Stable Matching Mechanism Research Articles

Can Everyone Benefit from Social Integration?

When several two-sided matching markets merge into one, it is inevitable that some agents will become worse off if the matching mechanism used is stable. I formalize this observation by defining the property of integration monotonicity, which requires that every agent becomes better off after any number of matching markets merge. Integration monotonicity is also incompatible with the weaker efficiency property of Pareto optimality. Nevertheless, I obtain two possibility results. First, stable matching mechanisms never hurt more than one-half of the society after the integration of several matching markets occurs. Second, in random matching markets there are positive expected gains from integration for both sides of the market, which I quantify.

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.

Incentive Compatibility of Large Centralized Matching Markets

We study the manipulability of stable matching mechanisms. To quantify incentives to manipulate stable mechanisms, we consider markets with random cardinal utilities, which induce ordinal preferences over match partners. We show that most agents in large matching markets are close to being indifferent of overall stable matchings. In one-to-one matching, the utility gain by manipulating a stable mechanism does not exceed the gap between utilities from the best and worst stable partners. Thus, most agents in a large market would not have significant incentives to manipulate stable mechanisms. The incentive compatibility extends to many-to-one matching when agents employ truncation strategies and capacity manipulations in a Gale—Shapley mechanism.

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.

Two-Sided Matching with Endogenous Preferences

We modify the stable matching problem by allowing agents' preferences to depend on the endogenous actions of agents on the other side of the market. Conventional matching theory results break down in the modified setup. In particular, every game that is induced by a stable matching mechanism (e.g., the Gale-Shapley mechanism) may have equilibria that result in matchings that are not stable with respect to the agents' endogenous preferences. However, when the Gale-Shapley mechanism is slightly modified, every equilibrium of its induced game results in a pairwise stable matching with respect to the endogenous preferences as long as they satisfy a natural reciprocity property. (JEL C78, D82)

Short Lists in Centralized Clearinghouses

Stable matching mechanisms are used to clear many two-sided markets. In practice, these mechanisms leave many agents on both sides unmatched. What factors determine the number of unmatched agents, and the quality of matches that do form? This paper answers this question, with a particular focus on how match outcomes depend on correlations in agent preferences. I consider three canonical preference structures: fully idiosyncratic preferences, common preferences (agents agree on the attractiveness of those on the opposite side), and aligned preferences (potential partners agree on the attractiveness of their match).I find that idiosyncratic preferences result in more matches than common preferences do. Perhaps more surprisingly, the case of aligned preferences results in the fewest matches. Regarding match quality, the story reverses itself: aligned preferences produce the most high quality matches, followed by common preferences. These facts have implications for the design of priority rules and tie- breaking procedures in school choice settings, as they point to a fundamental tradeoff between matching many students, and maximizing the number of students who get one of their top choices.

Manipulability of Stable Mechanisms

We study the manipulability of stable matching mechanisms and show that manipulability comparisons are equivalent to preference comparisons: for any agent, a mechanism is more manipulable than another if and only if this agent prefers the latter to the former. One important implication is that when agents on one side of the market have unit demand, no stable matching mechanism is less manipulable than another for all agents. (JEL C78, D82)

Investment Incentives in Labor Market Matching

We provide an illustration of how the design of labor market clearing mechanisms can affect incentives for human capital acquisition. Specifically, we extend the labor market matching model (with discrete transfers) of Kelso and Crawford (1982) to incorporate the possibility that agents may invest in human capital before matching. We show that in this setting, the worker-optimal stable matching mechanism incentivizes workers to make (nearly) efficient human capital investments. En route to our main result, we show that so long as the space of salaries is sufficiently rich, every stable outcome in the Kelso and Crawford (1982) setting is approximately efficient.

Optimal truncation in matching markets

Although no stable matching mechanism can induce truth-telling as a dominant strategy for all participants (Roth, 1982), recent studies have presented conditions under which truthful reporting by all agents is close to optimal (Immorlica and Mahdian, 2005; Kojima and Pathak, 2009; Lee, 2011). Our results demonstrate that in large, uniform markets using the Men-Proposing Deferred Acceptance Algorithm, each woman's best response to truthful behavior by all other agents is to truncate her list substantially. In fact, the optimal degree of truncation for such a woman goes to 100% of her list as the market size grows large. In general one-to-one markets we provide comparative statics for optimal truncation strategies: reduction in risk aversion and reduced correlation across preferences each lead agents to truncate more. So while several recent papers focused on the limits of strategic manipulation, our results serve as a reminder that without pre-conditions ensuring truthful reporting, there exists a potential for significant manipulation even in settings where agents have little information.

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.

Manipulability of Stable Mechanisms

We study the manipulability of stable matching mechanisms and show that manipulability comparisons are equivalent to preference comparisons: for any agent, a mechanism is more manipulable than another if and only if this agent prefers the latter to the former. One important implication is that no stable matching mechanism is less manipulable than another for all agents.

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).

Interdependent Utility and Truth-Telling in Two-Sided Matching

Mechanisms which implement stable matchings are often observed to work well in practice, even in environments where the stable outcome is not unique, information is complete, and the number of players is small. Why might individuals refrain from strategic manipulation, even when the complexity cost of manipulation is low? I study a two-sided, one-to-one matching problem with no side transfers, where utility is interdependent in the following intuitive sense: an individual's utility from a match depends not only on her preference ranking of her assigned partner, but also on that partner's ranking of her. I show that, in a world of complete information and linear interdependence, a unique stable matching emerges, and is attained by a modified Gale-Shapley deferred acceptance algorithm. As a result, a stable rule supports truth-telling as an equilibrium strategy. Hence, these results offer a new intuition for why stable matching mechanisms seem to work well in practice, despite their theoretic manipulability: individuals may value being liked.

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).

Optimal Truncation in Matching Markets

Since no stable matching mechanism can induce truth-telling as a dominant strategy for all participants, there is often room in matching markets for strategic misrepresentation (Roth). In this paper we study a natural form of strategic misrepresentation: reporting a truncation of one's true preference list. Roth and Rothblum prove an important but abstract result: in certain symmetric, incomplete information settings, agents on one side of the market (“the women”) optimally submit some truncation of their true preference lists. In this paper we put structure on this truncation, both in symmetric and general settings, when agents must submit preference lists to the Men-Proposing Deferred Acceptance Algorithm. We first characterize each woman's truncation payoffs in an incomplete information setting in terms of the distribution of her achievable mates. The optimal degree of truncation can be substantial: we prove that in a uniform setting, the optimal degree of truncation for an individual woman goes to 100% of her list as the market size grows large, when other women are truthful. In this setting, we demonstrate the existence of an equilibrium where all agents use truncation strategies. Compared to truthful reporting, in any equilibrium in truncation strategies, welfare diverges for men and women: women prefer the truncation equilibrium, while men would prefer that participants truthfully report. In a general environment, we show that the less risk averse a player, the greater the degree of her optimal truncation. Finally, when correlation in preferences increases, players should truncate less. While several recent papers have focused on the limits of strategic manipulation, our results serve as a reminder that without the pre-conditions ensuring truthful reporting, even in settings where agents have little information, the potential for manipulation can be significant.

Quantile Stable Matchings for Markets with Contracts

We study manipulability of stable matching mechanisms and show that manipulability comparisons are equivalent to preference comparisons: for any agent, a mechanism is more manipulable than another if and only if this agent prefers the latter to the former. In particular, this implies that no two stable mechanisms can be ranked in terms of manipulability for all agents. We then construct quantile stable mechanisms, establish conditions under which they are well defined, and observe that they can be ranked in terms of manipulability for one side of the market, but not for both. In particular, the typically used mechanism that selects the stable matching that is best for one side of the market has no incentive advantage over other quantile stable mechanisms when both sides of the market are strategic. As an auxiliary result, we provide sufficient conditions for the existence of median stable matchings, and more generally, show that quantile stable matchings exist when contracts are strong substitutes and satisfy the law of aggregate demand. This last result is of independent interest as experiments show that agents who match in a decentralized way tend to coordinate on the median stable matching, when it exists.

An analysis of the German university admissions system

This paper analyzes the sequential admissions procedure for medical subjects at public universities in Germany. Complete information equilibrium outcomes are shown to be characterized by a stability condition that is adapted to the institutional constraints of the German system. I introduce matching problems with complex constraints and the notion of procedural stability. Two simple assumptions guarantee existence of a student optimal procedurally stable matching mechanism that is strategyproof for students. In the context of the German admissions problem, this mechanism weakly Pareto dominates all equilibrium outcomes of the currently employed procedure. Applications to school choice with affirmative action are also discussed.

School choice: Impossibilities for affirmative action

This paper investigates the welfare effects of affirmative action policies in school choice. We show that affirmative action policies can have perverse consequences. Specifically, we demonstrate that there are market situations in which affirmative action policies inevitably hurt every minority student – the purported beneficiaries – under any stable matching mechanism. Furthermore, we show that another famous mechanism, the top trading cycles mechanism, suffers from the same drawback.

Impossibility of stable and nonbossy matching mechanisms

Stability is a central concept in matching theory, while nonbossiness is important in many allocation problems. We show that these properties are incompatible: there does not exist a matching mechanism that is both stable and nonbossy.

Two-sided matching with interdependent values

We introduce and study two-sided matching with incomplete information and interdependent valuations on one side of the market. An example of such a setting is a matching market between colleges and students in which colleges receive partially informative signals about students. Stability in such markets depends on the amount of information about matchings available to colleges. When colleges observe the entire matching, a stable matching mechanism does not generally exist. When colleges observe only their own matches, a stable mechanism exists if students have identical preferences over colleges, but may not exist if students have different preferences.