When Students Are Matched to Schools, Who Wins?

Research suggests that this system, versions of which school boards across the US have embraced, can increase overall welfare, on top of other advantages. For example, Duke’s Atila Abdulkadiroğlu, whose seminal work in the early 2000s with Tayfun Sönmez of Boston College prompted Boston Public Schools to rethink the mechanism it employed (see “School choice in practice,” below), argued in a 2011 paper with Yeon-Koo Che of Columbia and Yosuke Yasuda of Osaka University that it may be preferable to some other school-assignment systems in part because it breaks ties between students who want the same school by rewarding the student who ranked it higher. In other systems, such decisions are usually determined by chance, using a random lottery. And these situations are common: even if siblings or residential proximity give some students greater priority over others, many students will still have equal claim on the same school. (This dilemma also occurs at the college level, where universities sometimes use early-acceptance policies to help them choose between equally qualified candidates on the basis of the candidates’ passion for the school.) Abdulkadiroğlu and his coresearchers argue as well that a system that rewards strategizing might even benefit “naive” students ill-placed to game it, since strategic students would avoid putting popular schools at the top of their lists, leaving more spots for everyone else.

But the drawbacks are evident. First, there is significant room for regret if a student feels she has misjudged her chances at getting into one school or another. Second, the system rewards risk-taking, but for only select students: if you had a 1/100 chance of getting into your favorite school and a 1/10 chance at your second favorite, you should rank your second-favorite school at the top of your list—unless you had the option of opting out of the system entirely should you miss your 1/100 chance. Students for whom private school is an option can take a chance on the most popular public schools and go private if they don’t get the outcome they want.

More fundamentally, there is no easy rule of thumb for understanding your chances. Even if you could effectively parse historical data on levels of demand for one school or another, others might be doing the same, meaning that how your fellow students are ranking their choices in the same year you are ranking yours—the only year that matters—will remain opaque.

The system leaves even sophisticated individuals befuddled. “Occasionally we’d get notes from Yale faculty with school-aged kids asking us how it worked,” says Zimmerman of the New Haven lottery.

Even systems simpler than the so-called Boston Mechanism then being used by New Haven may be too complicated for many users. Research by Cornell’s Alex Rees-Jones and UPenn PhD candidate Samuel Skowronek finds that 23 percent of medical students mistakenly misrepresented their preferences in a strategy-proof system that mirrored the real-life residency match that they had only just taken part in. Chicago Booth’s Eric Budish and UPenn’s Judd Kessler have found similar results for course allocation at the Wharton School at the University of Pennsylvania (see “You can’t always say what you want,” below). These findings and experiences raise concerns that families lacking time or statistical know-how would be least prepared to navigate algorithmic matching processes—and perhaps more fundamentally, that such processes may not be as effective as anticipated if students and their parents don’t understand them.

Is honesty the best policy?

Last September, Kirsten Youngren attached a handwritten list of 27 high schools to her refrigerator. Over the following six months, the slip of paper became crowded with amendments: dates and times for school visits, and long, wiggly arrows noting schedule changes; checks, strike-throughs, and stars to code experiences at the schools; and highlighting to indicate long commutes.

Youngren’s daughter, Bridget, is a rising ninth grader in Lower Manhattan who was navigating a wide variety of options for high school. She had taken the controversial test that determines placement at eight of New York’s elite, or “specialized,” high schools, mostly under the influence of her best friend, who wanted to go to Stuyvesant, the fourth-best public high school in the US, according to national school-ranking website Niche, and the school at the center of the recent debate in New York over admissions policies and race. Bridget was also interested in “screened” schools—that is, public schools that prioritize interested students on the basis of academic performance, attendance records, or art portfolios—as well as nonselective public schools, which would admit her on the basis of New York’s strategy-proof lottery system, which Atila Abdulkadiroğlu and his collaborators, MIT’s Parag Pathak and Harvard’s Alvin Roth, helped design. (The previous system left tens of thousands of students unmatched until late summer and then often assigned them to schools they didn’t want; the new design helped Roth win the Nobel Prize in Economic Sciences in 2012.) And Bridget was weighing three Catholic high schools.

The fact that Bridget’s parents could pay Catholic-school tuition—though not the much-higher fees at other New York private schools—gave her more options than many of her peers had. But her mother felt the advantage was even more deeply rooted than that: Kirsten, an architect who works from home for a firm that accommodates flexible hours, was able to invest a huge amount of time in the process, researching the schools and the system itself. “There’s something to be said about the amount of work that needs to be done,” she says. “It was sometimes awkward being one of the few parents at Bridget’s middle school who had the flexibility to put in that time.”

And even though the middle school—a public school on the Lower East Side with a high number of low-income students—and others like it offered help to students and families navigating the lottery process, sometimes that advice was ill-conceived: Kirsten recalls someone at a presentation last fall saying that students should think twice about applying to the most popular schools, since their chances of getting in were small. Kirsten ignored the suggestion, but parents who took the advice to heart might have needlessly given up their child’s chance, since New York’s matching system rewards “truth telling,” or honest ranking of a student’s favorite schools, over strategy games.

In their research on New Haven’s schools, Zimmerman and his coresearchers wanted to quantify the effects of this sort of inequality, and particularly to understand how it played out in a system even more strategically complicated than New York’s. Past research assumed either that students and their families were strategizing correctly or that they were making one of a limited number of possible mistakes, such as not knowing their own priority group or playing naively by simply listing their choices in order of their preferences. With Princeton’s Adam Kapor and Christopher Neilson, Zimmerman surveyed 417 families who took part in the New Haven high-school lottery in 2015 and 2017, asking how they ordered their rankings and why, and then analyzed the matches that had resulted.

They find that people consistently under- or overestimated their odds of getting into a school and strategized poorly as a result. Moreover, the researchers’ models showed that unless mistakes could be reduced by a third for ninth-grade matches and by almost two-thirds for elementary-school placements, a system that rewards strategic play would perform worse than one in which families simply rank their preferences honestly.

“It’s important to think about mistakes,” Zimmerman says. “If you could get people to play the system perfectly, that would be the better matching mechanism. But we were far from that.” The researchers tried developing an app that would increase families’ understanding of their odds and help them strategize accurately. But they soon saw that simply adopting a truth-telling approach made more sense. Their work found an audience in New Haven Public Schools, and for the 2019–20 school year, the city began employing a matching algorithm similar to New York’s.

Districts that want to strategy-proof their matching systems—remove the incentive for students to report anything but their true school preferences—typically choose between two types of algorithms, top trading cycles and deferred acceptance. A top-trading-cycles mechanism prioritizes efficiency: it allocates spots at various schools in such a way that no two students would both be made better off by swapping their school assignments. Deferred acceptance creates stable matches; a match is unstable if a student at one school prefers another, and has higher admissions priority (based on the hierarchy of student traits the district uses to determine priority at different schools) at her target school than any of its admitted students does.

Because DA produces stable matches, DA systems are more restricted by districts’ priority structures than are TTC systems. This can prevent swaps of schools between students, even when both students would prefer the trade. The result may be lower student satisfaction, but the district might accept that cost if it sees a benefit in preventing swaps—for instance, to limit the average distance between students’ homes and schools, to keep bussing costs down.

Chicago Booth’s Jacob Leshno says that currently most districts don’t use TTC systems, and he suggests a potential reason many have opted for DA instead: TTC systems are harder to explain to students who don’t get the schools they want. In a DA system, when a student doesn’t get the match she wants, the explanation is straightforward: she didn’t have high enough priority at that school, based on whatever priority structure the district has created. But in TTC, it can be more difficult to explain in a nontechnical way why a student received a disappointing match, except that it was part of an efficient distribution of satisfactory matches overall.

However, Leshno and Stanford’s Irene Lo wanted to help administrators make full use of their options for school-matching systems by providing tools to help explain how TTC school-assignment algorithms work. Their research demonstrates it’s possible to explain matches under TTC systems to students and parents using the same palatable notion that applies to DA systems, removing a big impediment to their implementation.

In essence, a given student has an endowment based on her priority at each school, which the researchers represent as tokens. Her budget of tokens is different for each school, since some factors—such as whether she lives nearby—are school specific. Each school also has a “price,” or a minimum number of each token students need in order to afford admission. Because students can trade school assignments based on their preferences, a student with insufficient tokens for School A, her target school, could potentially still afford admission using tokens for School B, provided there was at least one student admitted to School A who preferred School B but didn’t have enough B tokens. After assignments are made, districts can even publish prices publicly, showing the minimum number of each token that was required for admission to each school, to help students and their families verify the student is matched to their favorite of all the schools they can afford. The researchers suggest that being able to frame outcomes in this way may make it easier for districts to adopt TTC mechanisms for school choice.

Horizontal versus vertical variation

One realization that struck Lauren Young periodically as she was considering Claudia’s middle-school options was that, for all the schools did to advertise their differences—art every day versus gym every day, a focus on math and science, a forward-thinking principal—they were ultimately more alike than not. “The system encouraged parents to overemphasize what’s special about a school, and you forget that they’re all teaching the same curriculum and they’re really only different at the margins,” she says. When she thought about it this way, the District 15 lottery was “a choice, but not a choice.”

In economics, vertical differentiation refers to variation within a set of goods such that all consumers have the same preferences—one product is of higher quality than the rest, and everyone prefers that product. Horizontal differentiation occurs when different consumers have preferences for different things. Generally speaking, school choice has the opportunity to make a bigger difference, Leshno says, when there’s greater horizontal differentiation—that is, when students don’t all want the same thing.

But Leshno and Lo’s larger finding is that the debate over which algorithm to use risks obscuring a more powerful lever for student happiness: a school district’s priorities. Should the sibling of a current student get priority at a particular school? Should children have first dibs on their neighborhood school? Should attendance at a school open house give students an advantage in a lottery—a factor that diversity advocates argue hurts poorer, nonwhite families? As districts decide which traits to prioritize and how much relative weight to give to each, they determine, to a large extent, the set of schools available to each student to choose from. That makes the choice of priority structure hugely important, no matter the algorithm: generally speaking, student welfare will be higher in a district with a sensible priority structure than in a district with a poorly designed one, regardless of the mechanism either is using.

“When you have priorities that make sense, that’s where things change, and so that’s where the discussion should be,” Leshno says. He emphasizes that districts need to be transparent about not only their choices of priorities, but also how those choices affect which sets of schools are available to which students.

Nor should the debate over matching mechanisms disguise the limits of market design. “All of this only helps to the extent that we create better matches,” Zimmerman says. “We’re still allocating students to the schools we have: creating better matches doesn’t fix the problem of bad schools.”

Carol Burris, a former high-school principal, executive director of the Network for Public Education, and author of the book On the Same Track: How Schools Can Join the Twenty-First-Century Struggle against Resegregation, is even more critical. Lotteries as the primary vehicle for admissions do not decrease segregation, she argues. And in the meantime, they elide problems related to curriculum, disparities in which can exacerbate segregation. Schools try to distinguish themselves using their varying offerings, when in fact every student deserves to attend a school with a rich and varied curriculum. Choice, says Burris, often becomes a substitute for the hard (and more expensive) work of improving individual schools.

School matching is not necessarily a zero-sum pursuit, Leshno says. School choice does not create additional educational resources, but with a well-designed system, a district can make the most of the resources it does have. When they’re functioning as they should, school-matching systems have the potential to better pair what schools have to offer with what individual students need, “even if you accept the fact that there aren’t enough good seats for everybody.”

Choice only goes so far

Lauren Young felt good about the list of schools Claudia finally submitted to District 15, but more ambivalent about the process itself. They had left off two schools that Young and her husband felt were entirely wrong for their daughter, and ranked all nine others. Private school was not a consideration, for financial and philosophical reasons, but other ways of opting out of the system did cross Young’s mind from time to time: “The whole experience made me realize why people move to the suburbs,” she says. Kirsten Youngren felt similarly, visiting friends in New Jersey last year and wondering whether her family shouldn’t perhaps uproot from Manhattan and start building a life in a place where kids go to the local public school, and can count on that local public school being sound.

In April, Claudia found out where she will be starting sixth grade this fall: Brooklyn Collaborative, which had been fourth on her list. The three schools she ranked higher were historically white-majority institutions with affluent student bodies and the best academic reputations in the district. Meanwhile, very few affluent children—a group into which Claudia falls—attended Brooklyn Collaborative in the past; its incoming class of sixth-graders will now be the district’s most affluent, a product of the district’s diversity priorities and the way parents and students ranked their choices.

Lauren is happy with the school’s pedagogy. Claudia, meanwhile, was disappointed not to have gotten into the arts-focused school she had ranked No. 1, and appealed the decision unsuccessfully. Given the strength of her desire to get into her No. 1 school, a strategic-play mechanism might have worked in her favor, but it is impossible to know for certain.

A month earlier and across the East River, Kirsten Youngren’s daughter, Bridget, learned that she had been matched with the 12th-ranked school on her list of 12 New York City public high schools. Bridget decided to enroll in a Catholic school this fall.