Mr. Roth designed a system in which students had an incentive to tell the truth about where they wanted to go. A centralized office could then assign them to a school best suited for them, based both on their own preferences and the preferences of the schools they were applying to.

The school systems he helped create use a “deferred acceptance algorithm,” which was developed by Mr. Shapley’s theoretical work.

The system works by tentatively accepting students to their top-choice school. It holds off on the final assignment until it has gone through all the other applications to make sure there are not other students who have a higher claim to a spot at that given school (because of higher test scores, a sibling at the school or whatever other criteria the school prioritizes), even if those students happened to rank the school lower on their list of preferences.

“The idea is to level the playing field,” Mr. Pathak said. “You want to make sure that not only do sophisticated players not have to spend the time learning the strategies and different heuristics that will get them ahead, but also that unsophisticated players are not hurt by the fact that they are not aware of all this information.”

This same sort of system is used to match new medical school graduates to medical residency programs, which was once a messy process that led to a lot of unhappy candidates. Now all residency assignments are posted simultaneously. In the mid-1990s Mr. Roth redesigned the system to help match married couples who were jointly looking for jobs at hospitals.

Mr. Roth has also helped build a system that assigns kidney donation swaps.

For example, a man needs a kidney, and his wife is willing to donate one of hers but she is not a match. Across the country there is a couple in the same position, and it turns out that the wives are a match for the husbands in the opposite couple. In this simple case, the two couples essentially barter their kidneys: Wife A gives her kidney to Husband B, and Wife B gives her kidney to Husband A. It is rare that two couples will serendipitously match each other’s kidney donation needs this way, and there are often more pairs of donor-recipients involved. Mr. Roth’s system helps find the most efficient exchange of organs so that the most patients can be saved with the fewest number of pairs involved in a given trade.