A Beautiful Mind

by Sylvia Nasar

  12. See, for example, Norman McRae, John von Neumann, op. cit., pp. 350–56.

  13. As told to Harold Kuhn, 4.17.97.

  14. John Nash, Les Prix Nobel 1994, op. cit.

  15. Silvano Arieti, Creativity, op. cit., p. 294.

  16. J. Nash to R. Leonard, e-mail.

  17. Ibid.

  18. The conversation between Nash and Gale was recounted by Gale in an interview, 9.20.95. Gale also suggested that Nash use Kakutani’s fixed point theorem instead of Brouwer’s to simplify the proof, a suggestion that Nash followed in the note in the National Academy of Sciences Proceedings.

  19. John F. Nash, Jr., “Equilibrium Points in N-Person Games,” communicated by S. Lefschetz, 11.16.49, pp. 48–49.

  20. Gale, interview.

  21. Tucker, interview, 10.94.

  22. Gian-Carlo Rota, interview, 12.12.95.

  23. Tucker’s account of Minsky’s thesis on computers and the brain, “Neural Networks and the Brain Problem,” is given in an interview with Stephen B. Maurer published in the Two Year College Mathematics Journal vol. 14, no. 3 (June 1983).

  24. Tucker, interview.

  25. Harold Kuhn, “Nobel Seminar,” Les Prix Nobel 1994, op. cit., p. 283.

  26. Tucker, interview, 10.94.

  27. Ibid.

  28. Ibid.

  29. John Nash, Les Prix Nobel 1994, op. cit.

  30. Tucker, interview.

  31. Letter from Albert W. Tucker to Solomon Lefschetz, 5.10.50.

  32. Ibid.

  33. See, for example, introduction, John Eatwell, Murray Milgate, and Peter Newman, The New Pal-grave, op. cit.

  34. “It so happens that the concept of the two-person zero-sum games has very few real life applications,” John C. Harsanyi, “Nobel Seminar,” Les Prix Nobel 1994, op. cit., p. 285.

  35. Ibid.

  36. Nobel citation.

  37. Avinash Dixit and Barry Nalebuff, Thinking Strategically, op. cit.

  38. Ibid.

  39. “Nowadays it almost seems to be obvious that the correct application of Darwinism to problems of social interaction among animals requires the use of non-cooperative game theory,” according to Reinhard Selten, “Nobel Seminar,” Les Prix Nobel 1994, op. cit., p. 288.

  40. “Game Theory,” in Eatwell, Milgate, and Newman, op. cit., p. xiii.

  41. Michael Intriligator, personal communication, 6.27.95.

  42. Selten, op. cit., p. 297.

  43. Von Neumann, as Nash always acknowledged, nonetheless helped to gain attention for Nash’s ideas. For example, the preface to the third edition (1953) of Theory of Games and Economic Behavior directs readers to Nash’s work on noncooperative games, p. vii.

  11: Lloyd

  1. T. S. Ferguson, “Biographical Note on Lloyd Shapley,” in Stochastic Games and Related Topics in Honor of Professor L. S. Shapley, edited by T. E. S. Raghavan, T. S. Ferguson, T. Parthasarathy, and O. J. Vrieze (Boston: Kluwer Academic Publishers, 1989).

  2. See, for example, Carl Sagan, Broca’s Brain (New York: Random House, 1979).

  3. David Halberstam, The Fifties, op. cit.

  4. The description of Shapley’s experiences during the war, at Princeton, and at RAND draw on the recollections of Harold Kuhn, 11.18.96; Norman Shapiro, 2,9.96; Martin Shubik, 9.27.95 and 12.13.96; Melvin Hausner, 2.6.96; Eugenio Calabi, 3.2.96; John Danskin, 10.19.96; William Lucas, 6.27.95; Hartley Rogers, 1.26.96; John McCarthy, 2.4.96; Marvin Minsky, 2.13.96; Robert Wilson, 3.7.96; Michael Intriligator, 6.27.95.

  5. Letter from John von Neumann, 1.54.

  6. Solomon Leader, interview, 6.9.95.

  7. Rogers, interview, 1.26.96.

  8. “It was like ESP. Shapley seemed to know where all of the pieces were all of the time,” Minsky, interview.

  9. Hausner, interview, 2.6.96.

  10. Danskin, interview, 10.19.95.

  11. Letter from Lloyd Shapley to Solomon Lefschetz, 4.4.49.

  12. Interviews with Nancy Nimitz, 5.21.96, and Kuhn, 4.4.96.

  13. Shapiro, interview, 12.13.96.

  14. Intriligator, interview, 6.27.95.

  15. Shubik, interview, 12.13.96.

  16. Lloyd S. Shapley, interview, 10.94.

  17. Ibid.

  18. Shubik, interview, 12.13.96.

  19. Interviews with Shapley, Shubik, McCarthy, Calabi.

  20. Calabi, interview.

  21. Ibid.

  22. Ibid.

  23. Shubik, interview, 9.27.95.

  24. Shubik, interview, 9.27.95.

  25. Letter from Nash to Martin Shubik, undated (1950 or 1951).

  26. McCarthy, interview.

  27. McCarthy, interview.

  28. Hausner, interview, 2.6.96; M. Hausner, J. Nash, L. Shapley, and M. Shubik, “So Long Sucker — A Four-Person Game,” mimeo provided by Hausner.

  29. Interviews with Shubik and McCarthy.

  30. John Nash and Lloyd Shapley, “A Simple Three-Person Poker Game,” Annals of Mathematics, no. 24 (1950).

  31. “To some extent there was a competition between Nash, Shapley, and me,” Shubik, interview, 12.13.96.

  32. Shapley, interview.

  33. Shapley, Additive and Non-Additive Set Functions, Ph.D. thesis, Princeton University, 1953. Shapley published his famous result — the so-called Shapley value — a value for n-person games, in 1953.

  34. Martin Shubik, “Game Theory at Princeton,” op. cit., p. 6: “We all believed that a problem of importance was the characterization of the concept of threat in a two person game and the incorporation of the use of threat in determining the influence of the employment of threat in a bargaining situation. [Nash, Shapley, and I] worked on this problem, but Nash managed to formulate a good model of the two person bargain utilizing threat moves to start with.” Shubik is referring here to Nash’s “Two-Person Cooperative Games,” published in Econometrica in 1953 but actually written in August 1950 during Nash’s first summer at RAND.

  35. Letter from Albert W. Tucker, 1953

  36. Ibid.

  37. Letter from Frederick Bohnenblust, spring 1953.

  38. Letter from John von Neumann, 1.54.

  39. Kuhn, interview, 11.18.96.

  40. Shapley, interview, 10.94.

  12: The War of Wits

  1. John McDonald, “The War of Wits,” Fortune (March 1951).

  2. William Poundstone, Prisoner’s Dilemma, op. cit.; Fred Kaplan, The Wizards of Armageddon, op. cit.; The RAND Corporation: The First Fifteen Years (Santa Monica, Calif.: RAND, November 1963) and 40th Year Anniversary (Santa Monica: RAND, 1963); John D. Williams, An Address, 6.21.50; Bruce L. R. Smith, The RAND Corporation (Cambridge: Harvard University Press, 1966); Bruno W. Augenstein, A Brief History of RANDs Mathematics Department and Some of Its Accomplishments (Santa Monica, Calif.: RAND, March 1993); Alexander M. Mood, “Miscellaneous Reminiscences,” Statistical Science, vol. 5, no. 1 (1990), pp. 40–41.

  3. Herman Kahn, On Thermonuclear War (Princeton: Princeton Universih’ Press, 1960), as quoted in Poundstone, op. cit., p. 90.

  4. Isaac Asimov, Foundation (New York: Bantam Books, 1991).

  5. Poundstone, op. cit.

  6. Kaplan, op. cit., p. 52.

  7. Ibid., p. 10.

  8. Oskar Morgenstern, The Question of National Defense (New York: Random House, 1959), as quoted in Poundstone, op. cit., pp. 84–85.

  9. McDonald, “The War of Wits,” op. cit.

  10. The account of RND’s beginnings is based on Poundstone, op. cit.

  11. Ibid., p. 93.

  12. See, for example, Stanislaw Ulam, Adventures of a Mathematician, op. cit.; Richard Rhodes, The Making of the Atomic Bomb (New York: Simon & Schuster, 1986); Hodges, Alan Turing: The Enigma, op. cit.

  13. Mina Rees, “The Mathematical Sciences and World War II,” op. cit.

  14. The sketch of RAND’s mathematics, economics, and computer groups is based largely on interviews with RAND staff and consultants from the earlv Cold War period, including Kenneth Arrow, 6.26.95; Bruno Au
genstein, 6.13.96; Richard Best, 5.22.96; Bernice Brown, 5.22.96; John Danskin, 10.19.95; Martha Dresner, 5.21.96; Theodore Harris, 5.24.96; Mario Juncosa, 5.21.96 and 5.24.96; William Karush, 5.96; William F. Lucas, 6.26.95; John W. Milnor, 9.95; John McCarthy, 2.4.96; Alexander M. Mood, 5.23.96; Evar Nering, 6.18.96; Nancy Nimitz, 5.21.96; Melvin Peisakoff, 6.3.96; Harold N. Shapiro, 2.20.96; Norman Shapiro, 2.29.96; Lloyd S. Shapley, 11.94; Herbert Simon, 10.16.95; Robert Specht, 2.96; Albert W. Tucker, 12.94; Willis H. Ware, 5.24.96; Robert W. Wilson, 8.96; Charles Wolf, Jr., 5.22.96.

  15. Augenstein, interview, 6.13.96.

  16. R. Duncan Luce, interview, 1996.

  17. The descriptions of Arrow’s contributions are taken from Mark Blaug, Great Economists Since Keynes (Totowa, N.J.: Barnes & Noble, 1985), pp. 6–9.

  18. Kenneth Arrow, professor of economics, Stanford University, interview, 6.26.95.

  19. McDonald, interview.

  20. Richard Best, former manager of security, RAND Corporation, interview, 5.22.96.

  21. Interviews with Alexander M. Mood, professor of mathematics, Universih of California at Irvine, former deputy director, mathematics department, RAND Corporation, 5.23.96, and Mario L. Juncosa, mathematician, RAND, 5.21.96 and 5.24.96.

  22. Kaplan, op. cit., p. 51.

  23. Bernice Brown, retired statistician, RAND, interview, 5.22.96.

  24. Augenstein, interview.

  25. Arrow, interview.

  26. Chronicle of the Twentieth Century, op. cit., p. 667.

  27. David Halberstam, The Fifties, op. cit.

  28. Ibid.

  29. Ibid., p. 46.

  30. Kaplan, op. cit.

  31. Martha Dresner, interview.

  32. Best, interview.

  33. Halberstam, The Fifties, op. cit., p. 45; Chronicle of the Twentieth Century, op. cit., p. 677.

  34. Halberstam, op. cit., p. 49.

  35. Chronicle of the Twentieth Century, op. cit., p. 750.

  36. Best, interview.

  37. Ibid.

  38. Letter from Col. Walter Hardie, U.S. Air Force, to RAND, 10.25.50.

  39. As told to Harold Kuhn, interview, 8.97.

  40. Letter from John Nash to John and Virginia Nash, 11.10.51.

  41. Best, interview.

  42. The Eisenhower guidelines refer to DOD directive 52206, 1953 and Executive Order 10450, 1953.

  43. Danskin, interview.

  44. Robert Specht, interview, 10.96.

  45. John Williams, The Complcat Strategyst, op. cit.

  46. The account of mathematicians’ work habits is based on interviews with Brown, Mood, Juncosa, Danskin, and Shapiro.

  47. Interviews with Mood and Juncosa.

  48. Juncosa, interview.

  49. Mood, interview.

  50. The description of Williams is based on interviews with Best, Brown, Mood, and Juncosa; Poundstone, op. cit.; and Kaplan, op. cit.

  51. Mood, interview.

  52. As quoted in Poundstone, op. cit., p. 95.

  53. Mood, interview.

  54. Danskin, interview.

  55. Arrow, interview.

  56. Mood, interview.

  57. Best, interview.

  58. Harold Shapiro, interview.

  59. Mood, interview.

  60. Danskin, interview.

  61. Ibid.

  62. Best, interview.

  13: Game Theory at RAND

  1. Kenneth Arrow, interview, 6.26.95.

  2. M. Dresher and L. S. Shapley, Summary of RAND Research in the Mathematical Theory of Games (RM-293) (Santa Monica, Calif.: RAND, 7.13.49).

  3. Arrow, interview.

  4. Fred Kaplan, The Wizards of Armageddon, op. cit.

  5. Thomas C. Schelling, The Strategy of Conflict (Cambridge: Harvard University Press, 1960).

  6. Ibid.

  7. Arrow, interview.

  8. See, for example, Martin Shubik, “Game Theory and Princeton,” op. cit.; William Lucas, “The Fiftieth Anniversary of TGEB,” Games and Economic Behavior, vol. 8. (1995), pp. 264–68; Carl Kaysen, interview, 2.15.96.

  9. John McDonald, “The War of Wits,” op. cit.

  10. For a humorous account of Prussian military’s romance with probability theory see John Williams, The Compleat Strategist, op. cit.

  11. McDonald, op. cit.

  12. Bernice Brown, interview, 5.22.96.

  13. Rosters, RAND Department of Mathematics.

  14. Dresher and Shapley, op. cit. For a lucid description of game theoretic analyses of duels, see Dixit and Skeath, op. cit.

  15. Dresher and Shapley, op. cit.

  16. For von Neumann’s views, see Clay Blair, Jr., “Passing of a Great Mind,” Life (February 1957), pp. 88–90, as quoted in William Poundstone, Prisoner’s Dilemma, op. cit., p. 143.

  17. Arrow, interview.

  18. See Poundstone, op. cit.; Joseph Baratta, interview, 8.12.97.

  19. Arrow, interview.

  20. John H. Kagel and Alvin E. Roth, The Handbook of Experimental Economics (Princeton: Princeton University Press, 1995), pp. 8–9.

  21. Albert W. Tucker, interview, 12.94.

  22. See, for example, Avinash Dixit and Barry Nalebuff, Thinking Strategically, op. cit.

  23. See, for example, Anatole Rappaport, “Prisoner’s Dilemma,” in John Eatwell, Murray Milgate, and Peter Newman, The New Palgrave, op. cit., pp. 199–204.

  24. Dixit and Nalebuff, op. cit.

  25. Harold Kuhn, interview, 7.96.

  26. Poundstone, op. cit.; also Kagel and Roth, op. cit.

  27. John F. Nash, Jr., as quoted in Kagel and Roth, op. cit.

  28. Martin Shubik, “Game Theory at Princeton, 1949–1955: A Personal Reminiscence,” in Toward a History of Game Theory, edited by E. Roy Weintraub (Durham, N.C.: Duke University Press, 1992).

  29. The first version of Nash’s analysis of the role of threats in bargaining was published as a RAND memorandum, “Two-Person Cooperative Games, P-172” (Santa Monica, Calif.: RAND, 8.31.50). A final version appeared under the same title in Econometrica (January 1953), pp. 128–40. Also “Rational Non-Linear Utility,” RAND Memorandum, D-0793, 8.8.50.

  30. Kaplan, op. cit.

  31. Ibid.

  32. Ibid.

  33. Ibid., pp. 91–92.

  34. Ibid.

  35. Bruno Augenstein, interview.

  36. R. Duncan Luce and Howard Raiffa as quoted in Poundstone, op. cit., p. 168.

  37. Thomas Schelling, The Strategy of Conflict (Cambridge, Mass.: Harvard University Press, 1960).

  14: The Draft

  1. Department of Mathematics, Princeton University.

  2. Recommendations of 5.11.50 by Solomon Lefschetz, chairman, mathematics department, to president, Princeton University, that John Forbes Nash, Jr., be appointed research assistant, three-quarters time, on A. W. Tucker’s ONR Contract A-727.

  3. See, for example, David Halberstam, The Fifties, op. cit.

  4. Proceedings of the International Congress of Mathematicians, August 30–September 6, 1950, vol. 1, p. 516.

  5. Letter from John Nash to Albert W. Tucker, 9.10.50. Letter from John Nash to Solomon Lefschetz, undated (probably written between April 10 and April 26, 1948), gives the clearest statement of why Nash wanted to avoid the draft: “Should there come a war involving the U.S. I think I should be more useful, and better off, working on some research project than going, say, into the infantry.”

  6. Letter from Fred D. Rigby, Office of Naval Research, Washington, DC, to Albert W. Tucker, 9.15.50.

  7. Letter from J. Nash to A. W. Tucker, 9.10.50.

  8. Letters from A. W. Tucker to Local Board No. 12, 9.13.50; Raymond J. Woodrow to Local Board No. 12, 9.15.50 and 9.18.50; Raymond J. Woodrow, Committee on Project Research and Inventions, Princeton University, to Local Board No. 12, Bluefield, W.Va., re occupational deferment for John F. Nash, Jr. (with reference to RAND consultancy).

  9. Letter from F. D. Rigby to A. W. Tucker, 9.10.50.

  10. Ibid.

  11. Halb
erstam, op. cit.

  12. Hans Weinberger, interview, 10.28.95.

  13. Harold Kuhn, interview, 9.6.96.

  14. Gottesman, Schizophrenia Genesis, op. cit., pp. 152–55; also Bruce Dohrenwind, professor of social psychology, Columbia University, interview, 1.16.98.

  15. H. Steinberg and J. Durrel, “A Stressful Situation as a Precipitant of Schizophrenic Symptoms,” British Journal of Psychiatry, vol. 111 (1968), pp. 1097–1106, as quoted in Gottesman, Schizophrenia Genesis; op. cit.

  16. Notes of telephone call from Alice Henry, secretary, department of mathematics, Princeton University, re I-A classification of John Nash and request that Dean Douglas Brown write a letter to ONR to be forwarded to the Bluefield draft board, 9.15.50.

  17. “Information Needed in National Emergency,” form filled out 9.50 by John F. Nash, Jr., refers to I-A status, pending application for II-A, ONR and RAND research roles.

  18. Letter from Raymond J. Woodrow, Committee on Project Research and Inventions, Princeton University, to commanding officer, Office of Naval Research, New York Branch, re deferment for John F. Nash, Jr., 9.18.50.

  19. Letter from W. S. Keller, Office of Naval Research, New York Branch, to Selective Service Board No. 12, Bluefield, W.Va., re deferment for John F. Nash, Jr., 9.28.50.

  20. Richard Best, interview, 5.96.

  21. Melvin Peisakoff, interview, 5.96.

  22. Best, interview.

  23. Letter from Raymond J. Woodrow to John Nash, 10.6.50.

  24. Ibid.; letter from L. L. Vivian, ONR, New York Branch, to commanding officer, ONR, New York Branch Office, re notification of Nash by draft board that active service postponed until June 30, 1951, and continued I-A status, 11.22.50.

  15: A Beautiful Theorem

  1. Richard J. Duffin, interview, 10.26.95.

  2. “He can hold his own in pure mathematics, but his real strength seems to lie on the frontier between mathematics and the biological and social sciences,” letter from Albert W. Tucker to Marshall Stone, 12.14.51.

  3. John Nash, “Algebraic Approximations of Manifolds,” Proceedings of the International Congress of Mathematicians, vol. 1 (1950), p. 516, and “Real Algebraic Manifolds,” Annals of Mathematics, vol. 56, no. 3 (November 1952; received October 8, 1951). For expositions of Nash’s result, see John Milnor, “A Nobel Prize for John Nash,” op. cit., pp. 14–15, and Harold W. Kuhn, introduction, “A Celebration of John F. Nash, Jr.,” Duke Mathematical Journal, vol. 81, no. 1 (1995), p. iii.