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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

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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

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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. of Math. equations, Amer. J. Math. 80 (1958), 931–954. Stud. 24, Princeton Univ. Press, 1950, pp. 105-116. [N17] ———, Le problème de Cauchy pour les équations dif- férentielles d’un fluide général, Bull. Soc. Math. France 90 [N5] ———, The bargaining problem, Econometrica 18 (1950), 155–162. (Also in [K2].) (1962), 487–497. [N18] , Analyticity of the solutions of implicit function prob- [N6] ———, Non-cooperative games, Ann. Math. 54 (1951), ——— 286–295. (Also in [K2].) lems with analytic data, Ann. Math. 84 (1966), 345–355. [N19] , Autobiographical essay, Les Prix Nobel 1994, Stock- [N7] ———, Real algebraic manifolds, Ann. Math. 56 (1952), ——— 405–421. (See also Proc. Internat. Congr. Math., 1950, holm: Norsteds Tryckeri, 1995. (AMS, 1952), pp. 516–517.) [N20] ———, Arc structure of singularities, Duke J. Math. 81 (1995), 31–38. (Written in 1966) [N8] J. F. NASH JR. (with J. P. MAYBERRY and M. SHUBIK), A com- parison of treatments of a duopoly situation, Econometrica [N21] J. F. NASH JR. (with H. KUHN, J. HARSANYI, R. SELTEN, 21 (1953), 141–154. J. WEIBULL, E. VAN DAMME, and P. HAMMERSTEIN), The work of John F. Nash Jr. in game theory, Duke J. Math 81 (1995), [N9] ———, Two-person cooperative games, Econometrica 21 (1953), 128–140. 1–29. [N10] J. F. NASH JR. (with C. KALISCH, J. MILNOR, and E. NERING), Some In addition, there were also a number of Rand Corporation experimental n-person games, Decision Processes, (Thrall, memoranda written by Nash on diverse subjects such as ma- Coombs and Davis, eds.), Wiley, 1954, pp. 301-327. chine memories and parallel control (Nasar, pp. 403, 407, 411, 1 [N11] ———, C -isometric imbeddings, Ann. Math. 60 (1954), 436), as well as an unpublished lecture at the World Congress 383–396. (See also Bull. Amer. Math. Soc. 60 (1954), 157.) of Psychiatry in Madrid in 1996.

off large portions of the electromagnetic spec- Despite Nash’s remarks in his thesis trum to commercial users. A multiple-round pro- about a possible evolutionary3 inter- cedure was carefully designed by experts in the pretation of the idea of a Nash equilib- game theory of auctions to maximize both the rium, attention at the time was focused payoff to the government and the utility of the pur- almost entirely on its interpretation as chased wavelengths to the respective buyers. The the only viable outcome of careful rea- result was highly successful, bringing more than soning by ideally rational players. … $10 billion to the government while guaranteeing Fortunately … Maynard Smith’s book an efficient allocation of resources. By way of con- Evolution and the Theory of Games di- trast, a similar auction in New Zealand, without rected game theorists’ attention away such a careful game-theoretic design, was a disaster from their increasingly elaborate defi- in which the government realized only about 15 nitions of rationality. After all, insects percent of its expected earnings and the wave- can hardly be said to think at all, and lengths were not efficiently distributed. (In one so rationality cannot be so crucial if case, a New Zealand student bought a television game theory somehow manages to pre- station license for one dollar!) dict their behavior under appropriate One totally unexpected triumph of equilibrium conditions. Simultaneously the advent theory has been its application to population ge- of experimental economics brought netics and evolutionary biology. Based on the pi- home the fact that human subjects are oneering work of Maynard Smith, game-theoretic no great shakes at thinking either. When ideas are now applied to the competition between they find their way to an equilibrium of different species or within a species. (Compare a game, they typically do so using trial- [MS], [HS], [W]. A more precise form of this theory, and-error methods. popularized by Dawkins [D1], holds that the com- 3 petition is rather between individual genes.) There Here Binmore does not refer to biological evolution, but rather to a dynamic process in which repeated plays of has also been an interesting reverse flow of ideas, a game converge to an equilibrium. Unfortunately, this from evolution back to economics. According to discussion from Nash’s thesis does not appear in his pub- Binmore (in [W]): lished work.

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In all of the applications one very important corol- [D1] R. DAWKINS, The Selfish Gene, Oxford Univ. Press, lary must be emphasized: Although equilibrium 1976. theory, as developed by Nash and his successors, [D2] ———, The Extended Phenotype, Oxford Univ. Press, seems to provide the best-known description of what 1982. [D3] , The Blind Watchmaker, Norton, 1986. is likely to happen in a competitive situation, an equi- ——— [D4] ———, River out of Eden, BasicBooks, Harper Collins, librium is not necessarily a good outcome for any- 1995. one. In contrast to the classical economic theory [DK] R. DAWKINS and J. R. KREBS, Arms races between and of Adam Smith, where free competition leads to within species, Proc. Royal Soc. London B 205 (1979), best-possible results, and in contrast to classical 489–511. Darwinian theory, where natural selection always [G1] S. J. GOULD, Full House, Harmony Books, 1996. leads to improvement in the species,4 the actual [G2] ———, Darwinian fundamentalism, New York Review dynamics of unregulated competition can be dis- of Books, June 12, 1997, 34–37. astrous. We all know that political conflict between [HS] P. HAMMERSTEIN and R. SELTEN, Game theory and evo- nations can lead to an arms race, which is bad for lutionary biology, Handbook of Game Theory with Economic Applications, vol. 2, (Aumann and Hart, everyone concerned, and in extreme cases can lead eds.), Elsevier, 1994, pp. 929-993. to totally unnecessary war. Similarly, in evolu- [Ka] V. KALOSHIN, Generic diffeomorphisms with superex- tionary theory an arms race within a species or be- ponential growth of numbers of periodic units, Prince- tween competing species over geological periods ton Univ., in preparation. of time can be extremely detrimental.5 Indeed, it [K1] H. KUHN, Introduction to “A celebration of John F. Nash seems perfectly conceivable that natural selection Jr.”, Duke J. Math 81 (1995), dedicated to Nash. may sometimes lead to a dead end and eventual [K2] ——— (ed.), Classics in game theory, Princeton Univ. extinction. Here is a mildly exaggerated version of Press, 1997. an example which goes back to Darwin. (Compare [MS] J. MAYNARD SMITH, Evolution and the theory of games, [D3], [D4].) Suppose that an amorous peahen will Cambridge Univ. Press, 1982. [M] J. MILNOR, A Nobel prize for John Nash,6 Math. Intel- always choose the peacock with the most splen- ligencer 17 (1995), 11–17; 56. did tail. This must lead to an evolutionary arms race [Nas] S. NASAR, A beautiful mind, Vanity Fair (June 1998), during which the tails get progressively larger, 196–201, 224–230. (A highly condensed version of until the males become so clumsy that they can- the book, with color pictures). not escape from predators. [No] A. NOBILE, Some properties of the Nash blowing-up, Similar comments apply to economic theory. In Pacific J. Math. 60 (1975), 297–305. this case, one hopes that carefully chosen gov- [OR] M. OSBORNEN and A. RUBINSTEIN, A course in game the- ernment regulation can modulate the negative ef- ory, MIT Press, 1994. fects of unbridled competition and lead to a bet- [S] L. S. SHAPLEY, A value for n-person games, Contri- butions to the Theory of Games II, Ann. of Math. Stud. ter outcome for all concerned. However, the vol. 28, Princeton Univ. Press, 1953, pp. 307-317. question as to just who will do the careful choos- (Also in [K2]) ing is of course a matter of politics and leads to [U] G. UMBHAUER, John Nash, un visionnaire de l’économie. an even more complicated problem for equilib- Gaz. Math. 65 (1995), 47–69. rium theory. [vN] J. VON NEUMANN, Zur Theorie der Gesellschaftspiele, Math. Ann. 100 (1928), 295–320. References [vNM] J. VON NEUMANN and O. MORGENSTERN, Theory of [AM]M. ARTIN and B. MAZUR, On periodic points, Ann. Math. games and economic behavior, Princeton Univ. Press, 81 (1965), 82–99. 1944. [C] A. A. COURNOT, Recherches sur les principes math- [W] J. WEIBULL, Evolutionary game theory, MIT Press, ématiques de la théorie des richesses, 1838 (trans- 1995. (Introduction by K. Binmore.) lation by N. T. Bacon: Researches into the Math- [Zer]E. ZERMELO, Über eine Anwendung der Mengenlehre ematical Theory of Wealth, McMillan, (1927). auf die Theorie des Schachspiels, Proc. 5th Interna- tional Congress of Mathematicians, vol. 2, Cam- 4According to Darwin, “As natural selection works solely bridge Univ. Press, 1913, 501–504. by and for the good of each being, all corporeal and men- [Zeu] F. ZEUTHEN, Problems of monopoly and economic tal endowments will tend to progress towards perfection.” welfare, Routledge, London, 1930. However, he also expressed a contrary view. Compare the discussion of “why progress does not rule the history of life” in Gould [G1]. Unfortunately, there has been much unnecessary misunderstanding and bad feeling between those like Maynard Smith who work with theoretical mod- els for evolution and those like Gould [G2] who empha- size that the real world is more complicated than any model. 5Compare [DK], [D2]. When interpreting the phrase 6Historical correction: I claimed in [M] (quoted by Nasar, “arms race” for an evolutionary contest, remember that p. 68) that Nash’s ideas on desingularizing algebraic va- reproductive success is more important than battle rieties date back to the early 1950s. In fact, the correct prowess. The best evolutionary strategy is often to “make dates are 1963–64 (Nasar, Chapter 42). Nash’s construc- love, not war”! tion was probably first published by Nobile [No] in 1975.

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