John Forbes Nash (1928–2015) Master of Games and Equations

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COMMENT OBITUARY John Forbes Nash (1928–2015) Master of games and equations.

ohn Forbes Nash, an exalted mathema- and nonlinear partial differential equations. tician whose life took dramatic turns He was not afraid to tackle the hardest between genius, mental illness and celeb- problems in the field, and he succeeded. In

J MIT MUSEUM rity status, made major contributions to game 1957, he — in parallel with Italian mathe- theory, geometry and the field of partial matician Ennio de Giorgi — solved Hilbert’s differential equations. nineteenth problem involving partial differ- Nash, who died on 23 May, was born in ential equations. Bluefield, West Virginia, in 1928. His father It was during a talk in 1959 on what is seen was an electrical engineer and his mother to be one of the hardest problems in maths — a schoolteacher. In 1945, after excelling in the Riemann hypothesis — that the audience mathematics at high school, he attended realized that there was something wrong with the Carnegie Institute of Technology (now Nash. His talk was incomprehensible. Carnegie Mellon University) in Pittsburgh, He was diagnosed with paranoid schizo- Pennsylvania. At first he studied chemi- phrenia that year. Over the next two decades, cal engineering, but soon after enrolling he Nash was in and out of hospitals. He under- switched to chemistry and then to maths. went therapy, and for a while left the United In Nash’s final year, one of his profes- States and sought asylum in Switzerland in sors wrote a recommendation letter for the an attempt to escape his imagined tormen- 19-year-old supporting his application to tors. For many years he wandered around the graduate school. It simply stated: “He is a Princeton campus. Throughout this period, mathematical genius.” In 1948, Nash was Alicia, who divorced Nash in 1963, oversaw accepted by Harvard University in Cam- a formidable task in a complex game. There much of his care. bridge, Massachusetts, and by Princeton is also the uncertainty over whether the In the late 1980s, Nash reappeared in University in New Jersey. He chose Princeton. person you are playing against is sufficiently academic circles, and in 1994 he was awarded As a PhD student, Nash proved the exist- rational to play the equilibrium strategy. If the Nobel Memorial Prize in Economic ence of the equilibrium that now carries his both players are rational and their rational- Sciences for his work on game theory. The name. His 1950 paper ‘Equilibrium points in ity is common knowledge, they would play Nobel and the 2001 film A Beautiful Mind, n-person games’, contains about 330 words, it. But experiments often reveal that people based on journalist Sylvia Nasar’s book of the two references and not one equation (J. F. Nash are not rational. Regardless of whether people same name, which recounted Nash’s strug- Jr Proc. Natl Acad. Sci. USA 36, 48–49; 1950). actually play the Nash equilibrium in social or gles, propelled him into the limelight. One of the citations is the 1944 book Theory economic interactions, working out what it In May this year, Nash received the Abel of Games and Economic Behavior — in which is (or what the Nash equilibria are) is the first prize from the Norwegian Academy of mathematician John von Neumann and econ- step to analysing any game. Science and Letters for his work on partial omist Oskar Morgenstern introduce game Although dismissed at the time by von differential equations. On the way back from theory, a mathematical approach for studying Neumann as a triviality, the Nash equilib- the celebration in Norway, John and Alicia strategic and economic decisions. rium has been used to analyse all sorts of (who had remarried in 2001) were killed in The Nash equilibrium is a position in competitive situations. As well as being key a car accident in a taxi on the New Jersey a game from which none of the players to decision-making in economics and poli- turnpike. John was 86. can change their strategy to improve their tics, the idea is important in biology. Here, I met John in 1998 at the Institute for pay-off. Imagine a game with two players the nearly equivalent concept, formulated by Advanced Study in Princeton. Over the years, (yourself and another person) and two strat- evolutionary biologist John Maynard Smith I gave several talks there that he attended. One egies, A and B. If you both choose A, your in the 1970s is called an evolutionarily stable summer’s day, when the usual sitting arrange- pay-off is 2. If you choose A and your oppo- strategy (ESS). If all members of a popula- ments for lunch were disrupted by the closure nent chooses B, you score 0. If you choose B tion adopt an ESS, then natural selection of the main kitchen, I noticed John, the physi- and the other player chooses A, your pay-off prevents a rare mutant from spreading. cist Edward Witten and Andrew Wiles, the is 3. If you both choose B, you score 1. The On completing his PhD, Nash joined the British mathematician who proved Fermat’s same applies to your opponent. Massachusetts Institute of Technology (MIT) last theorem, sitting down together at a small In this example, the Nash equilibrium in Cambridge in 1951. He worked — first as table. I wondered which of them would start occurs when both players choose B. If both an instructor and later as a professor — in the the conversation. None of them did. I seem to players choose B, their pay-off is 1; if either mathematics faculty until he resigned in 1959. remember that they ate their meal in silence. ■ player switches to A, their pay-off falls to 0. It was while he was at MIT that Nash met and In other words, neither player can indepen- married Alicia Lopez-Harrison de Lardé, a Martin A. Nowak is professor of dently switch their strategy and improve their physics student there. mathematics and biology, and director of pay-off. Observe that if both players select A, Among mathema- NATURE.COM the Program for Evolutionary Dynamics there is no Nash equilibrium because you ticians, Nash is best For more on John at Harvard University in Cambridge, could improve your pay-off by switching to B. known for his work in Nash’s life, see. Massachusetts, USA. Calculating the Nash equilibrium can be real algebraic geometry go.nature.com/bi7gxg e-mail: [email protected]