of a set of schools with priorities over students and a set of students with preferences over schools. Schools’ priorities are often based on multiple criteria, for example, merit-based test scores as well as minimal-access rights (siblings attending the school, students’ proximity to the school, etc.). Traditionally, minimal-access rights are incorporated into priorities by always giving minimal-access students higher priority over non-minimal-access students. However, stability based on such adjusted priorities can be considered unfair because a minimal-access student may be admitted to a popular school, whereas another student with a higher merit score but without a minimal-access right is rejected, even though the former minimal-access student could easily attend another of her minimal-access schools. We therefore weaken stability to minimal-access stability: minimal-access rights promote access to only at most one minimal-access school. Apart from minimal-access stability, we also would want a school choice mechanism to satisfy strategy-proofness and minimal-access monotonicity, that is, additional minimal-access rights for a student do not harm her. Our main result is that the deferred acceptance mechanism is the only mechanism that satisfies minimal-access stability, strategy-proofness, and minimal-access monotonicity. Because this mechanism is in fact stable, our result can be interpreted as an impossibility result: fairer outcomes that are made possible by the weaker property of minimal-access stability are incompatible with strategy-proofness and minimal-access monotonicity. Funding: This work was supported by the Spanish State Research Agency [Grant PID2020-114251GB-I00], AGAUR–Generalitat de Catalunya [Grant 2021-SGR-00416], the Severo Ochoa Programme for Centres of Excellence in R&D (Barcelona School of Economics) [Grant CEX2019-000915-S], and the Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung [Grant 100018_192583].

When do stable matching mechanisms fail? The role of standardized tests in Chinese college admissions

In this paper, we investigate matching problems where priorities and preferences are misaligned. In the case of centralized Chinese college admissions, students are matched based on their test scores in standardized tests, a noisy realization of their aptitudes due to measurement errors. We show that in this case any matching mechanism that is stable with respect to score is not stable with respect to aptitude. The resulting instability leads to colleges' incentive to participate in early admissions (zizhu zhaosheng), a form of market unraveling. However, a manipulable mechanism such as the Immediate Acceptance mechanism, combined with limited information about priorities, may succeed in mending this market failure. We then design and conduct a laboratory experiment where we compare the performance of two mechanisms (the Immediate Acceptance mechanism and the Deferred Acceptance mechanism), under two timing conditions of the submission of students' rank-ordered lists of colleges (before the exam and after the exam), using a two-stage matching market design with the possibility of early offers. In the experiment, a significant level of market unraveling occurred under mechanisms that are not stable with respect to aptitude, confirming theoretical predictions. We also find that the Immediate Acceptance mechanism under pre-exam submission condition significantly reduces such unraveling.

The Performance of School Assignment Mechanisms in Practice

Theory points to a potential trade-off between two main school assignment mechanisms; Boston and Deferred Acceptance (DA). While DA is strategy-proof and gives a stable matching, Boston might outperform DA in terms of ex-ante efficiency. We quantify the (dis)advantages of the mechanisms by using information about actual choices under Boston complemented with survey data eliciting students' school preferences. We find that under Boston around 8% of the students apply to another school than their most-preferred school. We compare allocations resulting from Boston with DA with single tie-breaking (one central lottery; DA-STB) and multiple tie-breaking (separate lottery per school; DA-MTB). DA-STB places more students in their top-n schools, for any n, than Boston and results in higher average welfare. We find a trade-off between DA-STB and DA-MTB. DA-STB places more students in their single most-preferred school than DA-MTB, but fewer in their top-n, for n >= 2. Finally, students from disadvantaged backgrounds benefit most from a switch from Boston to any of the DA mechanisms.

How to De-Reserve Reserves: Admissions to Technical Colleges in India

We study the joint implementation of reservation and de-reservation policies in India that has been enforcing comprehensive affirmative action since 1950. The landmark judgment of the Supreme Court of India in 2008 mandated that whenever the OBC category (with 27% reservation) has unfilled positions, they must be reverted to general category applicants in admissions to public schools without specifying how to implement it. We disclose the drawbacks of the recently reformed allocation procedure in admissions to technical colleges and offer a solution through “de-reservation via choice rules.” We propose a novel priority design—Backward Transfers (BT) choice rule—for institutions and the deferred acceptance mechanism under these choice rules (DA-BT) for centralized clearinghouses. We show that DA-BT corrects the shortcomings of existing mechanisms. By formulating India’s legal requirements and policy goals as formal axioms, we show that the DA-BT mechanism is unique for the concurrent implementation of reservation and de-reservation policies. This paper was accepted by Yan Chen, behavioral economics and decision analysis.

Reallocation with priorities and minimal envy mechanisms

We investigate the problem of reallocation with priorities where one has to assign objects or positions to individuals. Agents can have an initial ownership over an object. Each object has a priority ordering over the agents. In this framework, there is no mechanism that is both individually rational (IR) and stable, i.e. has no blocking pairs. Given this impossibility, an alternative approach is to compare mechanisms based on the blocking pairs they generate. A mechanism has minimal envy within a set of mechanisms if there is no other mechanism in the set that always leads to a set of blocking pairs included in the one of the former mechanism. Our main result shows that the modified Deferred Acceptance mechanism (Guillen and Kesten in Int Econ Rev 53(3):1027–1046, 2012), is a minimal envy mechanism in the set of IR and strategy-proof mechanisms. We also show that an extension of the Top Trading Cycle (Karakaya et al. in J Econ Theory 184:104948, 2019) mechanism is a minimal envy mechanism in the set of IR, strategy-proof and Pareto-efficient mechanisms. These two results extend the existing ones in school choice.

Tie-breaking and efficiency in the laboratory school choice

In school choice problems with weak priorities, the deferred acceptance (DA) mechanism may produce inefficient stable matchings due to tie-breaking. The stable-improvement-cycles (SIC) and choice-augmented deferred acceptance (CADA) mechanisms were proposed to remedy inefficiencies but they are manipulable. In a simple environment, we theoretically and experimentally analyze students' strategic behavior when DA, SIC, and CADA are implemented. We show that obtaining the efficiency gain relative to DA crucially depends on whether students report their preferences truthfully in SIC and whether they play a particular equilibrium strategy in CADA. Our laboratory experiment reveals that (i) non-negligible degrees of untruthful reporting are observed but they are not a major drawback for practical efficiency improvements of the mechanisms we consider; (ii) SIC achieves gains from trade whenever they exist, both on and off the equilibrium paths; and (iii) the additional layer of equilibrium coordination required by CADA makes it harder for CADA to fully produce the promised welfare advantage relative to DA. These findings are robust to various environments.

Responsiveness to priority‐based affirmative action policy in school choice

AbstractWe consider the priority‐based affirmative action policy in school choice. We weaken the responsiveness to affirmative action policy by requiring at least one minority student to be weakly better off when their priorities are improved. We find that under both the student‐proposing deferred acceptance mechanism and the top trading cycles mechanism at least one minority student becomes weakly better off when their priorities are improved. We also strengthen the responsiveness to affirmative action policy by requiring all minority students to be weakly better off when their priorities are improved. We call this property strict responsiveness to affirmative action policy. We find that there is no nonwasteful, individually rational, mutually best, and strategy‐proof mechanism that is strictly responsive to affirmative action policy. We then find a sufficient condition for the affirmative action policy to satisfy for the student‐proposing deferred acceptance mechanism to be strictly responsive to affirmative action policy by setting restrictions on the priority improvements made by the policy.

The iterative deferred acceptance mechanism

Lately, there has been an increase in the use of sequential mechanisms, instead of the traditional direct counterparts, in college admissions in many countries, including Germany, Brazil, and China. We describe these mechanisms and identify their shortcomings in terms of incentives and outcome properties. We introduce a new mechanism, which improves upon these shortcomings. Unlike direct mechanisms, which ask students for a full preference ranking over colleges, our mechanism asks students to sequentially make choices or submit partial rankings from sets of colleges. These are used to produce a tentative allocation at each step. If at some point it is determined that a student can no longer be accepted into previous choice, then she is asked to make another choice among colleges that would tentatively accept her. Participants following the simple strategy of choosing the most-preferred college in each step is an ex-post equilibrium that yields the Student-Optimal Stable Matching.

A comparison study on responsiveness of three mechanisms to affirmative action in school choice

This note provides a comparison study on responsiveness of three popular mechanisms to affirmative action in school choice. When the exogenous school choice environment gives full priority to the minority, the student-proposing deferred acceptance (DA) mechanism is responsive to both the quota-based and reserve-based affirmative action policies. In comparison with DA mechanism, the top trading cycles (TTC) mechanism is responsive to neither the quota-based nor the reserve-based affirmative action even when the exogenous school choice environment gives full priority to the minority. When each minority student has higher priority than each majority student at each school, the TTC mechanism is responsive to the quota-based affirmative action. However, under the environment as above, the Boston mechanism is not responsive to the quota-based affirmative action.

Application Costs and Congestion in Matching Markets

Abstract A matching market often requires recruiting agents, or ‘programmes’, to costly screen ‘applicants’, and congestion increases with the number of applicants to be screened. We investigate the role of application costs: higher costs reduce congestion by discouraging applicants from applying to certain programmes; however, they may harm match quality. In a multiple-elicitation experiment conducted in a real-life matching market, we implement variants of the Gale-Shapley deferred-acceptance mechanism with different application costs. Our experimental and structural estimates show that a (low) application cost effectively reduces congestion without harming match quality.

Centralized School choice with unequal outside options

We study how market design choices exacerbate or mitigate pre-existing inequalities among participants. We introduce outside options in a well-known school choice model, and show that students always prefer manipulable over strategy-proof mechanisms if and only if they have an outside option. We test for the proposed relationship between outside options and manipulability in a setting where we can identify students’ outside options and observe applications under two mechanisms. Consistent with theory, students with an outside option are more likely to list popular, highly-rated schools under the Boston mechanism, and this gap disappears after switching to a Deferred Acceptance mechanism.

Ex-Ante Welfare Superiority of the Boston Mechanism Over the Deferred Acceptance Mechanism

Ex-Ante Welfare Superiority of the Boston Mechanism Over the Deferred Acceptance Mechanism

Parallel markets in school choice

When applying to schools, students often submit applications to distinct school systems that operate independently, which leads to waste and distortions of stability due to miscoordination. To alleviate this issue, Manjunath and Turhan (2016) introduce the Iterative Deferred Acceptance mechanism (IDA). We design an experiment to compare the performance of this mechanism under parallel markets (DecDA2) to the classic Deferred Acceptance mechanism with both divided (DecDA) and unified markets (DA). Consistent with the theory, we find that both stability and efficiency are highest under DA, intermediate under DecDA2, and lowest under DecDA. While IDA is not strategy-proof, we show theoretically that strategic reporting can only lead to improved efficiency for all market participants. The experimental results are consistent with this prediction. Our findings cast doubt on whether strategy-proofness should be perceived as a universal constraint to market mechanisms.

Centralized Matching with Incomplete Information

We study the impacts of incomplete information on centralized one-to-one matching markets. We focus on the commonly used Deferred Acceptance mechanism (Gale and Shapley 1962). We show that many complete-information results are fragile to a small infusion of uncertainty about others’ preferences. (JEL C78, D11, D21, D47)

The Design of Teacher Assignment: Theory and Evidence

Abstract To assign teachers to schools, a modified version of the well-known deferred acceptance mechanism has been proposed in the literature and is used in practice. We show that this mechanism fails to be fair and efficient for both teachers and schools. We identify a class of strategy-proof mechanisms that cannot be improved upon in terms of both efficiency and fairness. Using a rich dataset on teachers’ applications in France, we estimate teachers preferences and perform a counterfactual analysis. The results show that these mechanisms perform much better than the modified version of deferred acceptance. For instance, the number of teachers moving from their positions more than triples under our mechanism.

Deduction Dilemmas: The Taiwan Assignment Mechanism

This paper analyzes the Taiwan mechanism used nationwide for high school assignment starting in 2014. In the Taiwan mechanism, points are deducted from an applicant’s score, with larger penalties for lower-ranked choices. Deduction makes the mechanism a hybrid of the Boston and deferred acceptance mechanisms. Our analysis sheds light on why Taiwan’s new mechanism has led to massive nationwide demonstrations and why it nonetheless remains in use. (JEL D47, I21, I28)