Lloyd Shapley (1923–2016) a Founding Father of Game Theory

Home , David Gale, Lloyd Shapley, Martin Shubik, RAND Corporation

COMMENT OBITUARY Lloyd Shapley (1923–2016) A founding father of game theory.

loyd Stowell Shapley made than one suitor. Marry. The result is profound contributions to stable, since the extramarital liaisons almost every area of game that were previously rejected will be Ltheory — a field of mathematics that disliked by the girl partners, while tries to understand how people’s all others will be disliked by the boy RAND CORPORATION choices influence others’. His findings partners.” have been applied to all sorts of set- Thus was born the ‘deferred tings, from politics to hospitals. His acceptance algorithm’, variants whimsically titled 1962 paper — ‘On of which are used today to assign College Admissions and the Stability US medical graduates to their first of Marriage’— published in The jobs, and children to state schools in American Mathematical Monthly and a growing number of US cities. co-authored with the late mathemati- Other work from the 1970s by cian David Gale, won Shapley the 2012 Shapley and the late economist Nobel prize in economics, which I Herbert Scarf on the money-free shared with him. exchange of indivisible goods Shapley, who died on 12 March, (‘barter exchange’) later helped to was born in 1923 in Cambridge, organize kidney transplants when Massachusetts, to the astronomer donors cannot directly donate to Harlow Shapley and his wife Martha. the patient of their choice because He pursed a mathematics degree at of incompatibilities. And starting Harvard University in Cambridge. In in the 1990s, Shapley’s ideas about 1943, during his third year and at the two-sided matching and extended height of the Second World War, he barter exchange led to a branch of was drafted into the US Army. In his economic engineering called market years of service, he worked at an air design, which seeks to find practical base in China and won the Bronze Star, a Not every game has a core outcome. ways to fix broken markets. US military decoration, for breaking a code But for those that do, it often indicates In the early 1960s, Shapley and John for Soviet weather reports. how competition will play out. Shapley’s Milnor (an undergraduate at Princeton After the war, Shapley graduated from paper with Gale explored this concept in when Shapley was a graduate student) Harvard and worked for two years at the the context of two-sided ‘matching games’ initiated the study of ‘oceanic games’. In RAND Corporation, which at the time (D. Gale and L. S. Shapley Am. Math. Mon. these, there is an ‘ocean’ of many small provided research and analysis to the 69, 6–15; 1962). In these situations, two players each alone having insignificant US military. There, he began his work on sets of players (in the paper, boys and girls influence, so only the actions of people en game theory and came to the attention of seeking marriages, and colleges seeking masse can affect the overall outcome. He the field’s founder, John von Neumann. students and students seeking colleges) later explored these with Robert Aumann, In 1949, Shapley entered the PhD have preferences about whom they would another Nobel economics laureate, in programme in mathematics at Princeton like to match with. their volume Values of Non-Atomic Games University in New Jersey — then a hotbed In a simple model of one-to-one match- (RAND Corporation, 1968). for game theory. There, he overlapped with ing — as applies when each player seeks one Although Lloyd and I shared the Nobel the mathematician John Nash and the econ- spouse, for instance — the core outcomes prize, we never worked together. But his omist Martin Shubik (who would become are those that are stable. After everyone in work was fundamental to my own — for his long-term collaborator), among many the game has chosen, there are no pairs (of instance, on the practical design of market others. Shapley rejoined RAND in 1954, and girls and boys, in the 1960s partnering exam- places. He was a forbidding presence at stayed with the organization for 27 years. In ple) who are not matched to each other but meetings; I suspect shyness was to blame 1981, he moved to the University of Califor- would both prefer to be. for his apparent fierceness. nia, Los Angeles, where he retired in 2001. In a note written to Shapley in 1960, There is a crater on the Moon named Game theory describes any situation Gale asked, “For any pattern of prefer- Shapley, in honour of Lloyd’s astronomer in which the pay-offs that participants ences, is it possible to find a stable set of dad. In game theory, Lloyd will likewise be receive from their actions are at least partly marriages?” Shapley gave his answer in a remembered for the mammoth impact he determined by the actions of other people. letter to him: “Let each boy propose to his had on the field. ■ Shapley was one of the first to formulate best girl. Let each girl with several pro- and study the ‘core of the game’ — the set of posals reject all but her favorite, but defer Alvin E. Roth is professor of economics at outcomes (consequences for everyone in the acceptance until she is sure no one better Stanford University, California, USA. He group) with the property that no coalition of will come her way. The rejected boys then shared the 2012 Nobel Memorial Prize in players can do better for themselves by coor- propose to their next-best choices, and Economic Sciences with Lloyd Shapley. dinating to produce a different outcome. so on, until there are no girls with more e-mail: [email protected]