John Forbes Nash, Jr
John Forbes Nash, Jr.
Awarded a Nobel Prize in Economics in 1994 for his pioneering work in game theory, John Nash (born 1928) distinguished himself as one of the foremost mathematical researchers and theorists of the twentieth century.
Game theory was the subject of Nash's doctoral dissertation at Princeton University. Expanding upon the initial game theory of John von Neumann and Oskar Morgenstern, published in their The Theory of Games and Economic Behavior, Nash developed what became known as "Nash's equilibrium" to explain how two or more competitors can arrive at a mutually beneficial yet noncooperative business arrangement. In 1951, he developed the theory that manifolds—objects containing various forms and components—can be described accurately using algebraic equations. He later developed what became known as the NashMoser theorem, which explained how it was possible to embed a manifold in a Euclidean space by employing differential calculus instead of algebra and geometry. Nash's subsequent career was diminished by severe mental illness, which was documented in Sylvia Nasar's biography of Nash, A Beautiful Mind, and the film of the same name directed by Ron Howard.
Early Interest in Math
Nash was born on June 13, 1928, in Bluefield, West Virginia, and raised by his parents, John Nash, Sr., an electrical engineer for the Appalachian Power Company, and Margaret Nash, a teacher who retired after her marriage and placed a high value on the education of her two children. As a young man he seemed disinclined to study but displayed a passion for electronics and chemistry experiments that he conducted in his bedroom.
When he was a young teenager, Nash read a book by E. T. Bell, Men of Mathematics, to which Nash attributed his eventual passion for number theory. While attending high school and concurrent classes at Bluefield College, Nash collaborated with his father on a paper titled "Sag and Tension Calculations for Cable and Wire Spans Using Catenary Formulas," which was published in a 1945 edition of Electrical Engineering. Nash also entered the George Westinghouse competition, winning one of ten nationally awarded full scholarships, which he used to enroll in the Carnegie Institute of Technology in Pittsburgh.
Mixed Success in Education
Initially aspiring to become an engineer like his father, Nash changed his major to chemistry after performing poorly in mechanical drawing. After he also had trouble with a physical chemistry class, he was convinced by his calculus instructor John Synge to major in mathematics. In 1948, Nash was awarded the John S. Kennedy Fellowship at Princeton University.
At Princeton, Nash was in close proximity to the Institute of Advanced Study, which attracted such notable mathematicians as Albert Einstein, Kurt Godel, Karl Oppenheimer, Hermann Weyl, and John von Neumann. According to Sylvia Nasar, "Princeton in 1948 was to mathematicians what Paris once was to painters and novelists, Vienna to psychoanalysts and architects, and ancient Athens to philosophers and playwrights." In 1949, Nash was awarded an Atomic Energy Commission fellowship to continue his doctoral studies at Princeton. The school's faculty and students admired Nash for his obvious intellect, but his academic career remained undistinguished.
While at Princeton, Nash invented two board games. The first, called "Nash" or "John," was a twoperson, zerosum game, meaning that one player's advantage must result in a proportional disadvantage for the opponent. Unlike other zerosum games such as chess and tictactoe, however, a tie or draw was impossible in Nash's game. The game had been invented independently from Nash and eventually was marketed in the 1950s as Hex. Nash also collaborated with several students to create the game "So Long, Sucker," a multipleplayer game that rewarded the player most skilled at deception.
Battled Mental Illness
After graduating from Princeton, Nash taught mathematics at the Massachusetts Institute of Technology in Cambridge. He had a son with Eleanor Stier before marrying Alicia Larde in 1957, with whom he also fathered a son. Along with teaching at MIT, Nash worked at the RAND Corporation think tank in Santa Monica, California. Nash was fired in 1954 after being arrested for indecent exposure in a public restroom during a Santa Monica police sting against homosexuals. Nasar wrote: "The biggest shock to Nash may not have been the arrest itself, but the subsequent expulsion from RAND." In 1957, he divided his time between the Institute for Advanced Study and the Courant Institute of Mathematical Sciences at New York University.
In early 1959, Nash began exhibiting symptoms of paranoid schizophrenia. After losing his ability to teach and do research, he underwent insulin coma therapy during several stays in psychiatric hospitals, including one where he shared a room with poet Robert Lowell. When not institutionalized, he made several trips to Europe, where he attempted to establish a world government and resign his United States citizenship because he was convinced he was a political prisoner. He also declared himself the emperor of Antarctica and tried to establish a defense fund for what he believed was an impending extraterrestrial attack.
In 1962, Alicia Nash filed for divorce, and Nash lived with his widowed mother until her death in 1969. He then moved back into the house he shared with Alicia Nash. For the next 15 years, Nash spent much of his time wandering freely on the Princeton campus. In the late 1980s, however, he showed signs of remission from mental illness. He accepted the Nobel Prize for economics in 1994 and spent much of the 1990s attending to his second son's schizophrenia. He and Alicia Nash eventually remarried.
Developed Game Theory
While at RAND, Nash participated in developing new technologies, theories, and strategies for the United States military through a private nonprofit organization that employed many of the nation's most prominent intellectuals. One of the strategies that RAND was beginning to explore for modern warfare was game theory, which expressed itself in such Cold War strategies as mutual deterrence and the arms race. Whereas John von Neumann and Oskar Morgenstern had conceived of game theory as a zerosum relationship between noncooperating competitors, Nash argued that some competitors could benefit from an adversarial relationship by seeking an equilibrium point that would either minimize negative repercussions or maximize positive outcomes. Jeremy Bernstein, writing in Commentary, noted: "Part of Nash's contribution was to allow one to relax the assumptions of von Neumann's theorem; the game does not have to be zerosum or involve only two players. … What he showed was that in a very wide range of such 'games,' there must be at least one such strategy leading to equilibrium, and if there are several, one must decide among them." Assuming that all competitors behave in a rational manner, Nash hypothesized that each party would apply its dominant strategy to yield mutually beneficial results.
In 1950, Nash submitted his equilibrium theory as his Princeton doctoral thesis. It has since become widely used in military and economic strategies, as well as in biology. According to animal behaviorist Peter Hammerstein, quoted by Robert Pool in Science, "The Nash equilibrium turns out to be terribly important in biology. … Such concepts are proving vital in analyzing a range of biological data, from sex ratios to animals' decisions about whether to fight each other for territory or food." The theory earned him the Nobel Prize, shared with fellow game theorists John Harsanyi and Reinhard Selten.
Following his work in game theory, Nash focused on, among other things, manifolds. According to Nasar: "In one dimension, a manifold may be a straight line, in two dimensions a plane, or the surface of a cube, a balloon, or a doughnut." Although the object remains the same, it appears different when viewed from different perspectives. Because of their mutability, manifolds seemingly defied accurate depictions until Nash employed polynomial algebraic equations to describe them in 1950 and 1951.
In September 1951, beginning his tenure at MIT, Nash combined his work with manifolds with an interest in fluid dynamics. Nash applied the results of this research to his next mathematical theory, which asserted that it is possible to embed a Riemannian manifold in a Euclidean space. An eighteenthcentury German mathematician, G.F.B. Riemann theorized that previous Euclidean notions of geography were inaccurate, due to the curvature of the earth's surface, therefore making all parallel lines subject to intersection and the sums of any triangle's angles unequal to 180 degrees. Rather than employ geometry or algebra to solve the problem, Nash developed a new method of applying 19thcentury differential calculus. Jurgen Moser later applied the breakthrough to celestial mechanics, resulting in its eventual name: the NashMoser theorem.
Books
Nasar, Sylvia, A Beautiful Mind: The Life of Mathematical Genius and Nobel Laureate John Nash, Simon & Schuster, 1998.
Periodicals
Commentary, August 1998.
Forbes, July 3, 1995.
Science, October 21, 1994.
Time, October 24, 1994.
Washington Post, December 18, 2001. □
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Nash, John Forbes, Jr.
John Forbes Nash, Jr., 1928–2015, American mathematician, b. Bluefield, W.Va., grad. Carnegie Institute of Technology (now CarnegieMellon Univ., B.A. and M.A. 1948), Ph.D. Princeton 1950. During a fiveyear period, beginning with his doctoral thesis in 1949, he established the mathematical principles of modern game theory (see games, theory of). In four papers published between 1950–53 he made seminal contributions to game theory and to bargaining theory, which led to a revolution in economic theory. Nash also made significant contributions to mathematics, including the Nash embedding theorems and work on elliptic partial differential equations and parabolic nonlinear partial differential equations. In 1959 he began to experience what he termed
"mental disturbances"
and remained in seclusion for the next 30 years, suffering from paranoid schizophrenia, which he blamed on the mental effort expended in resolving contradictions in quantum theory. Nash returned to his academic research once the disease was in remission, and for his landmark work on the mathematics of game theory he shared the 1994 Nobel memorial economics prize with HungarianAmerican economist John Harsanyi and German mathematician and economist Reinhard Selten. In 2015 he shared the Abel Prize (in mathematics) with Louis Nirenberg for their work on nonlinear partial differential equations. Nash wrote Essays on Game Theory (1997).
See biography by S. Nasar (1998); H. W. Kuhn, ed., A Celebration of John F. Nash, Jr, (1996).
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Nash, John Forbes, Jr.
NASH, JOHN FORBES, JR.
(b. Bluefield, West Virginia, 13 June 1928)
game theory, Nash equilibrium, bargaining, differential geometry, Riemannian manifolds, nonlinear differential equations.
Nash won a Nobel Prize for formulating the idea of a Nash equilibrium and proving that such equilibria always exist in finite games. He also founded modern bargaining theory and made substantial contributions in differential geometry. His unexpected recovery from a longstanding schizophrenic illness in time to be awarded his Nobel Prize made him something of a folk hero, celebrated in both a book on his life and an Oscarwinning movie.
The highs and lows of the life of John Nash are out of the range of experience of most human beings. As an undergraduate, he initiated the modern theory of rational bargaining. His graduate thesis formulated the idea of a Nash equilibrium, which is now regarded as the basic building block of the theory of games. He went on to solve major problems in pure mathematics, using methods of such originality that his reputation as a mathematical genius of the first rank became firmly established. But at the age of thirty he fell prey to a serious schizophrenic illness. Irrational delusions precipitated a variety of selfdestructive behaviors that wrecked his career and his marriage. With only occasional remissions, his illness persisted for many years, during which time he languished in obscurity, cared for by his exwife, Alicia, in spite of everything. By the early 1990s, he was no longer delusional, although this fact was not widely appreciated. Fortunately, his recovery was brought to the attention of the Nobel committee who were deciding to whom to award prizes for game theory, which had by degrees totally transformed the face of economic theory while Nash was out of action. Their award of the 1994 Nobel Prize in Economics to Nash (along with John Harsanyi and Reinhard Selten) was instrumental in making him something of a cultural hero in his old age, celebrated in Sylvia Nasar’s bestselling biography, and in the movie, A Beautiful Mind, in which Nash as a young man is played by the appropriately goodlooking Russell Crowe. Nash himself seems to think that the mental instability that is popularly thought to accompany genius may be a price worth paying. As he says of Zoroaster, without his “madness,” he would perhaps only have been another of the faceless billions who have lived and died on this planet.
Nash Bargaining Solution Nash intended to follow in his father’s footsteps and become an engineer, but the chemical engineering courses at Carnegie Tech (now Carnegie Mellon University) did not hold his attention, and he finally registered as a mathematics major. He took only one course in economics, but he reports that this was enough to inspire the idea that is nowadays referred to as the Nash bargaining solution. The originality of this work can be measured by the fact that the tradition among economists at this time was that the bargaining problem is indeterminate unless one has psychological information about the relative “negotiating skills” of the bargainers. For example, in the case of the classic problem of DividetheDollar, in which a dollar can be split between two players if and only if they agree on who should get how much, economists felt unable to say anything at all.
John von Neumann and Oskar Morgenstern’s pathbreaking Theory of Games and Economic Behavior endorsed this position as late as 1944, arguing that nothing more can be said beyond the fact that a rational bargain will be individually rational and Pareto efficient. The former simply means that both bargainers get as much from their agreed outcome as they would from refusing to agree at all. The latter means that nothing is wasted, in the sense that no other outcome is available that both bargainers prefer. In DividetheDollar, any split of the dollar satisfies both criteria.
In 1950, Nash argued to the contrary that the problem of rational bargaining under complete information (when both bargainers’ preferences are common knowledge) is determinate—although he did not disavow a “negotiating skills” interpretation until a later paper of 1953. His approach was based on von Neumann and Morgenstern’s 1944 proof that, under plausible assumptions, a rational decision maker will act as though maximizing the expected value of a function that assigns a real number called a utility (or a payoff) to each possible outcome. If this utility function is concave, the decision maker is said to be risk averse, since he then prefers a physical mixture of half of any two outcomes to a lottery in which he gets each of these two outcome with probability one
half. With two players, any outcome can be identified with a pair (u_{1},u_{2}). Nash therefore abstracted a bargaining problem to be a pair (X, d), in which the feasible set X is a convex, compact set whose points represent all possible bargaining outcomes, and the disagreement point d is a point inside X that corresponds to the result of a disagreement. The shape of the set X and the location of the point d within X are determined by the extent to which each player is averse to taking risks.
Nash proposed a set of axioms for the rational outcome of such a bargaining problem. They admit a unique solution that is called the Nash bargaining solution of the problem. It is the point (u_{1}, u_{2}) in X at which the Nash product (u_{1}d_{1}) (u_{2}_{}d_{2}) is maximized (subject to u_{1}>d_{1} and u_{2}>d_{2}). If the players have identical attitudes to taking risks, the Nash bargaining solution of DividetheDollar corresponds to a fiftyfifty split, but if we make one player more risk averse than the other, his share of the dollar will decrease.
The following are informal variants of Nash’s axioms:
Axiom 1. The outcome is individually rational and Pareto efficient.
Axiom 2. The outcome is independent of the calibration of the bargainers’ utility scales.
Axiom 3. If the bargainers sometimes agree on the payoff pair s when t is feasible, then they never agree on t when s is feasible.
Axiom 4. In symmetric situations, both bargainers get the same.
The second axiom recognizes that the choice of an origin and a unit for a utility scale is arbitrary, as in the case of a temperature scale. The fourth axiom is less a rationality assumption than a decision to confine attention to symmetric bargaining procedures. (In its absence, the Nash product is replaced by (u_{1}d_{1})^{a}(u_{2}d_{2})^{b}, where a > 0 and b > 0 are constants whose ratio characterizes the relative bargaining power of the two bargainers in an asymmetric procedure. The resulting bargaining outcome is said to be a generalized or asymmetric Nash bargaining solution.) The third axiom, which compares rational agreements in different bargaining problems, is an informal version of a principle called the Independence of Irrelevant Alternatives. For example, a committee of the prestigious Econometric Society was deciding which of A, B, or C to invite to give a fancy lecture. B was quickly eliminated, but it took a long time to agree that the invitation should go to A rather than C. Someone then pointed out that B could not make the event anyway. This observation provoked a renewal of the debate that ended up with the invitation going to C. This is a violation of the Independence of Irrelevant Alternatives, which says that the choice between A and C should be independent of the availability of B, who is an “irrelevant alternative” because he will not be chosen even if available.
In 1953, Nash extended his result to the case when a predetermined disagreement point d is not given, but each player has a number of strategies that might be used in the event of a disagreement. If each player can make an irrevocable threat to use some mixture of these strategies if the bargaining breaks down, Nash showed that the situation reduces to a game that can be solved by an appeal to von Neumann’s minimax principle. This result has limited application in practice because of the difficulty of making threats that are genuinely irrevocable.
Nash Program Nash was born in 1928, the same year in which John von Neumann created the subject of game theory by proving his minimax theorem. Not much notice was taken of this major creative step until von Neumann and Morgenstern published The Theory of Games and Economic Behavior in 1944. This book is divided into two very distinct parts, which are nowadays regarded as the origins of noncooperative and cooperative game theory respectively. In the noncooperative half of the book, the authors offered a general formulation of a game and analyzed the case of twoperson, zerosum games—the case to which the minimax theorem applies—in detail. In the cooperative half of the book, they observed that when the players in a game can sign binding preplay agreements that govern their future behavior, then the detailed strategic structure of the game becomes irrelevant. They then exploited this insight to study the problem of coalition formation in games with many players, but their results are nowadays often thought only to be fully applicable in zerosum cooperative games.
Nash’s 1950 axiomatic characterization of his bargaining solution was received by those who took note of it as a new approach to cooperative game theory. His axiomatic methodology became the standard tool in this area among the small school of mathematicians who followed up his ideas in the 1960s. However, Nash’s alternative defense of his bargaining solution was largely overlooked until considerably later.
His alternative defense consisted of a brief analysis of an explicit noncooperative bargaining model. In this Nash Demand Game, each of the two players simultaneously makes a binding commitment to a takeitorleaveit utility demand. If the pair of demands lies in the feasible set X of the bargaining problem, then each player receives his demand. Otherwise, each player receives his payoff at the disagreement point d. Anticipating his 1951 paper, Nash observed that any Paretoefficient, individually rational outcome of the bargaining problem corresponds to a Nash equilibrium. He was therefore faced with an equilibrium selection problem. Which of this infinite class of Nash equilibria should be regarded as the solution of the game? To deal with this problem, Nash introduced an element of doubt about the precise nature of the feasible set into his model. He replaced X by a smooth probability density function that differs from 1 or 0 only in a small band containing the frontier of X. Under mild conditions, the Nash equilibria of this smoothed Nash Demand Game are then all close to the Nash bargaining solution of the original bargaining problem.
A typically laconic sentence in Nash’s 1951 paper on Nash equilibria proposes using the study of such noncooperative negotiation models more generally. In consequence, the idea that the range of applicability of cooperative solution concepts should be explored by investigating the type of noncooperative negotiation models that implement them has become known as the Nash program. A big success in this program came in 1982, when Ariel Rubinstein showed that bargaining models in which the players can exchange demands forever until an agreement is reached have a unique subgameperfect equilibrium, provided that the players both discount the unproductive passage of time at a positive rate. (Selten defines a subgameperfect equilibrium to be a pair of strategies that is not only a Nash equilibrium in the whole game, but also induces Nash equilibria in all subgames, whether reached in equilibrium or not.) When the interval between successive demands in Rubinstein’s model approaches zero, it turns out that the unique subgameperfect outcome converges on a generalized Nash bargaining solution in which the bargaining powers a and b are the reciprocals of the respective rates at which the two players discount time (Binmore, 1987). Impatience therefore joins risk aversion as a characteristic that inhibits bargaining success. This result is commonly thought to represent a striking vindication of both the Nash bargaining solution and the Nash program in general.
Nash Equilibrium After completing his undergraduate degree, Nash received offers of fellowships from both Harvard and Princeton. It was fortunate that he chose to go to the mathematics department at Princeton after receiving an encouraging letter from Albert Tucker, who became his thesis advisor. His fellow students, notably John Milnor and Lloyd Shapley, were a brilliant group that flourished in the hothouse atmosphere that followed the mass emigration of European mathematicians from oppression in their own countries. Harold Kuhn remained a loyal friend through Nash’s long illness, and was later to prove instrumental in bringing Nash’s recovery to the attention of the Nobel committee.
Nash’s short thesis began by defining the notion of a Nash equilibrium for a noncooperative game. An nplayer game can be idealized as a bundle of strategy sets and a payoff function. Each player independently chooses a strategy from his or her strategy set. The payoff function then maps the resulting strategy profile to a vector of real numbers that specifies who gets what payoff when the strategy profile is used in the game. A Nash equilibrium is a profile of strategies, one for each player, in which each player’s strategy is a best reply to the strategies chosen by the other players. Nash went on to show that all finite games have at least one Nash equilibrium if mixed strategies are allowed. (Even before von Neumann, Émile Borel had formulated the notion of a pure strategy as a plan of action for a player that specifies his behavior under all possible contingencies in a game. He also drew attention to the importance of mixed strategies, in which a player selects a pure strategy using a carefully chosen random device. Mixed strategies become relevant when it is important to keep your opponent guessing.) It is for this work that Nash was awarded a Nobel Prize in 1994.
There are two factors that make Nash equilibria important in game theory. The first depends on the notion of a rational solution of a game. A book that offers advice on how to play a game when it is common knowledge that all the players are rational would need to recommend the play of a Nash equilibrium in order to be authoritative. If it recommended the play of a strategy profile that is not a Nash equilibrium, then at least one player would elect not to follow the book’s advice if he believed that the other players would. The book would then fail to be authoritative.
The second reason that Nash equilibria are important is evolutionary. If players are repeatedly drawn at random from a very large population to play a particular game, then the strategies that they are planning to use will vary over time if they keep adjusting their behavior in the direction of a better reply to whatever is currently being played in the population at large. Such an adjustment process can only cease to operate when the population reaches a Nash equilibrium. (With this interpretation, a mixed equilibrium can be realized as a polymorphic equilibrium, in which different players in the population all plan to play a pure strategy, but these pure strategies need not be the same.)
Nash referred to the second interpretation as “mass action” in his thesis, but the editors of Econometrica, where his thesis was published in 1951, asked for this section to be removed. However, it is nowadays generally acknowledged that it is the evolutionary interpretation that explains the very considerable predictive power of Nash equilibrium for economic data obtained in laboratory experiments with experienced human subjects who are sufficiently well paid. For similar reasons, Nash equilibrium is also important in explaining biological data. (An evolutionarily stable strategy [ESS], as introduced by John Maynard Smith and George Price [1972], is simply a refinement of a symmetric Nash equilibrium.) The title of Richard Dawkins’s Selfish Gene explains the biological success of the idea of a Nash equilibrium in a nutshell. One can use the rational interpretation of a Nash equilibrium to predict the outcome of an evolutionary process, without needing to follow each enormously complicated twist and turn that the process might take.
The reason why some papers prove to be culturally pivotal is a matter for historians of science. The idea of an equilibrium is certainly not original to Nash’s 1951 paper. It is implicit in David Hume’s famous Treatise on Human Nature of 1739, and explicit in Augustin Cournot’s 1838 work on the market games played between two rival manufacturers (for which reason a Nash equilibrium is sometimes called a CournotNash equilibrium). Von Neumann was also aware that the strategy profiles that satisfy his minimax principle for twoperson, zerosum games are necessarily Nash equilibria for this special class of games. Nor was Nash’s use of a fixedpoint theorem to prove his existence theorem unprecedented. Von Neumann had made a similar use of the Brouwer fixedpoint theorem to prove his minimax theorem. Shizuo Kakutani was moved to prove his generalization of the Brouwer theorem after hearing von Neumann lecture on the subject. However, what cannot be contested is that it was Nash’s work that eventually converted the economics profession to game theory, albeit after a gestation period of more than a quarter of a century. Nowadays, the idea of a Nash equilibrium is regarded as the basic tool of microeconomic theory, and all its recent successes, notably the design of bigmoney auctions in the telecom industry and elsewhere, can be traced back to Nash’s 1951 paper.
Nash was not shy about taking his ideas to the big names in the academic world. He famously proposed a scheme for reinterpreting quantum physics to Albert Einstein, who responded by suggesting that he first learn some physics. It is unfortunate that Nash got similar treatment from John von Neumann, when he showed him his existence theorem. Von Neumann apparently dismissively observed that he saw how the result could be proved using a fixedpoint theorem.
Why did von Neumann not see the significance of Nash’s theorem? One possibility is that von Neumann recognized that the bestreply criterion is only a necessary condition for a strategy profile to count as the rational solution of a game, but that the equilibrium selection problem would need to be solved—as von Neumann had implicitly solved it for twoperson, zerosum games— before one could claim to have a sufficient condition for the rational solution of a general game. Perhaps von Neumann would have taken more interest if he had considered the evolutionary implications of Nash equilibria, or if he had been aware of Nash’s application (with Lloyd Shapley) of the idea to threeplayer poker models. However, uninformed enthusiasts do Nash no favors when they make von Neumann’s atypical lack of insight on this occasion a reason for belittling von Neumann’s own achievements in game theory. It is similarly no criticism of Isaac Newton that he stood on the shoulders of giants.
Nash Embedding After completing what eventually turned out to be one of the most successful theses ever written, Nash spent time on and off at RAND in Santa Monica, California. RAND is a private foundation set up at the onset of the Cold War with the Soviet Union, for the purpose of maintaining the input from scientists and mathematicians that had proved very valuable at some pivotal points in World War II. A mythology has grown up that attributes an absurdly unrealistic influence on political and military strategy to game theorists at this time, especially those associated with RAND, but Nash himself seems to have contributed nothing of military value at all. He therefore deserves none of the coals of fire heaped, for example, on the head of von Neumann— supposedly the inspiration for title character in the movie Dr. Strangelove—for being thought to have created game theory for evil purposes.
Although Nash had such a large impact on economic theory, he never thought of becoming an economist. He was anxious to make his mark as a creative mathematician. The idea that first brought him the kind of recognition he was seeking was that the apparently very general shapes to which mathematicians refer when speaking of manifolds are fundamentally no more general than the shapes determined by polynomial equations, provided that one operates in a Euclidean space of high enough dimension. Even to propose such a conjecture was thought to be a wild venture by mathematicians of the time. In spite of such skepticism, Nash turned down the offer of a permanent position at RAND in order to return to Princeton in 1950, where he worked on his idea with Donald Spencer, who proved to be an invaluable aid to Nash as he sought to put his intuitions into the form of an acceptable mathematical proof—a task that he always found difficult. Nash published the completed paper under the title “Real Algebraic Manifolds” in 1952. Its second theorem asserts that:
Theorem. A closed differential manifold always has a proper algebraic representation in the Euclidean space of one more than twice its number of dimensions.
Certain problems in differential geometry can therefore be reduced to counting the number of solutions to polynomial equations (Artin and Mazur).
Nash was disappointed at not being offered a position at Princeton, but accepted the offer he received from the Massachusetts Institute of Technology (MIT), to which he relocated in 1951. Although the Mathematics Department at MIT boasted Norbert Wiener, it had not yet acquired the prestige it currently enjoys, and Nash was one of a number of young men hired with the deliberate intention of putting the department on the map. The atmosphere seems to have been almost absurdly competitive, and Nash’s endeavors to assert his superiority in this new environment made him popular neither with his colleagues nor his students, who doubtless felt that he should put up or shut up. However, it is hard to believe that Nash really proved one of the major mathematical theorems of the twentieth century in response to a testy challenge from a colleague, as he joked when first presenting the work in 1955. His earlier work would naturally have focused his attention on the problem that had been considered by Georg Friedrich Bernhard Riemann long before.
The question is whether the abstract shapes called Riemannian manifolds are really as abstract as they were thought to be. Nash argued that they are really nothing more than submanifolds of an ordinary flat space, but a proof of this claim would need to show how to construct a sufficiently smooth embedding of any given Riemannian manifold in a Euclidean space of sufficiently high dimension. Nash astonished the mathematical community by describing such a construction in his paper “The Imbedding Problem for Riemannian Manifolds,” which was published in the Annals of Mathematics in 1956. Its final theorem asserts that:
Theorem Any Riemannian nmanifold with C^{k} positive metric, where k > 2, has a C^{k} isometric embedding in a Euclidean space of (3n^{3}+14n^{2}+11n)/2 dimensions.
The proof incorporates a result that J. Schwartz refers to in his influential Nonlinear Functional Analysis as the “hard” implicit function theorem. This theorem applies, for example, to functions from one Banach space to another, even when their Gateaux derivatives may be unbounded as linear operators and have an unbounded linear inverses. (See also Moser [1961] and Lang [1962].)
This work opened a window on the properties of nonlinear partial differential equations, which subject Nash pursued in 1956 while ostensibly on leave at the Advanced Institute at Princeton, but spending much of his time at the Courant Institute in New York. He returned to MIT the next year with novel results on local existence, uniqueness, and continuity. In spite of previous frictions, his MIT colleagues were generous in the help they gave Nash in putting his ideas into a publishable form. The paper appeared in 1958 with the title “Continuity of Solutions of Parabolic and Elliptic Equations.” However, Nash was disappointed to find that some of the results of this paper had been anticipated by Ennio de Giorgi, who had been working independently on similar problems. This coincidence, which he still feels may have lost him his chance at a Fields Medal, together with the failure of his next project—which was a wildly ambitious attempt to rewrite the foundations of quantum theory— is thought to have been partly instrumental in precipitating the breakdown that followed.
Breakdown Nash was a bookish loner as a boy. He became more obviously eccentric when he began to mix in academic circles, the subject of comment even by colleagues whose own behavior would be regarded as decidedly odd by normal standards. His adolescence seems to have been delayed or extended, so that he remained sexually ambivalent, fiercely competitive, and overly anxious to impress into his late twenties. His schizophrenia was presaged by his apparent unawareness of his responsibilities, notably toward his students at MIT and the illegitimate child he fathered in 1953. The bright spot in his personal life came with his marriage to Alicia, whom he married in 1957. However, Nash’s life began to fall apart in 1958, when he reports that the same intuition that had
served him so well in solving mathematical problems began to feed him delusions that led to his becoming increasingly dysfunctional. Nothing is gained by itemizing the selfdestructive behaviors that led to his being forcibly hospitalized on several occasions. With occasional remissions, matters continued in this way until 1970, when Alicia—whom he had divorced for her part in his hospitalizations—took him in to prevent his becoming homeless. He then famously survived as a phantom haunting the Princeton campus, engaged in arcane research comprehensible to nobody but himself. Only in the early 1990s did his old Princeton acquaintances begin to notice signs of a recovery that is apparently unusual in such serious cases as his. Nash believes that he eventually learned to distinguish between his rational and irrational intuitions using the power of his intellect. To the extent that this is true, there is therefore a message of hope for those similarly afflicted.
Nobel Laureate In 1993, a symposium on game theory was held of the kind that the Nobel committee for economics sometimes use to help them decide to whom to award a prize in a particular area. Most of the game theorists who attended assumed that Nash’s illness ruled him out as a candidate. (It is not clear why the fact that someone is thought to be ill should disbar him from an academic honor, but it had even proved difficult to get Nash nominated as a fellow of the Econometric Society some years earlier, although the vote when taken was overwhelmingly favorable.) However, word of Nash’s recovery got to the committee in time for it to be possible for him to be awarded the 1994 Nobel Prize along with John Harsanyi and Reinhard Selten, who developed his ideas in the context of games with incomplete information and games with a dynamic structure. (Reports of dissent in the committee and resentment elsewhere would seem to overdramatize the actual events, although it is perhaps a pity that the Nobel citation should have left his work on bargaining unmentioned.)
John and Alicia Nash are now remarried, and take pleasure in the modicum of fame that the book and movie about their lives has brought them. Nash’s switchback career therefore ends on an upbeat note, although few would agree with his own assessment that his intellectual achievements were an adequate compensation for all the accompanying pain and suffering.
BIBLIOGRAPHY
WORKS BY NASH
“The Bargaining Problem.” Econometrica 18 (1950): 155–162.
With Lloyd Shapley. “A Simple ThreePerson Poker Game.” In Contributions to the Theory of Games, edited by Harold Kuhn and Albert Tucker. Annals of Mathematics Studies, no. 24. Princeton, NJ: Princeton University Press, 1950. Reprinted in Essays on Game Theory.
“NonCooperative Games.” Annals of Mathematics 54 (1951):
286–295. “Real Algebraic Manifolds.” Annals of Mathematics 56 (1952):
405–421. “TwoPerson Cooperative Games.” Econometrica 21 (1953): 128–140.
“The Imbedding Problem for Riemannian Manifolds.” Annals of Mathematics 63 (1956): 20–63.
“Continuity of Solutions of Parabolic and Elliptic Equations.”
American Journal of Mathematics 80 (1958): 931–954.
Essays on Game Theory. Cheltenham, U.K.: Edward Elgar, 1996.
“Autobiography.” In The Essential John Nash, edited by Harold Kuhn and Sylvia Nasar. Princeton, NJ: Princeton University Press, 2002.
OTHER SOURCES
Artin, M., and B. Mazur. “On Periodic Points.” Annals of Mathematics 81 (1965): 82–99.
Binmore, Ken. “Nash Bargaining Theory II.” In Economics of Bargaining, edited by Ken Binmore and P. Dasgupta. Cambridge, U.K.: Cambridge University Press, 1987.
Cournot, A. Researches into the Mathematical Principles of the Theory of Wealth. London: Macmillan, 1929. First published 1838.
Dawkins, Richard. The Selfish Gene. Oxford: Oxford University Press, 1976.
Hume, David. A Treatise on Human Nature, edited by L. A.
SelbyBigge. Revised by P. Nidditch. Oxford: Clarendon Press, 1978. First published 1739.
Kuhn, Harold, and Sylvia Nasar, eds. The Essential John Nash.
Princeton, NJ: Princeton University Press, 2002.
Lang, S. Introduction to Differentiable Manifolds. New York: Wiley, 1962.
Maynard Smith, John, and G. Price, “Logic of Animal Conflict.” Nature 246 (1972): 15–18.
Milnor, J. Differentiable Topology. Princeton, NJ: Princeton University Press, 1958.
_____. “The Game of Hex.” In The Essential John Nash, edited by Harold Kuhn and Sylvia Nasar. Princeton, NJ: Princeton University Press, 2002.
Moser, Jurgen. “A New Technique for the Construction of Solutions of Nonlinear Differential Equations.” Proceedings of the National Academy of Sciences of the United States of America 47 (1961): 1824–1831.
Nasar, Sylvia. A Beautiful Mind. New York: Faber and Faber, 1998.
Rubinstein, Ariel. “Perfect Equilibrium in a Bargaining Model.”
Econometrica 50 (1982): 97–109.
Schwartz, J. Nonlinear Functional Analysis. London: Gordon and Breach, 1969.
Selten, Reinhard. “Reexamination of the Perfectness Concept for Equilibrium Points in ExtensiveGames.” International Journal of Game Theory 4 (1975): 25–55.
Shapley, Lloyd. “A Value for nPerson Games.” In Contributions to the Theory of Games, vol. 2, edited by Harold Kuhn and Albert Tucker. Annals of Mathematics Studies, no. 28. Princeton, NJ: Princeton University Press, 1953.
von Neumann, John. “Zur Theorie der Gesellschaftsspiele.”
Mathematische Annalen 100 (1928): 295–320.
von Neumann, John, and Oskar Morgenstern. The Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press, 1944.
Ken Binmore
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