One of the issues in organizing a centralized matching mechanism is to make the market thick, by including all or most of the relevant choices in the centralized system. If not, there will be transactions outside of the centralized marketplace, and some of them may be costly to the system.
Here's a paper that explores that in the context of college admissions in Chile.
Aftermarket Frictions and the Cost of Off-Platform Options in Centralized Assignment Mechanisms, by Adam Kapor, Mohit Karnani, and Christopher Neilson, Journal of Political Economy, forthcoming.
Abstract: "We study the welfare and human capital impacts of colleges’ (non)participation in Chile’s centralized higher-education platform, leveraging administrative data and two policy changes: the introduction of a large scholarship program and the inclusion of additional institutions, which raised the number of on-platform slots by approximately 40%. We first show that the expansion of the platform raised on-time graduation rates. We then develop and estimate a model of college applications, offers, wait lists, matriculation, and graduation. When the platform expands, welfare increases, and welfare, enrollment, and graduation rates are less sensitive to off-platform frictions. Gains are larger for students from lower-socioeconomic-status backgrounds."
"in virtually every practical implementation there exist many off-platform options that are available to participants of the match. In primary and secondary education, these include private schools or charter schools that do not participate in the centralized system. In other cases, such as higher education, some providers may be excluded from the platform by regulation, while others may choose not to participate. When off-platform options exist, applicants may renege on their assigned matches in favor of programs that did not participate in the centralized process. In turn, these decisions lead to the use of wait lists and aftermarkets, which may be inefficient due to the presence of congestion and matching frictions and can be inequitable if some students are better able to navigate this partially decentralized process, negating some of the benefits of the match.
"In this paper, we study the empirical relevance of the configuration of on- and off-platform options for students’ welfare and for persistence and graduation in higher-education programs. We document the importance of negative externalities generated by off-platform options and quantify a measure of aftermarket frictions that contribute to generating them in practice. Our empirical application uses data from the centralized assignment system for higher education in Chile, which has one of the world’s longest-running college assignment mechanisms based on the deferred-acceptance (DA) algorithm.2 We take advantage of a recent policy change that increased the number of on-platform institutions from 25 to 33, raising the number of available slots by approximately 40%. We first present an analysis of the policy, which shows that when these options are included on the centralized platform, students start college sooner, are less likely to drop out, and are more likely to graduate within 7 years. Importantly, these effects are larger for students from lower-socioeconomic-status (SES) backgrounds, suggesting that the design of platforms can have effects on both efficiency and equity.
...
"We find that when students are allowed to express their preferences for a larger variety of options on the platform, welfare increases substantially, as does the share of students graduating on time.
...
"Intuitively, when a desirable program is not on the platform, it can cause some students who would have placed in that program to instead receive a placement in a different program available on the platform. These students may then decline that placement in favor of the off-platform program, creating vacancies that in turn lead to increased reliance on wait lists, which may be subject to frictions. Moreover, the absence of a particular program may distort the placements of other students, even if the students whose placements are affected would never enroll in that program. These students may also be less satisfied and more likely to decline their placement.
"Taken together, our results show empirically that the existence of off-platform options affects the equity and efficiency of centralized assignment systems. "
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Earlier (with some links to still earlier papers):
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