milnor.qxp 9/9/98 4:07 PM Page 1329 John Nash and “A Beautiful Mind” John Milnor ohn Forbes Nash Jr. published his first paper earth material far beyond what one might expect. with his father at age seventeen. His thesis, She gives detailed descriptions of the delibera- at age twenty-one, presented clear and ele- tions, not only for the 1958 Fields Medals, where mentary mathematical ideas that inaugu- Nash had been one possible candidate, but even Jrated a slow revolution in fields as diverse for the 1994 Nobel Prize in Economics—delibera- as economics, political science, and evolutionary tions that were so explosive that they led to a rad- biology. During the following nine years, in an ical restructuring of the prize and a complete amazing surge of mathematical activity, he sought change in the nominating committee. In general her out and often solved the toughest and most im- sources are carefully identified, but in these par- portant problems he could find in geometry and ticular cases they remain anonymous. analysis. Then a mental breakdown led to thirty lost Although Nasar’s training was in economics and painful years, punctuated by intermittent hos- rather than mathematics, she is able to provide pitalization, as well as occasional remission. How- background, rough descriptions, and precise ref- ever, in the past ten years a pronounced reawak- erences for all of Nash’s major work. Also, she ening and return to mathematics has taken place. gives a great deal of background description of the Meanwhile, the importance of Nash’s work has places and persons who played a role in his life. been recognized by many honors: the von Neu- (Mathematical statements and proper names are mann Prize, fellowship in the Econometric Society sometimes a bit garbled, but the astute reader can and the American Academy of Arts and Sciences, usually figure out what is meant.) Thus we find fas- membership in the U.S. National Academy of Sci- cinating information about the history of Carnegie ences, culminating in a Nobel Prize. Tech, Princeton, the Rand Corporation, MIT, the In- stitute for Advanced Study, and the Courant In- A Beautiful Mind stitute, and also information about many well- Sylvia Nasar’s biography, A Beautiful Mind,1 tells known and not so well-known mathematical this story in carefully documented detail, based on personalities. The discussion leads into many in- hundreds of interviews with friends, family, ac- teresting byways: her description of MIT is inter- quaintances, and colleagues, as well as a study of woven with a discussion of the McCarthy era, while available documents. Indeed, she is a highly tal- her description of the Rand Corporation and of von ented interviewer and in some cases seems to un- Neumann leads to a discussion of the relation of game theory to cold war politics. (Von Neumann, John Milnor is director of the Institute for Mathematical who advocated a preemptive strike against the So- Sciences at the State University of New York, Stony Brook. viet Union, may have been the original model for His e-mail address is [email protected]. Kubrick’s Dr. Strangelove.) 1Sylvia Nasar, A beautiful mind: A biography of John Any discussion of Nasar’s book must point out Forbes Nash Jr., Simon & Schuster, 1998, $25.00 hard- a central ethical dilemma: This is an unauthorized cover, 459 pages, ISBN 0684819066. (See also [Nas].) biography, written without its subject’s consent or NOVEMBER 1998 NOTICES OF THE AMS 1329 milnor.qxp 9/9/98 4:07 PM Page 1330 cooperation. However, when mathematics is applied to other Nash’s math- branches of human knowledge, we must really ask ematical activity a quite different question: To what extent does the was accompa- new work increase our understanding of the real nied by a tangled world? On this basis, Nash’s thesis was nothing personal life, short of revolutionary. (Compare [N21], as well as which Nasar de- [U].) The field of game theory was the creation of scribes in great John von Neumann and was written up in collab- detail. This ma- oration with Morgenstern. (One much earlier paper terial is certainly had been written by Zermelo.) The von Neumann- of interest to a Morgenstern theory of zero-sum two-person games wide audience. was extremely satisfactory and certainly had ap- (Oliver Sacks, plication to warfare, as was amply noted by the mil- quoted in the itary. However, it had few other applications. Their publisher’s efforts to develop a theory of n-person or non-zero- blurb, writes that sum games for use in economic theory were really the book is “ex- not very useful. (Both Nash and the reviewer par- traordinarily ticipated in one experimental study of n-person moving, remark- games [N10]. As far as I know, no such study has able for its sym- ever been able to detect much correlation between pathetic insights von Neumann-Morgenstern “solutions” and the Photo by Robert P. Matthews, courtesy of Communications Dept., Princeton University. into both genius real world.) John Forbes Nash Jr., 1994. and schizophre- Nash in his thesis was the first to emphasize the nia”.) Inevitably, distinction between cooperative games, as studied however, the publication of such material involves by von Neumann and Morgenstern (roughly speak- a drastic violation of the privacy of its subject. ing, these are games where the participants can sit The book is dedicated to Alicia Nash, first his around a smoke-filled room and negotiate with wife and later his steadfast companion, whose each other), and the more fundamental noncoop- support through impossible difficulties has clearly erative games, where there is no such negotiation. played a major role in his recovery. In fact, the cooperative case can usually be re- Nash’s Scientific Work duced to the noncooperative case by incorporat- ing the possible forms of cooperation into the for- Pure mathematicians tend to judge any work in the mal structure of the game. Nash made a start on mathematical sciences on the basis of its math- the cooperative theory with his paper [N5] on the ematical depth and the extent to which it intro- Bargaining Problem, to some extent conceived duces new mathematical ideas and methods, or while he was still an undergraduate. (A related, solves long-standing problems. Seen in this way, much earlier study is due to Zeuthen.) As one re- Nash’s prize work is an ingenious but not sur- mark in this paper, Nash conjectured that every co- prising application of well-known methods, while operative game should have a value which ex- his subsequent mathematical work is far more presses “the utility to each player of the rich and important. During the following years he proved that every smooth compact manifold can opportunity to engage in the game.” Such a value be realized as a sheet of a real algebraic variety,2 was constructed by Shapley a few years later. proved the highly anti-intuitive C1-isometric em- However, the major contribution, which led to bedding theorem, introduced powerful and radi- his Nobel Prize, was to the noncooperative theory. Nash introduced the fundamental concept of equi- cally new tools to prove the far more difficult C∞- isometric embedding theorem in high dimensions, librium point: a collection of strategies by the var- and made a strong start on fundamental existence, ious players such that no one player can improve uniqueness, and continuity theorems for partial dif- his outcome by changing only his own strategy. ferential equations. (Compare [K1] and [M] for (Something very much like this concept had been some further discussion of these results.) introduced by Cournot more than a hundred years earlier.) By a clever application of the Brouwer Fixed Point Theorem, he showed that at least one equilibrium point always exists. (For more detailed 2 Artin and Mazur used this work [N7] to prove the im- accounts, see [OR], [M].) portant result that every smooth self-map of a compact Over the years the developments from Nash’s manifold can be approximated by one for which the num- seemingly simple idea have led to fundamental ber of periodic points of period p is less than some expo- nential function of p. For more than thirty years, no other changes in economics and political science. Nasar proof was known. However, Kaloshin has recently given illustrates the dollars and cents impact of game- a much more elementary argument, based on the Weier- theoretic ideas by describing “The Greatest Auc- strass Approximation Theorem. tion Ever” in 1994, when the U.S. government sold 1330 NOTICES OF THE AMS VOLUME 45, NUMBER 10 milnor.qxp 9/9/98 4:07 PM Page 1331 Publications by John Nash [N12] ———, Results on continuation and uniqueness of fluid flow, Bull. Amer. Math. Soc. 60 (1954), 165–166. [N13] , A path space and the Stiefel-Whitney classes, Proc. [N1] J. F. NASH JR. (with J. F. NASH SR.), Sag and tension calcula- ——— tions for wire spans using catenary formulas, Elect. Engrg. Nat. Acad. Sci. USA 41 (1955), 320–321. (1945). [N14] ———, The imbedding problem for Riemannian manifolds, Ann. Math. 63 (1956), 20–63. (See also Bull. Amer. Math. [N2] J. F. NASH JR., Equilibrium points in n-person games, Proc. Nat. Acad. Sci. USA 36 (1950), 48–49. (Also in [K2].) Soc. 60 (1954), 480.) [N15] , Parabolic equations, Proc. Nat. Acad. Sci. USA 43 [N3] ———, Non-cooperative games, Thesis, Princeton University, ——— May 1950. (1957), 754–758. [N16] , Continuity of solutions of parabolic and elliptic [N4] J. F. NASH JR. (with L. S. SHAPLEY), A simple three-person poker ——— game, Contributions to the theory of games, Ann.