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A matching market assigns objects to individuals, or individuals to each other. Typically, the different objects are indivisible, and individuals differ in how much they value each of them, so that the assignment has important implications for the well-being of the individuals. Moreover, relevant applications involve markets where the use of monetary payments is limited or infeasible, such as public school choice, assignment of graduate students, or the exchange of live-donor kidneys for transplantation. In these markets, exhausting all opportunities for mutually beneficial exchange with the limited means available is important for the well-being and, in the case of the last example, the health of those involved. This research paper will demonstrate how economic theory can offer some guidance for the design of markets in order to solve such problems of assignment.
Several problems may arise in assignment problems that impede the attainment of a satisfactory outcome, where “satisfactory” could refer to Pareto efficiency or to other welfare criteria. Best understood among these problems are unraveling, strategic behavior and a failure to arrive at a stable allocation; they will be defined and discussed later in greater detail. Indeed, a growing body of economic research on market design is concerned with developing mechanisms that ensure that outcomes with desirable welfare properties are reached, while ensuring that individuals have adequate incentives to participate and to truthfully reveal their preferences over how much they value the objects to be assigned. Two such mechanisms, the Gale-Shapley mechanism and the top trading cycles mechanism, will be presented here, as well as applications to assigning students to colleges and schools and to the exchange of live-donor kidneys for transplantation.
This research paper is organized as follows. The next section gives an introduction to the theory of matching markets, discussing characteristics that typically distinguish matching from competitive markets, such as heterogeneity, indivisibility, and a lack of market prices. Then the concepts of stability, strategy-proofless, and optimality of an outcome are presented. Also, two algorithms that can be used to achieve outcomes with these properties are briefly discussed. To illustrate the practical relevance of matching markets, three applications of real-world assignment problems and the methods that have been employed to solve them are then described. A short discussion of policy considerations and a brief survey of further topics in matching market theory follow, and several areas for future research are reviewed that promise to eventually generate interesting and much-needed results. The research paper concludes with a summary. Also included is a detailed list of references for the interested reader.
The following gives a brief introduction in the theory of matching markets, with an emphasis on stable outcomes and methods to implement such outcomes, while ensuring that participants have no interest to misrepresent their preferences. This is demonstrated both in a marriage market model and a housing market setup.
Principles and Terminology
Assigning individuals to objects or, to a lesser extent, to other individuals appears to be a feature common to most markets. There are, however, some characteristics that typically differentiate matching markets from competitive markets in general. For one thing, objects and individuals in a matching market are usually heterogeneous and indivisible. This means that “goods” on a matching market (e.g., individuals) are typically supplied and demanded in quantities of 1 that cannot be further broken down. In a competitive market, a good that is demanded by many traders will be divided to satisfy the traders’ demands at the market price.
Indeed, the use of the word market seems to imply that a market price will be used to equate demand and supply for goods or individuals. While this might often be a viable method of assignment (e.g., one used in auctions), there are a number of applications where reaching a market price might be infeasible. This is true, for example, when prices are regulated, as in the choice of public schools or of dormitories at a university where places are provided essentially for free and as in admission into certain professions with rigid wage-setting conventions. This is also true when prices and monetary payments cannot be used on ethical grounds, such as for exchanges of kidneys from live donors. Moreover, a market price may not be informative about preferences, for instance, when some participants in a matching market are subject to credit constraints, as in the cases of university choice and the job market. In an extreme case, when individuals have no access to loans but need to make a sizable investment in tuition fees or special training, personal wealth will at least partially determine individual rankings of schools or jobs. Absence or limitations of a market price and monetary payments is often referred to as a situation with nontransferable utility.
If a matching market is not cleared by the market price, its outcome, also referred to as the matching allocation, will depend crucially on the method used to assign individuals to objects. For instance, the order in which individuals are allowed to choose may matter, as earlier choosers have a better chance that their preferred match is still on the market (think of the drafts in American professional sports leagues). Hence, a relevant issue is how to evaluate and compare different conceivable outcomes of matching markets. One important concern is that a matching allocation should be final in the sense that all feasible profitable exchanges are exhausted and there is no mutually profitable opportunity for rematching among individuals. This property is called stability. If an assignment scheme violates stability, individuals who expect that they will have such a profitable rematching opportunity given an assignment might decide they are better off bypassing the assignment scheme, for instance, by attempting to secure a favorable match before the market takes place. This is known as unraveling of a matching market and can lead to quite unsatisfactory matching allocations (see, e.g., Li & Rosen, 1998; Roth & Xing, 1994).
Having identified a desirable property of allocations such as stability, the next step is to find an assignment scheme that actually reaches a stable outcome. Such an assignment scheme is often simply an algorithm specifying step-by-step instructions for the assignment based on the preferences of market participants. Because the algorithm is based on information announced by participants (i.e., their stated preferences), quality of outcomes can be evaluated only with respect to reported information. Therefore, it is of great importance to elicit truthful revelation of information by individuals. Devising algorithms, or mechanisms, that reach desirable outcomes from a social point of view while ensuring that agents do not benefit by strategically misrepresenting their preferences is the object of the field of market design (see Roth, 2002, for an overview).
Matching market mechanisms are usually centralized in the sense that assignments are generated by a central matchmaker or a clearinghouse. This is in contrast to what is typically the case in the literature on search and matching in the labor market. There matching is decentralized, often meaning that participants in the market meet and match randomly.
Two-Sided Matching Markets
The simplest instance of a matching market is the marriage market model. In a marriage market, economic agents belong to one of two different groups, called the market sides (e.g., men and women or students and colleges). Suppose, following the first example, that there are n women to be assigned to m men, such that every woman is matched to one man or stays solitary, and likewise every man is matched to one woman or stays solitary. That is, there are two disjoint market sides, and an individual on one market side can only be assigned to a member of the opposite market side. The restriction of allowable matches to those between two market sides is the defining characteristic of a two-sided matching market, as opposed to a one-sided matching market where there is only one market side in the sense that a priori any individual is free to match with any other individual. Furthermore, here each man is matched to at most one woman, and vice versa, so that matching is one to one. In the context of students and colleges, typically many students may be assigned to one college, so that matching is many to one.
Men and women have preferences in the form of complete rankings of all individuals from the other market side that they would prefer to be matched with over staying solitary. Suppose that preferences are strict, so that no individual is exactly indifferent between being matched with any two members of the opposite side (or between being matched with any member of the opposite side and staying solitary).
Stability of a Matching Allocation
As was mentioned above, a desirable property for an outcome of a matching market is that it is final in the sense that all mutually profitable matches have been exhausted.
To capture this, say that an outcome can be blocked if there is an individual or a pair of individuals who can make themselves strictly better off by altering their own assignment when mutual consent is required. More precisely, an outcome can be blocked by an individual, if he or she is assigned to somebody in the outcome but would prefer to stay solitary. An outcome can be blocked by a pair, if a pair of individuals, one from each side, is not assigned to each other in the outcome but would both strictly prefer this to their actual assignment in the outcome. Consequently, an outcome is called stable if there is neither an individual nor a pair that can block the outcome. That is, to borrow the image of a marriage market once more, given a stable outcome, no mutually desired marriage is left un-brokered, and nobody will demand a divorce. This illustrates well the appeal of a stable allocation: In a stable allocation, an individual cannot gain by choosing differently because all matches preferred to the current one would not accept a proposal. For this reason, stability is also sometimes said to eliminate justified envy, where envy is considered only justified when an envied match would actually accept a proposal.
The Gale-Shapley Algorithm
Fortunately, existence of a stable outcome in a marriage market is always guaranteed, as David Gale and Lloyd Shapley (1962) found. Since their proof is highly instructive for the understanding of a great variety of assignment problems, a sketch of it is given here. They prove existence constructively by proposing a very simple algorithm that never fails to achieve a stable matching allocation in a marriage market. A matching algorithm provides a detailed set of instructions for how to assign and possibly rematch individuals on both market sides. It proceeds sequentially in steps until a stopping condition is met, for instance, when there is no activity in the current step. Then it terminates, and the current assignment becomes the matching allocation. The Gale-Shapley algorithm is as follows:
(i) Each man proposes to his first choice, if this is preferred to remaining solitary. Otherwise, he does not propose.
(ii) Each woman tentatively accepts the most preferred proposal made to her, or stays solitary if that is preferred, and rejects all other proposals.
(i) Each man who has been rejected at step k – 1 proposes to his kth choice, if this is preferred to remaining solitary. Otherwise, he does not propose.
(ii) Each woman tentatively accepts the most preferred proposal made to her (out of all proposals in the current step and her most recent previous tentative acceptance), or stays solitary if that is preferred, and rejects all other proposals (including the supplanted previous tentative acceptance if applicable).
The algorithm stops when a step is reached in which no proposals are made. All tentative acceptances by women become permanent, and the resulting matches constitute the outcome.
The key characteristic of this algorithm is that women hold on to the best proposals they receive but are free to reject any previous proposal should a better one arrive later on. This means acceptances by the women are not binding as long the algorithm has not terminated. This is why this algorithm is also sometimes called the deferred acceptance algorithm.
To verify that a marriage market has a stable outcome, one can proceed in two steps: It has to be confirmed first that the algorithm always produces an outcome and second that the outcome produced is indeed stable. For the first statement, one simply notes that both men and women are finite in number. Therefore, after a finite time, the algorithm must have exhausted all entries in all individual rankings so that nobody proposes anymore, or, if not, there must have been a step when nobody proposed. Moreover, the outcome of the algorithm does not depend on the order in which men are allowed to propose in each step or on the order in which women are allowed to decide which proposals to reject. It does, however, depend on whether men or women propose, as will become clear below.
That the matching outcome produced by a Gale-Shapley algorithm must be stable can be seen as follows. Suppose there are a man and a woman who can block the outcome. Then, under the rules of the algorithm, the man must necessarily have proposed to the woman before and been rejected. But then, again according to the rules of the algorithm, the woman cannot prefer a match with this man to the match she is assigned in the outcome. Hence, the initial assumption that there is a pair that can block the outcome must be false. Neither can any individual block the outcome since men only propose to women if this is preferred to staying solitary, and women only hold on to such proposals that are preferred to staying solitary. Therefore, neither an individual nor a pair can block the outcome, and so the outcome of a Gale-Shapley algorithm is stable.
Optimal Matching Allocations
There may, however, be more stable outcomes than just the one arrived at by this algorithm. Furthermore, different stable outcomes may differ widely in how well-off the individuals find themselves. Gale and Shapley (1962) provide a useful result: There is always a stable outcome that is at least weakly preferred by all men to all other stable outcomes, which is typically called the men-optimal (or student-optimal in the case of college admission) stable outcome. Likewise, there is always a stable outcome that all women at least weakly prefer to all other stable outcomes, which is called the women-optimal stable outcome. Moreover, Gale and Shapley show that the men-optimal outcome is reached by a deferred acceptance algorithm in which the men propose (as outlined above) while, if instead the roles are reversed and women propose, such an algorithm reaches the women-optimal outcome.
This result has an immediate implication for how policy makers would want to implement the algorithm if one market side were of more concern than the other. For instance, suppose one market side consists of organizations rather than individuals—say, in a student-college interpretation of the marriage market, where men can be relabeled as students and women as colleges. Then the student-optimal outcome seems clearly desirable from a social point of view, when assuming that the well-being of colleges is not a crucial social concern. This suggests that it would be desirable from a social point of view to implement the algorithm such that it is the students who make the proposals.
Truthful Information Revelation
Using the Gale-Shapley algorithm, a social planner can implement a stable outcome and furthermore, simply by choosing which market side will play what role, can implement the stable outcome that is optimal for a given market side. However, the social planner must apply the algorithm on the basis of submitted preference rankings from both market sides. This means that the outcome that is reached will be stable only with respect to the stated preferences. If individuals do not report their preferences truthfully, however, the outcome might not be stable with respect to the true preferences, which may lead to Pareto inefficiency, secondary markets, or unraveling. Therefore, a further desirable property of any assignment mechanism is to ensure that individual participants cannot gain by misrepresenting their preferences. An assignment mechanism is called strategy-proof if it ensures that, for all possible combinations of individual strict preferences, participants at least weakly prefer to reveal information on their preferences truthfully. The interested reader may consult the textbook on two-sided matching by Roth and Sotomayor (1990) for a more exhaustive treatment of the following results and many others.
First, a negative result has to be mentioned: There is no assignment mechanism that generally implements a stable outcome in a marriage market and is strategy-proof (Roth, 1982). To see this, suppose men- and women-optimal outcomes do not coincide, and an algorithm is in place where men propose (as described above). Then all women prefer an outcome other than is reached by the algorithm. In particular, the following situation may arise. A woman misrepresents her preferences by eliminating her match in the men-optimal outcome from her submitted list. Then this man is rejected and tentatively assigned to another woman who prefers that man to her match in the men-optimal outcome. The newly rejected man in turn proposes to the first woman (who lied) and is tentatively accepted, if this woman prefers him to her match in the men-optimal outcome. That is, women exchange men such that all participating women are better off and all exchanged men worse off. Because this possibility cannot be precluded in general, strategy-proofiiess cannot be ensured for both market sides.
Lester Dubins and David Freedman (1981) present a slightly more encouraging result in finding that a mechanism that yields the men-optimal outcome on a marriage market ensures that no man can gain by misrepresenting his true preferences. That is, such a mechanism is strategy-proof among the men. This must be the case because by misrepresenting preferences, a man may induce a different stable outcome, but the men-optimal outcome is preferred by all men among all stable outcomes. A similar property holds for a mechanism inducing a women-optimal outcome: It is strategy-proof among the women. This is in fact a very useful result in case one is interested in a matching market where participants on one market side can be considered to be not self-interested. This may be the case, for instance, when assigning pupils to schools, students to universities or courses, or airplanes to landing slots. Some caution is appropriate, however, when the matching is many to one, as the result on strategy-proofness for one market side holds only for the market side having one partner each. To illustrate this, consider a school choice problem where schools have multiple places. A matching mechanism yielding the pupil-optimal allocation is strategy-proof among the pupils, but a matching mechanism yielding the school-optimal allocation is not necessarily strategy-proof among the schools.
In contrast to the two-sided marriage market discussed so far, an individual participating in a one-sided matching market may be assigned to any other individual participating in that market. Examples are team or group formation or marketplaces where indivisible objects are exchanged, such as markets for houses or dorm rooms. In these markets, every individual may initiate a trade with any other individual or group of individuals. The basic housing market model due to Lloyd Shapley and Herbert Scarf (1974) considers an exchange economy where n individuals trade in an indivisible good, say houses. Each individual owns one house when entering the market, has need of exactly one house, and possesses strict preferences over all existing houses. Potential matches in this market are between individuals and indicate house trades.
An outcome in this market is an allocation of houses among individuals such that each individual holds at most one house. That is, potential trades, or matches, among (groups of) individuals generate an allocation of houses to individuals, the market outcome. Note that also a school choice matching market could potentially be formulated as a one-sided market, if pupils are endowed with a place at certain school. Shapley and Scarf (1974) use the core as a solution concept to determine the outcome. An outcome is in the core if there is no group of individuals (of size one or greater) that could make every group member weakly and at least one strictly better off by reallocating the houses owned by group members among members of the group. This is equivalent to saying that an outcome exhausts all opportunities for mutually beneficial trade among any number of individuals (not only between pairs of members of opposing market sides).
The Top Trading Cycles Algorithm
When all individuals have strict preferences, there is always at least one matching allocation in the core (Roth & Postlewaite, 1977). Shapley and Scarf (1974) show that an outcome in the core can be reached by following a specific algorithm, the top trading cycles algorithm, which they attribute to David Gale. Before presenting the algorithm, it will be useful to explain what is meant by cycles. A cycle is a sequence of individuals such that each individual is followed by, or points to, the owner of his or her most preferred house, and the last individual in the sequence most prefers the first individual’s house. A cycle may consist of one individual only. Note that, among a finite set of individuals who have strict preferences, there must always be at least one cycle. For cycles of more than one individual, if all members of a cycle give their houses to their successors in the sequence and the last individual to the first, all individuals in the cycle will be strictly better off. That is, cycles identify opportunities for mutually strictly beneficial trades in groups. The top trading cycles algorithm exhausts all such opportunities and works as follows.
Let all individuals point to the owners of their most preferred houses. Identify all cycles, effect the implied exchange of houses, and remove the individuals in the cycles from the market.
Let all individuals that remain in the market after step k – 1 point to the owners of their most preferred houses. Identify all cycles, effect the implied exchange of houses, and remove the individuals in the cycles from the market.
The algorithm stops when there are no more individuals in the market.
This means that the top trading cycles algorithm iteratively identifies all trading cycles, executes the trades, and removes the individuals in the cycles until the market is cleared. The outcome generated by this algorithm is in the core for all housing markets (Shapley & Scarf, 1974). That is, given a matching allocation resulting from this mechanism, there is no opportunity for mutually strictly beneficial exchange among any group of agents. This means the outcome is Pareto efficient and essentially replicates the competitive equilibrium allocation (Roth & Postlewaite, 1977). Hence, from a social planner’s perspective, outcomes of a top trading cycles mechanism on a housing market have highly desirable properties. Moreover, the mechanism does not require monetary payments and can therefore be deployed in situations in which competitive market prices may not be feasible. To make it a desirable mechanism, it should also ensure that participants have no incentive to misreport their preferences. Roth (1982) shows that this is indeed the case.
In the following, a number of real-world assignment problems are presented, along with the matching markets that have arisen in correspondence. The aim of the following presentation is twofold. On one hand, assignment mechanisms that have been employed on matching markets are described and analyzed with respect to shortcomings such as unraveling or preference misrepresentation. On the other hand, possible remedies, or policies, are presented to amend any such shortcomings identified in the analysis.
The Market for Medical Interns and Residents
The market for medical interns and residents in the United States is a two-sided matching market. On one side are colleges offering internship and residency positions, and on the other side are graduate students seeking such positions. Both colleges and students are very heterogeneous in academic quality, both are indivisible (as parttime employment is usually infeasible), and wages for students are fixed before the application process begins. Alvin Roth (1984) gives an account both of the problems encountered in this market and the theoretical reasons. Before a viable assignment mechanism was introduced in 1951, matching was conducted in a de-centralized manner by private initiatives of market participants. Because the number of positions exceeded that of students, colleges had an incentive to try to preempt competitors in securing good candidates by making offers earlier. Indeed, whereas students tended to obtain a position about half a year before graduation in the 1930s, they were typically offered a contract about 2 full years before graduation by 1944. Such early timing, of course, forgoes a lot of information on the quality of candidates, which is very likely to adversely influence the quality of the matches. This is an instance of unraveling. Others include markets for law clerks and gastroenterologists (see Avery, Jolls, Posner, & Roth, 2001; Niederle, Proctor, & Roth, 2006).
In 1945, an attempt was made to remedy the shortcomings by disclosing information on candidates only shortly before graduation. This prevented unraveling but created another problem, as offers and acceptances were binding.
It became costly for colleges to hold offers open for any amount of time because by the time an offer was rejected, the market was quite likely to be depleted. Not surprisingly, while offers remained open for 10 days in 1945, this time fell to about 10 minutes by 1950.
As this development was unsatisfactory, a clearinghouse was formed in 1951 that used a centralized mechanism, the National Internship Matching Program (NIMP). This mechanism works as follows. Students and colleges submit their preferences over the other market side in the form of rankings to the clearinghouse. An algorithm is then used to determine the matching outcome. The principal characteristic of the algorithm employed is that it uses deferred acceptance, as with the Gale-Shapley algorithm. Therefore, the outcome reached by the NIMP is stable with respect to the stated rankings by students and colleges (Roth, 1984). Hence, given the outcome, no participant can obtain a better match—that is, the allocation is envy free—and there should be no reason for a secondary market to emerge. This may explain why the program, although participation was voluntary, was quite successful. Indeed, the mechanism is still in use today, albeit renamed the National Residency Matching Program (NRMP) and in a slightly modified form. Some changes were designed to accommodate concerns such as differences in medical training programs and the desire of couples to end up in similar locations. More significantly, the algorithm was changed from a college-proposing to a student-proposing format to ensure student optimality of the matching outcomes after complaints arose that students had been treated badly. Potential gains to students from misrepresenting their preferences appear to be small in the NRMP, however (see Roth & Peranson, 1999, for further details).
Public School Choice
The assignment of pupils to public schools provides another informative real-world example of a two-sided matching market. In many U.S. school districts, the assignment mechanism used to match pupils and schools often fails to implement a stable outcome, which can generate incentives to misreport preferences. For instance, the city of Boston employed the following mechanism to assign pupils to public schools in the years 1999-2005. Parents submitted a ranking of their top schools. Schools assigned priority to pupils who lived within walking distance, higher priority to those who had a sibling already enrolled, and highest priority to pupils who satisfied both criteria. Further details can be found in the studies by Abdulkadiroglu and Sdnmez (2003) and Ergin and Sdnmez (2006). The so-called Boston Student Assignment Mechanism (BSAM) proceeds as follows.
For each school, pupils who have listed that school as their first choice are assigned in order of priority while the school still has free capacity. Ties are broken using a random procedure (such as flipping a coin).
For each school, pupils who have not been matched to a school in a previous step and who have listed that school as their kth choice are assigned in order of priority while the school still has free capacity. Ties are broken using a random procedure.
The algorithm stops, when either all pupils are assigned or there is no school left with remaining places.
The key feature of this algorithm is that assignments at each step are permanent—that is, when a school accepts a pupil, this acceptance is binding. Note that this is in contrast to deferred acceptance, which is characteristic of the Gale-Shapley algorithm. Ultimately, the fact that acceptances are permanent gives pupils an opportunity for strategic behavior. This implies in turn that the BSAM is not strategy-proof. This is because being rejected by one’s top-ranked school may waste one’s priority at another school if the other school is over-demanded and fills to capacity in a previous step. For example, if some pupil most prefers an over-demanded school at which he or she does not have priority but does have priority at his or her second most preferred school, it is strictly better for the pupil to list the second most preferred school in the first place on the submitted ranking to minimize the chances of being stuck with his or her third choice or worse. That is, under the BSAM, pupils can strictly gain by misrepresenting their preferences.
This observation has spawned a lively debate on possible implications for policy, in particular on whether to replace the existing mechanism by one that is strategy-proof, such as the Gale-Shapley algorithm. To offer theoretical guidance, Ergin and Sdnmez (2006) investigate a game of school choice and find that the outcome of a pupil-optimal Gale-Shapley algorithm is more desirable from an efficiency perspective than the one of the BSAM (as long as the preferences of both market sides are strict). Furthermore, Adulkadiroglu, Pathak, Roth, and Sdnmez (2006) and Pathak and Sdnmez (2008) put forward the argument that a Gale-Shapley algorithm places lower cognitive and computational burden on participants than does the BSAM. This may be desirable, as evidence suggests that while some participants actually behave strategically, others announce their rankings sincerely. The BSAM systematically disadvantages these “naive” individuals if there are sophisticated players who behave strategically.
In 2006, public authorities changed the mechanism used in the assignment of pupils to public schools in Boston and introduced a version of the Gale-Shapley algorithm. New York City has also undergone a similar change in its assignment mechanism of pupils to high schools (see Abdulkadiroglu, Pathak, & Roth, 2005). There, a chief problem that had to be dealt with was strategic behavior of high schools.
As a final example, consider the problem of assigning kidneys from live donors to patients in need of transplantation. Often, patients who are in desperate need of a kidney transplant have a willing donor. (It is perfectly possible to live a healthy life with only one kidney.) Several compatibility conditions need to be met, however, to make a transplant feasible, such as having common blood types. Suppose that a patient has found a willing donor but is incompatible with that donor. Finding another patient who is compatible with the first donor and who has a live donor who is compatible with the first patient would make both patients better off (and presumably the donors, too). Obviously, it is of tremendous interest to identify all possible opportunities for mutually beneficial exchange because forgone opportunities to exchange will very likely result in the loss of human life.
Alvin Roth, Tayfun Sdnmez, and Utku Unver (2004) provide a theoretical analysis of this matching market and propose a mechanism for the exchange of kidneys from live donors. The key insight is that the market for kidney exchange is a one-sided matching market, similar to the housing market presented above. For this type of matching market, a mechanism is known that achieves a Pareto efficient allocation—that is, a matching outcome that exhausts all possible profitable trades—and is strategy-proof: the top trading cycles algorithm outlined above. Because not all patients necessarily have a donor to bring to the market, the appropriate theoretical model is a housing market with existing tenants, as studied by Abdulkadiroglu and Sdnmez (1999), who explicitly allow for a waiting list. Roth et al. (2004) extend this theoretical work by allowing for the possibility of a waiting list for kidneys from deceased donors and derive a top trading cycles and chains (TTCC) mechanism that is Pareto efficient and strategy-proof.
The number of live-donor kidney exchanges that are actually carried out appears to be quite small relative to that which could be achieved by exhausting all potential for mutually beneficial exchange, pointing to a clear need for a centralized matching institution. In 2005, a clearinghouse (the New England Program for Kidney Exchange) for gathering data on donors and patients and generating the actual assignment was founded in New England (Roth, Sdnmez, & Unver, 2005, 2007). By 2008, this clearinghouse had already enabled a substantial number of two-way and even some three-way kidney exchanges. A similar program has since been started in Ohio.
Recently, some further issues have been raised in public debate about some of the matching markets mentioned above. In the case of the market for medical residents, complaints have been made that the use of the NRMP depresses salaries for residents and fellows. An antitrust lawsuit in this matter was brought forward in 2002 but dismissed in 2004. Some theoretical support for the claim comes from Bulow and Levin (2006). They find that, when explicitly accounting for the fact that wage setting by colleges takes place before the matching market does, wages can indeed be depressed compared to the competitive outcome. Other studies (e.g., Kojima, 2007; Niederle, 2007) argue that this result hinges on particular assumptions that do not describe the real market for medical residents well.
As for school choice, recently it has been suggested that a version of the BSAM that modifies the random tie-breaking procedure would be able to better respect the intensity of pupils’ preferences for different schools (see Abdulkadiroglu, Che, & Yasuda, 2008; Miralles, 2008). Thus, schools could be allocated to the pupils who have the highest valuations, which is different from merely ensuring that schools are assigned to pupils who prefer them to the next best school. This concern becomes more relevant when the pupils’ preferences over schools are closely aligned, which seems to be the case in this market.
The cases mentioned above raise two additional issues. First, it is not clear whether the desirability of a mechanism is better judged based on whether it exhausts all potential mutually profitable exchanges or on whether it maximizes the sum of individuals’ valuations of the outcome, that is, aggregate surplus. Aggregate surplus, also sometimes interpretable as output, may be a relevant criterion, for example, when analyzing economy-wide policies, such as labor market regulations and their effects on growth. Second, often there will be some means of compensation between partners, for instance, through the exchange of favors or gifts.
These issues have at least been partially addressed. When preferences are perfectly aligned—that is, the rankings of all individuals in the market agree—Becker (1974) shows that stable matching allocations need not maximize aggregate surplus when the possibility to compensate a partner in any form is completely excluded. Indeed, independently of which matching allocation maximizes aggregate surplus, a stable outcome in his model always assigns individuals whose attributes are more attractive to individuals or objects that have more attractive attributes. This is known as positive assortative matching. The general case, where compensation within a match is possible but costly in terms of surplus, has only recently been characterized. Legros and Newman (2002, 2007) provide conditions for assortative matching and confirm that stability and surplus maximization need not coincide.
When asymmetric information about individuals’ attributes is a concern (e.g., concerning the productivity of firms and workers in labor markets), some form of segmentation and randomization of the matching within segments may be desirable (see, e.g., Jacquet & Tan, 2007;McAfee, 2002).
Directions for Future Research
While the design of matching markets is already sufficiently well understood to offer useful contributions in many important assignment problems, there are a number of issues in which further theoretical progress would appear to be highly valuable. A first topic worth mentioning for being understudied is the forming of and properties of individuals’ preferences. In large markets, constructing a complete preference ranking over every member of the other market side may be extremely costly because all potential matches would first need to be evaluated. In the labor market, for instance, this would require time-consuming interviews and assessments of and by all individual applicants and employers. A possible alternative may consist of designing a mechanism with a prematching stage in which participants indicate a crude preliminary ranking based on easily observable information (e.g., grades, public rankings). Such a mechanism has been tried out in the entry-level job market for economists since 2007. Whether this is indeed the optimal approach remains an open theoretical question.
Second, note that most of the theoretical results presented above rely on strict preferences, meaning that market participants cannot be indifferent between any two matches. This is indeed a limitation because the assumption cannot be easily discarded without affecting the theoretical properties of the mechanisms yet seems unreasonable in many relevant contexts. Some work has been done in identifying strategy-proof mechanisms for situations in which individuals may be indifferent between multiple matches. In such cases, the manner of breaking the tie and choosing the actual match may affect incentives for strategic behavior (see, e.g., Abdulkadiroglu, Pathak, & Roth, 2009; Erdil & Ergin, 2008; Miralles, 2008). Exploiting the possibility of indifference through the use of stochastic mechanisms— that is, matching mechanisms that use random procedures for assignment—appears to be a very promising field for future research. This is because such mechanisms allow respecting the intensity of individuals’ preferences over matches (e.g., by setting odds to obtain a certain match that are acceptable only for individuals with intense preference for that match).
A third limitation of the theory of matching markets is that many of the theoretical results described above do not necessarily apply when individuals care not only about the match they obtain (e.g., the instructor, school, university, firm) but also about who else obtains the same match. This is, for instance, the case when thinking about couples searching jointly for jobs in the labor market. Although evidence suggests that a hands-on approach to market design works well in the market for medical residents (see Roth & Peranson, 1999), sound theoretical guidance for generating mechanisms that can also account for preferences over the outcomes of individuals on the same market side would be very welcome.
Another important area for future research is analyzing the effects of the anticipated outcome of a matching market (possibly generated by a mechanism) on individuals’ behavior before entering the market. Often the attributes that the preferences of market participants are based on are subject to individual choices and investments. For example, the choice of how much education to acquire appears to affect individual outcomes in the labor market quite substantially. Some recent work suggests that, when partners in a match cannot compensate each other without incurring a loss in efficiency or when there is asymmetric information concerning attributes, stable outcomes need not maximize aggregate surplus and may distort individuals’ prematching choices when these are based on anticipating the matching outcome (Gall, Legros, & Newman, 2006; Hoppe, Moldovanu, & Sela, 2009).
There is a growing interest in and demand for applying economic theory to the design of mechanisms that solve real-world assignment problems. Cases in point are entry-level job markets, school choice, and exchange of live-donor kidneys. This research paper has presented an overview of such matching markets, typically characterized by heterogeneity and indivisibility of individuals and objects, as well as by limited possibilities for compensation of matching partners through side payments. A brief introduction to the theory of matching outlined important concepts such as stability of a matching allocation and strategy-proofness of an assignment mechanism. Two assignment mechanisms have been discussed in greater detail: the Gale-Shapley or deferred acceptance algorithm for two-sided matching markets and the top trading cycles algorithm for one-sided markets. Both algorithms satisfy two desired properties: stability of the resulting outcome and strategy-proofness.
The research paper proceeded to consider several real-world applications of matching markets in detail. In the market for medical residents and the choice of public schools, both two-sided matching markets, appropriate versions of the Gale-Shapley algorithm have been implemented with a high degree of success. An example of a one-sided matching market was provided in the exchange of kidneys for transplantation from live donors through a centralized facility using a version of the top trading cycles algorithm.
Further issues arise when it is in the social interest to choose a matching allocation that maximizes aggregate surplus. Because the stable outcome does not necessarily maximize aggregate surplus, a tension between surplus efficiency and stability may arise. In such cases, it is important to identify surplus-maximizing outcomes and to design and implement mechanisms that reliably reach such outcomes.
Finally, the research paper considered some relevant issues and questions that demand further investigation. The theory of matching markets needs to achieve higher levels of sophistication to provide results that are more applicable in empirically relevant situations, such as when individuals are indifferent between multiple matches or have preferences concerning who else obtains the same match as they do. This is needed to be able to devise adequate matching mechanisms that can be successfully employed to address a wider range of assignment problems. Promising directions for future research lie in the use of stochastic mechanisms, which may use fine-tuned random procedures for the assignment, and in considering the dynamic environment of an assignment problem to evaluate effects on choices made prior to entering the matching market, particularly those that affect attributes relevant for the resulting matching allocation.
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