Roth, who is new at Stanford this fall after several years at Harvard Business School, is an expert at "market design," or the creation of matching systems or other mechanisms in situations in which normal markets are, for whatever reason, impracticable. He is perhaps best known for designing the system that matches graduating American medical students with residency programs and the mechanism the New York public school system uses to pair students and schools, the latter of which he worked on alongside Atila Abdulkadiroglu and Tayfun Sonmez.
He is also notable, with Sonmez, for creating an innovative "kidney swap" program. Normally, people who want to donate kidneys to a sick family member with an incompatible blood type are stuck. But Roth had the idea of pairing up such families and having them swap organs. For example, suppose Jane Smith (who's B+) wants to give her brother John (who's A-) her kidney. But John would reject the organ because it's the wrong blood type. Roth's system would match the Smiths with the Lees: George (who's A-), who wants to donate, and Jill (who's B+) who needs a kidney. Jane Smith gives Jill Lee one of her kidneys, and George Lee gives John Smith one of his, both of which match. Instead of neither getting a kidney, both do.
Roth has been the subject of two excellent profiles, one by Forbes's Susan Adams and another by the Boston Globe's Leon Neyfakh, focusing on his efforts to apply his work in practice, through programs like the kidney swap. He also writes a blog aimed at a general audience — suitably titled "Market Designer" — which he updated Monday morning to warn that the "Blog may be delayed today…Count me as surprised."
Shapley, 89, may not yet know that he's won the prize. He lives on the West Coast, and the Nobel committee could not reach him before they announced the award. While more theoretical than Roth's work, Shapley's research has also focused on using game theory to resolve situations in ways that benefit all parties. He received his Ph.D in mathematics from Princeton in 1953, when the founders of game theory, John von Neumann and Oskar Morgenstern, were on the faculty there and at the nearby Institute for Advanced Study. Robert Aumann, a game theorist, frequent coauthor of Shapley's and corecipient of the 2005 Nobel prize in economics, said in his Nobel lecture that he considers Shapley "the greatest game theorist of all time."
Shapley is perhaps best known for the "Shapley value," which refers to the cumulative benefits reaped by all participants in a game in which participants are cooperating rather than competing. One case where such a value is important is in designing matches like those that Roth specializes in; indeed, Roth contributed a chapter to a book in Shapley's honor which explains what Shapley values are in more detail.
He is also notable for having solved, with David Gale, the "stable marriage problem": that of pairing a certain number of men and women, each of whom have ranked the members of the opposite sex, such that there is no partner with whom anyone would rather be than the one with whom they are paired. Berkeley has a fun Web game that explores the problem in more detail. This case can also be generalized to apply to other situations where one needs to pair two people, or a person and an institution, such that both are satisfied. That includes cases like pairing medical students and hospitals, or public school students and schools — exactly the sorts of situations in which Roth specializes.
Adequately summarizing all of Shapley's mathematical accomplishments would take several blog posts, but one other concept of his that has political and economic relevance is the "Shapley-Subik power index," which is used to determine which member of a decision-making body, such as a Congress or a parliament, holds the most power to determine outcomes. It often finds that members one wouldn't expect to hold a huge amount of sway are actually the most important actors.