Matching and Market Design: An Introduction to Selected Topics Fuhito Kojima1 and Peter Troyan2

Abstract This paper provides a survey of the research in matching and market design. We discuss both basic theories and recent advances. An emphasis is placed on applications of the theories to practical applications. JEL Classification Numbers: C70, D61, D63, D78.

1: Introduction Market design is a rapidly growing branch of microeconomics that uses insights from economic analysis to aid in the practical design of markets. In the past, the most successful economic theory took market institutions as given and focused on describing how they work. Market design, on the other hand, asks “What would an optimal institution look like?”. Using insights from game theory and mechanism design, as well as empirical, experimental and case studies, researchers in market design have aided in the design of many practical market institutions. Broadly defined, applications of market design include radio spectrum and procurement auctions, matching markets for medical interns/residents, student placement mechanisms in the context of public school choice, and organ transplant networks. Readers in Japan may be familiar with the 2003 introduction of the Japan Resident Matching Program (JRMP) for matching medical students with hospitals as part of a comprehensive reform in health care policy. (We will discuss a policy issue related to the Japanese residency market later, based on Kamada and Kojima (2010)). As seen from these examples, market design is a very exciting area in which academic economists can use their expertise to improve real-world policies.3 This paper focuses on one particular branch of market design: matching theory.4 Matching theory can be further divided into two broad sub-fields: (1) matching two types of agents (e.g., men and women or workers and firms), and (2) matching (allocating) objects to agents (e.g., Stanford University, Department of Economics, 579 Serra Mall, Stanford, CA, 94305. Email: [email protected]. 1

Stanford University, Department of Economics, 579 Serra Mall, Stanford, CA, 94305. Email: [email protected]. 2

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See Roth (2007, 2008) and Sönmez and Ünver (2009) for survey of market design, and Al Roth’s website (http://kuznets.fas.harvard.edu/~aroth/alroth.html) for a list of references. 4

Although the design of auctions is a very active and successful area of market design, we will not discuss this topic in detail here. See Milgrom (2004) and Cramton, Shoham, and Steinberg (2006) for an overview of the auction literature. 1

college dorm rooms to students, school seats to children, or kidneys to transplant patients). Since the publication of the seminal paper by Gale and Shapley (1962), matching theory has been studied mainly by mathematicians and computer scientists. However, beginning with Roth (1984), who observed that a labor market clearinghouse for U.S. medical residents had been using a mechanism equivalent to Gale and Shapley’s theoretical construct, economists have been increasingly interested in the study of matching theory and have recently begun applying the theory to many new practical applications. For example, Abdulkadiroğlu and Sönmez (2003) formalized the school choice framework, while the application to organ transplant networks was first discussed in Roth, Sönmez, and Ünver (2004). Based on these papers and subsequent works, economists are now designing institutions coping with these problems.5 Although matching market design is currently taught at only a few universities, the number is growing rapidly.6 These facts suggest that matching market design is still a relatively small field but is growing at a stunning rate. In what follows, we try to minimize the notation used. There are two reasons for this: first, it will be easier for readers unfamiliar with the field to understand the main ideas if they are not forced to remember cumbersome notation; second, we believe that much of matching theory can be presented without complicated notation. In fact, the reader will find that much of the discussion presented in this paper, though mathematical in nature, requires very little mathematical background beyond basic logic at an elementary level. This point was actually made by Gale and Shapley in their seminal paper, where they write that “mathematics need not be concerned with figures,” and “any argument which is carried out with sufficient precision is mathematical.” They go on to argue that their analysis of the matching problem serves to make this point. In addition to its practical appeal, one of the great aspects of matching theory is that anyone can grasp the arguments if followed carefully enough. In this sense, matching theory is open to anyone who is interested in the subject and willing to make the effort to understand it.

2: Two-Sided Matching This section describes the theory of two-sided matching, which will be applied to labor markets consisting of medical residents and hospital residency programs.

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See Abdulkadiroğlu, Pathak, and Roth (2005) and Abdulkadiroğlu, Pathak, Roth, and Sönmez (2005) for the design of school choice mechanisms, and Roth, Sönmez, and Ünver (2005) for the design of kidney exchange network. 6

Al Roth’s website (http://kuznets.fas.harvard.edu/~aroth/alroth.html) keeps a list of links to class notes and materials in various market design courses. 2

Medical residency is practical training for new doctors who have just graduated from medical school. This practice is very popular and is found in many countries. In the United States, the practice began around the start of the 20th century and has become an integral part of medical education for students and an important component of labor supply for hospitals. In Japan, residency was made compulsory beginning in 2004 as part of a comprehensive health care reform program to produce better doctors. The matching system was started as part of this larger reform effort and is still in use today (with some changes in the details). The Japanese mechanism is based on the system that has been used in the U.S. since the 1950s. As mentioned before, the U.S. mechanism has theoretical support starting with Gale and Shapley (1962), and subsequent work has more fully analyzed its advantages and disadvantages. In this section, we discuss the basic ideas of two-sided matching with a close eye on its application to residency markets. Let us begin by describing a basic two-sided matching model, the college admissions problem.7 Consider a market for medical school students and hospitals. We assume that each student has a preference relation over hospitals and an outside option (being unmatched). For instance, when we write student 1: hospital A, hospital B, we mean “Student 1 prefers A the most and B second. These are the only hospitals acceptable (i.e., preferred to being unmatched) to her.” Likewise, hospitals have preferences over students and leaving a position vacant; in addition, each hospital has a capacity that cannot be exceeded.8 For example, when we write hospital A: student 1, student 2, student 3,

capacity = 2,

we mean “Hospital A prefers student 1 the most, 2 second, and 3 third. All other students are unacceptable.” A matching is expressed as a mapping specifying which student is employed by which hospital (or remaining unmatched). For instance, when we write hospital A – student 1, student 2, 7

The term “College Admissions Problem” follows Gale and Shapley (1962). They framed the problem in terms of a matching between students and colleges. Needless to say, the mathematical structure is general enough that many other problems, including residency programs, can be analyzed with the same framework. 8

This assumption excludes hospital preferences involving substitutability and/or complementarity among students. There are situations where such considerations are important. See chapters 5 and 6 of Roth and Sotomayor (1990). 3

hospital B – student 3, we mean “Hospital A hires students 1 and 2 and hospital B hires student 3.” The college admissions problem is also called the many-to-one matching problem, because it is assumed that a hospital can be matched to more than one student but a student can be matched to at most one hospital. There are also strands of the literature on one-to-one matching (e.g., a marriage market in a monogamous society) and many-to-many matching (as is the case for some labor markets in which a worker may work for more than one employer (see Roth 1991)). For the purposes of this paper, it suffices to consider many-to-one matching problems. Now that we know how to formalize a matching problem mathematically, we must introduce the solution concept we will use to choose “reasonable” matching outcomes consistent with agent behavior. The solution concept we (and much of the matching literature) use is called stability. The concept of stability is an intuitively simple one. Essentially, a matching is stable if no agent (student or hospital) is matched to an unacceptable partner and no two agents (one student and one hospital) would mutually prefer to be matched to each other than to their current partners. To define stability more formally, we introduce the concepts of individual rationality and blocking pairs. Definition: A matching is individually rational if no agent is matched to an unacceptable partner. The idea behind this definition is simple: if the matching is not individually rational, at least one agent (a doctor, for example) is matched to an unacceptable partner (hospital). However, it seems reasonable to assume that this doctor can simply refuse a job at this hospital and remain unmatched (unemployed) instead. That means that the suggested matching is unsustainable. Thus individual rationality seems to be a minimal requirement for any reasonable solution concept. Is individual rationality enough for a reasonable solution concept? When one considers the market for residents and hospitals, there is a sense in which it is not satisfactory. To see this point, imagine that there is a student who is dissatisfied with her partner in the prescribed matching and would like to match with another hospital. What if this hospital has a vacant position? Or what if all the positions are filled, but the hospital prefers the student to one of the students it is currently employing? In such cases, the student-hospital pair can ignore the suggested matching and instead match with each other. Such a pair consisting of a student and a hospital is called a blocking pair, which is defined as follows: Definition: A matching is blocked if there is a student i and a hospital h satisfying the following conditions: (1) student i prefers hospital h to the hospital prescribed by the matching,

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(2) student i is acceptable to hospital h and either (i) there is a vacant position or (ii) h

prefers i to one of the students h is matched with. i and h are called a blocking pair. With these concepts in hand, we can formally define stability. Definition: A matching is stable if it is individually rational and there are no blocking pairs. The concept of stability seems intuitively very reasonable, and has been found to be useful both in the field and in laboratory experiments. For instance, Roth (1991) looks at a natural experiment in the United Kingdom in the 1960s where several regional markets for medical residents adopted centralized clearinghouses to produce matchings. He finds that those mechanisms that survived are those that produced stable matches, while those that produced unstable matches quickly unraveled and were abandoned. He also discusses the market for medical residents in the United States, and notes that the introduction of a stable mechanism in the 1950s lead to improved outcomes and is still in use (with modifications) today. Kagel and Roth (2000) conduct an experiment and find that experimental subjects participate in a clearinghouse that produces stable matches while they abandon a clearinghouse that produces unstable matches. Though many factors may lead to the success or failure of any given market, stability provides a simple and powerful theoretical explanation for why certain mechanisms succeed and others do not, and thus is an important consideration when designing markets. Agents cannot improve upon the outcomes of stable mechanisms via recontracting after the matching is announced, and so a stable matching is very robust. Stability also coincides with the notion of the core in an appropriately defined cooperative game.9 As any outcome in the core is Pareto efficient, any stable matching is also Pareto efficient, providing an important normative justification for stability. The next questions that naturally arise are (1) “Does a stable matching always exist?” and (2) “Is there any easy way to find stable matchings?” Note that desirable solutions may not exist in many problems in economics. For instance, pure strategy Nash equilibria may not exist in many non-cooperative games and the core is often empty in cooperative games. The existence and computation of stable matchings in the college admissions problem are the very questions answered by Gale and Shapley (1962). Theorem 1: Every college admissions problem has at least one stable matching. One of the (potentially multiple) stable matchings can be found by the deferred acceptance (DA) algorithm, described below. 9

In this paper we will neither define the core formally nor discuss cooperative game theory any further. See, for instance, Osborne and Rubinstein (1994) for an excellent introduction to cooperative game theory. 5

Next, we describe the deferred acceptance (DA) algorithm mentioned in the theorem that can be used to actually find a stable matching. Imagine a situation where the match organizer has collected information about the preferences of all doctors and hospitals and knows hospital capacities. With this information, execute the following procedure (although the process is described in terms of “applications” and “rejections”, we use these words only as a metaphor to intuitively describe the mathematical algorithm that produces the matching.) Step 1: Each student applies to her most preferred hospital. Each hospital tentatively keeps its most preferred acceptable students up to its capacity and rejects everyone else (if any). Step t: Each student who was rejected at step (t-1) applies to her most preferred hospital among those that have not rejected her (the student stops applying if there is no such hospital left). Each hospital considers the combined pool of both the new applicants at this step and all the students tentatively kept from the previous step. From this combined pool, each hospital tentatively keeps its most preferred acceptable students up to its capacity and rejects everyone else (if any). The algorithm terminates at a step in which no new rejection occurs. This happens when each student is either tentatively matched to a hospital or has been rejected by all hospitals acceptable to her. At that point, each tentative matching between a student and a hospital is made final. One can define a variant of this algorithm by switching the roles of hospitals and students, i.e., hospitals make offers and students tentatively keep offers. When needed, we refer to the studentproposing DA and the hospital-proposing DA to distinguish these two versions. However, unless explicitly stated otherwise, the term “DA algorithm” always means the student-proposing DA algorithm in this paper. Since at least one application is made at each step and each student has only a finite number of hospitals acceptable to her, the algorithm terminates in a finite number of steps. It is also clear that the algorithm specifies a matching for any input. It is easy to see that the DA algorithm produces a stable matching. To see this, first consider individual rationality. Observe that a student never applies to a hospital that is unacceptable to her, so no student can possibly be matched to an unacceptable hospital. Similarly, hospitals only (tentatively) keep acceptable students, so there is no way that the final match can include an unacceptable student. Lastly, we show that the matching produced by the DA algorithm has no blocking pairs. To show this, suppose that a student, say 1, prefers hospital A to the hospital (or the outside option) prescribed under the DA algorithm, which we will call hospital B. This means that student 1 has applied to hospital A and was rejected at some step of the algorithm (since, according to the algorithm, she would only apply to B after being rejected by A). This means that either student 1 is unacceptable to hospital A or, at the step where student 1 was rejected by hospital A, all positions at A were filled with more preferred students. In the former case, student 1 is certainly

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not part of a blocking pair. In the latter case, note that at the next step of the DA algorithm, hospital A will only keep students who are at least as good as those it had at the previous step (and thus these students are all still preferred to student 1). This holds for every later step, and so, at the end of the algorithm, the hospital still fills all its positions with students it prefers to student 1. Thus, student 1 and hospital A cannot be a blocking pair. This completes the proof, showing that the matching produced by the DA algorithm is stable. Theorem 1 demonstrates that a stable matching always exists, but there may be more than one stable matching in general. To see this, consider the following preference and capacity profile. student 1: hospital A, hospital B, student 2: hospital B, hospital A, hospital A: student 2, student 1,

capacity = 1,

hospital B: student 1, student 2,

capacity = 1.

The DA algorithm lets student 1 apply to A and student 2 apply to B. A and B (tentatively) keep 1 and 2, so, no more applications are made, and this matching is made final. Thus, the resulting matching is hospital A – student 1, hospital B – student 2. This matching is stable because (i) no agent is matched to any unacceptable partner, and (ii) both students 1 and 2 are matched with their first choice hospitals, so they will not be part of a blocking pair. However, the matching, hospital A – student 2, hospital B – student 1, is also stable because (i) no agent is matched to any unacceptable partner, and (ii) both hospitals A and B are matched with their first choice students, so they will not be a part of a blocking pair. Given that there can be multiple stable matchings, how can we compare them? It turns out that the matching produced by the DA algorithm satisfies an interesting property (see Theorem 2.12 of Roth and Sotomayor (1990) for the proof). Theorem 2: The result of the DA algorithm is the “student-optimal” stable matching. That is, each student weakly prefers the hospital she is matched with under DA to the hospital she is matched with by any other stable matching.

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Moreover, the student-optimal stable matching is hospital-pessimal; that is, every hospital weakly disprefers it to any stable matching (Theorem 2.13 of Roth and Sotomayor 1990). Similarly, the hospital-proposing variant of the DA algorithm results in the hospital-optimal and student-pessimal stable matching. In the last example, it is easy to verify that the second stable matching is the one that is produced by the hospital-proposing DA and is hospital-optimal and student-pessimal. The theorem says that different stable matchings may benefit different market participants. In particular, each version of the DA algorithm favors one side of the market at the expense of the other. The DA algorithm has been successfully applied to several labor markets. The most well known application is perhaps to the National Resident Matching Program (NRMP), which is a matching system for medical residents and hospital residency programs in the United States. The NRMP places more than 25,000 students at about 4,000 hospitals every year, and has been used since the 1950s. A pioneering paper by Roth (1984) found that the precursor of the NRMP, the National Intern Matching Program (NIMP), produced the exact same matching as the hospitalproposing version of the DA algorithm (although the NIMP algorithm was different from the DA algorithm). Gale and Shapley discovered the DA algorithm independently, motivated by a more academic interest – finding a core outcome, a theoretically appealing solution– while the organizers of the NIMP discovered their algorithm by a trial-and-error procedure. It is quite amazing that theoretically oriented economists/mathematicians and practically motivated market organizers discovered essentially the same procedure, and this fact suggests that there is something truly deep in the theory. At this point we note that Theorem 2 is related to a policy debate in the NRMP in the 1990s. Recall that the previous NIMP algorithm was the hospital-proposing DA. Some medical students argued that this system favored hospitals at the expense of the students and called for a reconsideration of the mechanism. This issue motivated work by Roth and Peranson (1999), Immorlica and Mahdian (2005), and Kojima and Pathak (2009), although a detailed discussion is beyond the scope of this paper. As we have seen so far, the DA algorithm finds a stable matching for any reported preference profile. However, this does not automatically mean that the DA is a practically useful algorithm for real markets. Why? A big concern is incentive compatibility: that is, the mechanism designer needs to elicit information from the market participants because preferences are private information. So if the mechanism is manipulable – meaning that players may benefit by lying – then the resulting matching may not be stable with respect to the true preferences. As it turns out, the DA algorithm is strategy-proof for doctors; i.e., it is in every doctor’s best interest to always report her true preferences, no matter what everyone else does. Strategy-proofness is often considered to be one of the most important desiderata in market design, and the fact that DA satisfies it is an important strength of the algorithm.

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While we do not present the proof of this result in this paper, it is actually quite intuitive.11 To see this point, note that there is no reason for a student not to apply to her first choice in the first round because, even if she is rejected, she will be able to apply to her second choice in the next round and still be given full consideration by that hospital. This happens in all future rounds as well. In other words, the “deferred” acceptance feature of the algorithm guarantees that she will be treated equally if she applies to a position later than others. What about hospitals? Do they have incentives to misreport their preferences? Unfortunately, there are cases in which misreporting benefits a hospital. To see this, consider the following preference and capacity profile (the same example as before). student 1: hospital A, hospital B, student 2: hospital B, hospital A, hospital A: student 2, student 1,

capacity = 1,

hospital B: student 1, student 2,

capacity = 1.

If everyone reports preferences truthfully, then the DA algorithm matches 1 to A and 2 to B, just as before. What if hospital A lies and reports that “student 2 is the only acceptable student, and student 1 is unacceptable?” Then, in the DA algorithm, student 1 is rejected by hospital A in the first step. This makes student 1 apply to hospital B in the second step. B keeps 1 and rejects student 2. This in turn induces student 2 to apply to hospital A in the third step. A accepts 2, terminating the algorithm. That means that hospital A is made better off by lying, because now it is matched to its choice student 2. Thus we conclude that the DA algorithm does not necessarily incentivize hospitals to report true preferences. Unfortunately, the lack of strategy-proofness for hospitals is not an isolated shortcoming of the DA algorithm, but is more fundamental. The following theorem, due to Roth (1982), shows a general impossibility result (see Theorem 4.4 of Roth and Sotomayor (1990) for the proof). Theorem 3: There exists no mechanism that is strategy-proof and produces a stable matching for all possible reported preference profiles. The policy implication of this result is immense. The theorem implies that the lack of incentive compatibility is not specific to the DA algorithm but is a problem that is impossible to solve. No matter how clever and careful the match organizer is, she cannot achieve both of these goals simultaneously.

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For the proof see Roth and Sotomayor (1990). Generalizations of this result are found in papers such as Hatfield and Milgrom (2005) and Hatfield and Kojima (2008, 2009). 9

2.1: Overcoming the incentive compatibility problem with large markets Given the general impossibility result discussed above, should we give up on designing good mechanisms? Not necessarily. First, note that the impossibility result says only that it is impossible to obtain exact strategy-proofness for all possible markets, and it is silent about how probable it is that market participants will actually misreport their preferences. If manipulation is possible in theory but very hard to conduct in practice, perhaps the DA algorithm is still viable (note that the DA algorithm is indeed used in many markets without failure, so this conjecture seems reasonable). Motivated by these questions, Roth and Peranson (1999) analyzed data obtained from the NRMP. Specifically, they looked at data on the submitted preferences of market participants and computed how many of the participants could have misreported their preferences to obtain a better assignment (assuming that the agent knows exactly what other agents’ submitted preferences are).12 They found that, out of around 4,000 hospitals, there were only about 20 to 30 that could have profitably manipulated the mechanism. Following their simulation analysis, Immorlica and Mahdian (2005) and Kojima and Pathak (2009) provided theoretical arguments for this phenomenon. In particular, they demonstrated that the probability that a hospital can profitably misreport preferences under the DA algorithm converges to zero when the market size (as measured by the number of participating hospitals) approaches infinity. The idea behind their results is that in large markets, there are so many hospitals that a misreport of a particular hospital has only a very small effect on how the algorithm proceeds. Therefore, the kind of manipulations suggested by the above example are very unlikely to be profitable. Note that their large market assumptions are motivated by markets like the NRMP and the JRMP (which, recall, is the Japanese analogue of the NRMP), in which there are more than 1000 hospitals and tens of thousands of students. Thus, these results all suggest that the DA algorithm is likely to work well in many practical applications, despite the impossibility result. Market size seems to be a new and important factor that is useful when studying the performance of matching markets. Another example of this is the residency market with married couples studied by Kojima, Pathak and Roth (2010). It is fairly common for doctors to marry other doctors and seek a pair of jobs in the same region. In the NRMP, for example, over 1000 residents register to the matching system as couples, accounting for over 4 percent of all participating residents. It is well-known that couples can result in the non-existence of stable matchings (Roth (1984) and unpublished work by Sotomayor). Despite this fact, stable 12

Incidentally, Peranson is president of the National Matching Services Inc., a company helping the NRMP and other matching markets by providing algorithms, interfaces and other services. He and Roth modified the NRMP algorithm in the mid 1990s in response to criticisms from some market participants (see Roth and Peranson (1999) for details). One of the concerns expressed by participants was whether the mechanism is fair in the sense that reporting the true preferences is a best action for them. This anecdote seems to suggest that interesting studies in market design are often motivated not only by theoretical curiosity but also by real concerns of market participants. 10

matchings have been found in markets such as the NRMP and the Association of Psychology Postdoctoral and Internship Centers (APPIC), which organizes the matching market for clinical psychologists in the U.S. In fact, Roth and Peranson (1999) run a number of matching algorithms using submitted preferences from 1993, 1994 and 1995 and find no instance in which any of these algorithms failed to produce a stable matching. Kojima, Pathak and Roth (2010) studied APPIC data for the years 1999-2007 and found a stable matching in each case. Kojima, Pathak and Roth (2010) set up a large market model in which the probability that a stable matching exists converges to one as the market size approaches infinity. The above works highlight the importance of examining real-life institutions when writing theory. In particular, the researchers observed a few notable facts seen in the data: stable matchings seemed to exist even when they were not guaranteed, the lack of strategy-proofness did not seem to cause failures of the mechanisms in applications, and the markets generally had large numbers of participants. Using these observations, they were able to construct a theory to better understand these markets. Such large market results can now be applied to overcome traditional impossibility results and successfully design real-world institutions.

2.2: The geographical distribution of doctors As another example of research focusing on practical issues, we discuss the problem of the geographical distribution of doctors, following Kamada and Kojima (2010). The geographical distribution of medical doctors is a contentious issue in health care. One problem is that many hospitals, especially those in rural areas, do not attract sufficient numbers of medical residents to meet their demands. The previous literature on two-sided matching suggests that a solution is elusive, as the “rural hospital theorem” by Roth (1986) shows that any hospital that fails to fill all its positions in one stable matching is matched to an identical set of doctors in all stable matchings. This result implies that a hospital that cannot attract enough residents under one stable matching mechanism will not be able to attract enough residents no matter what stable mechanism is used. The shortage of residents in rural hospitals is a serious problem in Japan, where the deferred acceptance algorithm has placed around 8,000 graduating medical students to about 1,000 residency programs each year since 2003. The Japanese residency matching started in 2003 as part of a comprehensive reform of the medical residency program. Prior to the reform, clinical departments in university hospitals called ikyoku had strong authority to allocate doctors. The system was criticized because it was viewed as giving clinical departments too much power and resulted in opaque, inefficient and unfair allocations of doctors against their will. To cope with these problems, the new system introduced a centralized matching procedure using the deferred acceptance algorithm. However, this system has also been criticized because 11

some hospitals, especially those in rural areas, believe that the new matching mechanism leaves them with fewer residents than they would have been assigned under the old system. In response, the Japanese government recently introduced “regional caps” which restrict the total number of residents that can be assigned to each of the country’s 47 prefectures. The government modified the deferred acceptance algorithm incorporating the regional caps beginning in 2009 in an effort to attain its distributional goals while keeping the advantages of the deferred acceptance algorithm. More specifically, in the Japan Resident Matching Program (JRMP) mechanism, there is a government-imposed target capacity for each hospital. The target capacities are defined in such a way that (i) for each hospital, the target capacity is at most the hospital’s true capacity, and (ii) the sum of the target capacities of hospitals in a region is less than or equal to the regional cap for that region. The JRMP mechanism is a system that produces the matching resulting from the deferred acceptance algorithm except that it uses the each hospital’s target capacity instead of its true capacity. The JRMP mechanism is based on a simple idea: use the standard deferred acceptance algorithm, but, in order to satisfy regional caps, simply force hospitals to be matched to a smaller number of doctors than their true capacities. Kamada and Kojima (2010) show that the current Japanese mechanism may result in avoidable inefficiency and instability despite its resemblance to the deferred acceptance algorithm. They then use these insights to propose a better mechanism. More specifically, they first introduce concepts of stability and (constrained) efficiency that take regional caps into consideration. They point out that the current Japanese mechanism does not always produce a stable or efficient matching and propose a mechanism called the flexible deferred acceptance mechanism which does. They show that the mechanism is strategy-proof for doctors. The flexible deferred acceptance mechanism matches more doctors to hospitals (in the sense of set inclusion) and makes every doctor weakly better off than the JRMP mechanism. These results suggest that replacing the current mechanism with the flexible deferred acceptance mechanism will improve the performance of the matching market. Methodologically, the study investigates a particular market in detail and offers practical solutions addressing market-specific problems. In that sense, this paper contributes to the general research agenda of matching and market design that emphasizes the importance of addressing problems appearing in practical allocation problems.

3: School Choice For a somewhat different application of matching theory, we next look at school choice programs. Public school districts usually assign students to the school located closest to their 12

home. Beginning in the 1980s, however, an increasing number of school districts in various countries began a system often called “school choice,” under which children or their parents can choose schools that they prefer even if the school is not located in their neighborhood. However, seats in the most desirable schools are often limited, so not everyone can attend their most preferred school. The problem, then, is how to fairly and efficiently assign students to overdemanded schools, so that the largest number of students receive their highest possible choices. We can use matching theory to provide answers to these questions.

3.1: Similarities to and differences from the college admissions problem The basic school choice model was formalized by Balinski and Sönmez (1998) and Abdulkadiroğlu and Sönmez (2003). The model is almost the same as the college admissions model described in the last section, but there are a few differences. Similarly to the college admissions model, each student has a preference relation over the schools and being unmatched. Schools have an ordering over students, but we call the relation a priority ordering rather than a preference relation. This is because in many public school districts, priorities are set by law and so we do not consider them to be the “preferences” of the schools. Since the priorities are set by law, schools do not have to report them, and thus we do not have to worry about manipulations by the schools, as we did with hospitals in the previous section. Despite this difference in interpretation, we can treat the priority ordering in the same way as a preference relation in the college admissions problem, so that the two problems are mathematically isomorphic. In this context, the concept of stability must also be given a new interpretation. Suppose that a matching is blocked by a pair consisting of student 1 and school A. This means that (i) the student is denied admission to school A even though she prefers it to the school she would attend under the matching, and (ii) the school either has a vacant seat or is matched to a student whose priority is lower than student 1. Such a situation may be considered to be unfair because student 1 is not admitted to school A even though she prefers it and has priority for it. In other words, student 1 has justified envy. Thus in school choice, we call a matching stable if (1) (individual rationality) no student is forced to attend an unacceptable school, and no school admits a student who is not eligible for it, and (2) (no blocking pair) no student with high enough priority is denied admission to a preferred school In this sense, stability is as an important fairness concept. Since the problem is mathematically the same as the college admissions problem, we can directly apply the theorems from the previous section. Theorem 1 tells us that a stable matching exists in the school choice problem once we interpret school priorities and stability in the manner described above. Theorem 2 says that DA finds the student-optimal stable matching. If one takes the viewpoint that it is only 13

students whose preferences should be taken into account when judging social welfare – which is reasonable here because every student has a right to public school seats, which are then objects to be consumed by children for the purpose of education – then the DA algorithm finds the most desirable outcome for all students among all fair (stable) matchings. Moreover, the fact that the DA algorithm is strategy-proof for students, although not for schools, is also interesting in this context. Recall from above that priority orderings are often set by law, so that schools need not “report” their priority. Since there is no possibility for manipulation by the schools, we can say that the DA algorithm is fully strategy-proof. 13 Recent advances in matching market design have had an impact on the design of school choice mechanisms in many public school systems. Reforms in Boston and New York City are two prominent examples. In both cases, old school choice mechanisms were replaced by the DA algorithm in the first half of the 2000s, in collaboration with economists. A reform of the school choice mechanism is also under way in San Francisco, and here again, economists are working closely with policy makers to design the system. We have argued that when writing theory, it is important to closely study real-life markets. We can give another example of this in the context of school choice. So far in our discussion, we have assumed that the priority order of each school is strict. In a typical school district, however, many students may have the same priority level, meaning the priority ordering is not strict, but is divided into very broad priority classes. In Boston, for instance, each school has four priority classes for new students (in addition to a special class for children who already attend the school): (1) children who live in the school’s neighborhood and whose siblings attend the school, (2) children whose siblings attend the school, (3) children who live in the school’s neighborhood, and (4) all remaining children Thus in reality, there are hundreds of children who have the same priority at any given school. To run the DA algorithm, however, we must have strict preferences and priorities. The current mechanism in Boston solves this problem by breaking ties in some way to produce a strict priority order for each school. There actually do exist school districts in which school principals may express preferences, New York City being one example. In such a case, the impossibility theorem of Roth (1982) suggests that school manipulation is a potential issue because there is no mechanism that is strategy-proof and produces a stable matching. However, in large school districts like New York City, manipulations will once again be difficult under the DA algorithm (recall our discussion of Roth and Peranson (1999), Immorlica and Mahdian (2005) and Kojima and Pathak (2009) in the last section). 13

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At first glance, this “tie-breaking DA” may seem to be a good idea. However, Erdil and Ergin (2008) point out that the efficiency of the matching can be harmed by artificial tie-breaking. Consider their example, reproduced below: Student 1: school B, school A, school C, Student 2: school C, school B, school A, Student 3: school B, school C, school A, School A: student 1, (students 2 and 3 have the same priority), capacity=1, School B: student 2, (students 1 and 3 have the same priority), capacity=1, School C: student 3, (students 1 and 2 have the same priority), capacity=1. Assume, for instance, that the tie-breaking is conducted in such a way that a student with a lower index is granted higher priority than a student with a higher index. In other words, if two students i and j with i
capacity=1,

School B: student 2, student 1, student 3,

capacity=1,

School C: student 3, student 1, student 2,

capacity=1.

Now, let us run the DA algorithm with this priority profile. The resulting matching prescribes School A – student 1, School B – student 2, School C – student 3. However, consider the following matching: School A – student 1, School B – student 3, School C – student 2. This matching is also stable with respect to the original priority profile (before ties were broken). Moreover, the latter matching is preferred by all students to the former (student 1 is 14

Other tie-breaking procedures, such as a random lottery, lead to similar results. 15

indifferent while 2 and 3 are strictly better off). This example shows that DA with tie-breaking can be very inefficient. Given the above, Erdil and Ergin construct an algorithm called the stable improvement cycles algorithm. Their algorithm finds a matching that is constrained Pareto optimal among all stable matchings with respect to the intrinsic priorities. First, they find a stable matching, say by using DA with tie-breaking. Then, they allow students to exchange their assigned schools if doing so profits everyone involved in the exchange and does not violate stability. The algorithm terminates when no additional such trades are possible. The term “cycle” comes from the fact that the students involved in such an exchange form a cycle. Kesten (2009) also proposes a method to find a matching that is constrained Pareto efficient among all stable matchings. Abdulkadiroğlu, Pathak, and Roth (2009) use data from Boston and New York City to calculate the efficiency gained by using stable improvement cycles (compared to standard DA). They find that there is little welfare improvement in the Boston data, while the improvement is large in the NYC data. Despite this welfare improvement, it is not obvious that DA should be replaced by Erdil and Ergin’s method, because their mechanism is no longer strategy-proof. Moreover, Abdulkadiroğlu, Pathak and Sönmez show that there is no strategy-proof mechanism that Pareto dominates DA with tie-breaking for all possible preference profiles. Thus, which of these mechanisms (or other, as yet unknown mechanisms) is the most desirable in a school choice context is an open question.

3.2: Lotteries in school choice and ex-ante efficiency Indifferences in priorities open up another interesting question. In the previous discussions, efficiency was evaluated solely from an ex post standpoint. In other words, a mechanism was deemed stable as long as the final matching did not allow any Pareto improvements. However, with indifferences, lotteries are regularly used to determine allocations. For example, in Boston’s school choice mechanism, ties in priorities are broken using a fair lottery, after which the DA algorithm is run. This implies that what children receive is not a deterministic matching but a random assignment, that is, a lottery over school seats. Thus, we should also consider welfare from the ex-ante point of view by looking at students’ welfare over lotteries rather than over deterministic matchings. Ex ante efficiency and ex post efficiency do not coincide in general, as shown by the following example, modified from Bogomolnaia and Moulin (2001). Student 1 and 2: school A, school B, Student 3 and 4: school B, school A, Let us assume that there are no intrinsic priorities, i.e., all students have the same priority at both schools. Let us also assume that there is one (uniform) lottery used to order the students, and the result of this lottery becomes the priority structure at both schools. For example, if the priority 16

realization is 1, 2, 3, 4, then 1 receives A, 2 receives B, and 3 and 4 remain unmatched. If the realized ordering is 4, 3, 2, 1, then 4 receives B, 3 receives A, and 2 and 1 remain unmatched. Note that student 2 is admitted to B, which is her second choice school, with a strictly positive probability (when the ordering is 1,2,3,4 for example). Meanwhile, student 3 is also admitted to his second choice school A with a strictly positive probability. Observe that B is the first choice for 3 and A is the first choice for 2. Thus, if these two students could exchange probability shares ex ante, both students would be better off. That is, before the lottery is realized, 3 agrees that whenever he receives A, he will give up his place to 2, and, in exchange, 2 agrees that whenever he receives B, he will give his place to 3. With this trade, both students have the same chance of being unmatched, but now, when they are matched, they are always matched to their first-choice school instead of sometimes being matched to their first-choice and sometimes being matched to their second. In other words, the exchange of probabilities results in a random assignment that first-order stochastically dominates the outcome of DA with tie-breaking. With this observation as a starting point, Bogomolnaia and Moulin propose the (symmetric simultaneous) eating algorithm, also called the probabilistic serial mechanism. The mechanism is based on an intuitive idea of “eating.” Imagine that each student is given a unit of time and is allowed to eat with speed one. What do they eat? They are allowed to eat small (in fact, infinitesimal) amounts of probability to get into a school of their choice. That is, at each point of time, each student increases the probability to get into her first choice school with speed 1. If many students desire the same school, the sum of consumed probability shares of that school may reach the capacity of the school. In such a case, students can no longer eat probability shares from that school, and each student who has been eating from that school begins to eat from her most preferred school which is unfilled. In discrete time, we can intuitively think of probability shares as marbles. Each school starts with, for example, 100 marbles, with each marble representing a 1 percent chance at being admitted to that school. At every period, each student gets to take (“eat”) one marble from a school of her choice, as long as that school has marbles remaining. When their favorite school has no marbles left, they take a marble from their next most preferred school. At the end of the algorithm, when no more marbles remain, we run a lottery in which a student’s chance at being admitted to a particular school is equivalent to the number of marbles she has collected for that school. The mechanism is actually run in continuous time and at time one, each student is assigned probabilities over all schools which indeed add up to 1. The probabilistic serial mechanism gives this profile of probabilities as the resulting random assignment. Bogomolnaia and Moulin show that this method eliminates the ex ante inefficiency of DA with tie-breaking. In the above example, for instance, students 1 and 2 are admitted to their first choice school A with probability 50 percent and students 3 and 4 are admitted to their first choice school B with probability 50 percent.

17

The drawback of this mechanism is that it is not strategy-proof. Many papers have studied these tradeoffs to decide between DA with tie-breaking and the probabilistic serial mechanism. For instance, Pathak (2008) conducted simulations (based on data obtained from the New York City school district) using DA with tie-breaking. He finds that the probabilistic serial mechanism does indeed show a welfare gain compared to DA with tie-breaking, but that this gain is “small”. He argues that DA is more desirable since the probabilistic serial mechanism is not strategy-proof and leads to only minimal welfare gains. By contrast, Kojima and Manea (2009) show that students have incentives to report their preferences truthfully in the probabilistic serial mechanism if the school district is sufficiently large. More specifically, if the number of seats at each school is sufficiently large, then reporting preferences truthfully is a dominant strategy for students, which gives a justification for the use of the probabilistic serial mechanism. Although the exact proof relies on some subtle arguments, we can discuss the intuition here. Note that manipulations have two effects. First, given the same set of available schools, reporting false preferences may prevent a student from eating her most preferred available school. Second, reporting false preferences can delay the expiration dates of some schools (that is, the period at which a school has no “marbles” left). The first effect always hurts the misreporting student because it prevents optimal consumption at any given time, while the second effect can benefit the student because it may allow her to eat probability shares of her preferred school later in the algorithm while eating more “high demand” schools earlier, before they are gone. Intuitively, the second effect becomes small as the school district becomes large because in a large market, the effect of any one agent’s eating behavior on the overall amount of eating by a large number of students is small. In the formal proof, the authors make this argument precise. The main idea is to show that the second effect becomes very small relative to the first effect when the environment becomes large, so the misreporting student hurts herself overall. Thus, an exact dominant strategy (rather than an approximate dominant strategy result) is obtained. Che and Kojima (forthcoming) look into these issues further. They show that the two mechanisms become asymptotically equivalent as the market size approaches infinity. So, as long as the size of a school district is large enough, there will be little difference between probabilistic serial and DA with tie-breaking. So there is a sense in which the market designer can feel safe using either mechanism: both the inefficiency of DA and the incentive problems of the probabilistic serial mechanism are likely to be negligible in large school districts. Here again, as in the two-sided matching environment, market size seems to help the designer by allowing her to achieve many desirable properties that might not be present in smaller markets.

4: Conclusion In this paper, we discussed recent developments in matching theory and market design, focusing on work that is directly motivated by, and provides insights into, the design of practical 18

market institutions. The field has been rapidly growing in recent years, but there are still numerous other applications where the insights gained may prove useful. It will likely continue to be an exciting area of research. Acknowledgment This paper has been prepared for an invited lecture delivered by Kojima at the 2010 Annual Meeting of the Japanese Economic Association. A large part of this survey is based on Kojima (2009) and Kojima and Yasuda (2009, 2010) as well as lecture notes from Kojima’s lectures (regular classes and minilectures) at Princeton, Stanford, Tokyo, Yale, and Yonsei. We are grateful to Mutafa Oguz Afacan and Tarun Galagali for their research assistance.

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References Abdulkadiroğlu, A., P. Pathak and A. Roth (2005) “The New York City High School Match”, American Economic Review Papers and Proceedings, Vol. 95, pp. 364-367. Abdulkadiroğlu, A., P. Pathak and A. Roth (2009), “Strategy-Proofness versus Efficiency in Matching with Indifferences: Redesigning the NYC High School Match”, American Economic Review, Vol. 99, pp. 1954-1978. Abdulkadiroğlu, A., P. Pathak, A. Roth, and T. Sönmez (2005), “The Boston Public School Match”, American Economic Review Papers and Proceedings, Vol. 95, pp. 368-371. Abdulkadiroğlu, A. and T. Sönmez (2003), “School Choice: A Mechanism Design Approach”, American Economic Review, Vol. 93, pp. 729-747. Balinski, M. and T. Sönmez (1999), “A Tale of Two Mechanisms: Student Placement”, Journal of Economic Theory, Vol. 84, pp. 73-94. Bogomolnaia, A. and H. Moulin (2001), “A New Solution to the Random Assignment Problem”, Journal of Economic Theory, Vol. 100, pp. 295-328. Che, Y.- K. and F. Kojima (2008), “Asymptotic Equivalence of Random Priority and Probabilistic Serial Mechanisms”, forthcoming, Econometrica. Cramton, P., Y. Shoham, and R. Steinberg (2006), Combinatorial Auctions, Cambridge: MIT Press. Erdil, A. and H. Ergin (2008), “What’s the Matter with Tie-breaking? Improving Efficiency with School Choice” American Economic Review, Vol 98, pp. 669-689. Gale, D. and L. Shapley (1962), “College Admissions and the Stability of Marriage”, American Mathematical Monthly, Vol. 69, pp. 9-15. Hatfield, J. and F. Kojima (2009), “Substitutes and Stability for Matching with Contracts”, forthcoming, Journal of Economic Theory. Hatfield, J. and F. Kojima (2009), “Group Incentive Compatibility for Matching with Contracts”, Games and Economic Behavior, Vol. 67, pp. 745-749 Hatfield, J. and P. Milgrom (2005), “Matching with Contracts”, American Economic Review, Vol. 95, pp. 913-935. Immorlica, N. and M. Mahdian (2005), “Marriage, Honesty, and Stability”, in Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 53-62.

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Kagel, J. and A. Roth (2000), "The dynamics of reorganization in matching markets: A laboratory experiment motivated by a natural experiment," Quarterly Journal of Economics, Vol. 115, pp. 201-235. Kamada, Y. and F. Kojima (2010), “Improving Efficiency in Matching Markets with Regional Caps: The Case of the Japan Residency Matching Program”, mimeo. Kojima, F. (2009), “Game Theory and Market Design,” The Nikkei (in Japanese). Kojima, F. and M. Manea (2010), “Incentives in the Probabilistic Serial Mechanism”, forthcoming, Journal of Economic Theory, Vol. 145, pp. 106-123. Kojima, F. and P. Pathak (2009), “Incentives and Stability in Large Two-Sided Matching Markets”, American Economic Review, Vol. 99, pp. 608-627. Kojima, F., P. Pathak and A. Roth (2009), “Matching with Couples”, mimeo. Kojima, F. and Y. Yasuda (2009), “Matching Market Design,” Keizai Seminar (in Japanese). Kojima, F. and Y. Yasuda (2010), “Frontiers of the School Choice Problem,” in Y. Yasuda (ed.), Designing School Choice Systems, NTT Press, Tokyo (in Japanese). Milgrom, P. (2004), Putting Auction Theory to Work, Cambridge: Cambridge University Press. Osborne, M. and A. Rubinstein (1994), A Course in Game Theory, Cambridge: MIT Press. Pathak, P. (2008), “Lotteries in Student Assignment: The Equivalence of Queuing and a MarketBased Approach”, mimeo. Roth, A. (1982), “The Economics of Matching: Stability and Incentives”, Mathematics of Operations Research, Vol. 7, pp. 617-628. Roth, A. (1984), “The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory”, Journal of Political Economy, Vol. 92, pp. 991-1016. Roth, A. (1986), “On the Allocation of Residents to Rural Hospitals: a General Property of TwoSided Matching Markets”, Econometrica, Vol. 54, pp. 425-427. Roth, A. (1991), “A Natural Experiment in the Organization of Entry-Level Labor Markets: Regional Markets for New Physicians and Surgeons in the United Kingdom”, American Economic Review, Vol. 81, pp. 415-440. Roth, A. (2008), “Deferred Acceptance Algorithms: History, Theory, Practice, and Open Questions”, International Journal of Game Theory, Special Issue in Honor of David Gale on his 85th birthday Vol. 36, 537-569.

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Roth, A. (2008), “What have we learned from market design?” Hahn Lecture, Economic Journal, Vol. 118, pp. 285-310. Roth, A. and E. Peranson (1999), “The Redesign of the Matching Market for American Physicians: Some Engineering Aspects of Economic Design”, American Economic Review, Vol. 89, pp. 748-780. Roth, A., T. Sönmez and U. Ünver (2004), “Kidney Exchange”, Quarterly Journal of Economics, Vol. 119, pp. 457-488. Roth, A., T. Sönmez and U. Ünver (2005), “Kidney Exchange Clearinghouse in New England”, American Economic Review Papers and Proceedings, Vol. 95, pp. 376-380. Roth, A. and M. Sotomayor (1990), Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis, Econometric Society Monographs No.18, Cambridge: Cambridge University Press. Sönmez, T and U. Ünver (forthcoming), “Market Design for Kidney Exchange”, in Z. Neeman, M. Niederle, N. Vulkan (eds.) Oxford Handbook of Market Design.

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