A Beautiful Mind (80 page)

3
. Isbell, e-mails.

4
. Letter from John Nash to Martin Shubik, undated (1950 or 1951); Hausner, interviews and e-mails.

5
. William Poundstone,
Prisoner’s Dilemma,
op. cit.; John Williams,
The Compleat Strategist
(New York: McGraw Hill, 1954).

6
. Poundstone, op. cit.

7
. Solomon Leader, interview, 6.9.95.

8
. Martha Nash Legg, interview, 8.1.95.

9
. Isbell, e-mails.

10
. Hartley Rogers, interview, 1.26.96.

11
. Ibid.

12
. Ibid.

13
. Nash may have had the idea while he was at Carnegie. This, in any case, is Hans Weinberger’s recollection, interview, 10.28.95.

14
. Martin Gardner,
Mathematical Puzzles and Diversions
(New York: Simon & Schuster, 1959), pp. 65–70.

15
. Gardner’s comment, in 1959, was that Hex “may well become one of the most widely played and thoughtfully analyzed new mathematical games of the century.”

16
. Gale, interview, 9.20.95.

17
. Dinner at which John Nash, David Gale, and the author were present, January 5, 1996, San Francisco.

18
. Gale, interview.

19
. Ibid.

20
. Phillip Wolfe, mathematician, IBM, interview, 9.9.96.

21
. John Milnor, “A Nobel Prize for John Nash,” op. cit.

22
. Ibid.; Gardner, op. cit.

23
. Gale, interview.

24
. Ibid.

25
. Ibid.

26
. Kuhn, interview.

27
. Ibid.

28
. Milnor, interview, 9.26.95.

7: John von Neumann

1
. See, for example, Stanislaw Ulam, “John von Neumann, 1903–1957,”
Bulletin of the American Mathematical Society,
vol. 64, no. 3, part 2 (May 1958); Stanislaw Ulam,
Adventures of a Mathematician
(New York: Scribner’s, 1983); Paul R. Halmos, “The Legend of John von Neumann,”
American Mathematical Monthly,
vol. 80 (1973); William Poundstone,
Prisoner’s Dilemma,
op. cit.; Ed Regis,
Who Got Einstein’s Office?,
op. cit.

2
. Poundstone, op. cit.

3
. Ulam, “John von Neumann,” op. cit.; Poundstone, op. cit., pp. 94–96.

4
. Harold Kuhn, interview, 1.10.96.

5
. In remarks at a Nobel luncheon at the American Economics Association meeting on 1.5.96, Nash traced a lineage from Newton to von Neumann to himself. Nash shared von Neumann’s interest in game theory, quantum mechanics, real algebraic variables, hydrodynamic turbulence, and computer architecture.

6
. See, for example, Ulam, “John von Neumann,” op. cit.

7
. Norman McRae,
John von Neumann
(New York: Pantheon Books, 1992), pp. 350–56.

8
. John von Neumann,
The Computer and the Brain
(New Haven: Yale University Press, 1959).

9
. See, for example, G. H. Hardy,
A Mathematician’s Apology (Cambridge,
U.K.: Cambridge University Press, 1967), with a foreword by C. P. Snow.

10
. Ulam, “John von Neumann,” op. cit.

11
. Poundstone, op. cit.

12
. Poundstone,
Prisoner’s Dilemma,
p. 190.

13
. Clay Blair, Jr., “Passing of a Great Mind,”
Life
(February 1957), pp. 89–90, as quoted by Poundstone, op. cit., p. 143.

14
. Poundstone, op. cit.

15
. Ulam, “John von Neumann,” op. cit.

16
. Harold Kuhn, interview, 3.97.

17
. Paul R. Halmos, “The Legend of John von Neumann,” op. cit.

18
. Ibid.

19
. Poundstone, op. cit.

20
. Halmos, op. cit.

21
. Ibid.

22
. Poundstone, op. cit.

23
. Ulam,
Adventures of a Mathematician,
op. cit.

24
. Ulam, “John von Neumann,” op. cit.

25
. Ibid.

26
. Ibid., p. 10; Robert J. Leonard, “From Parlor Games to Social Science,” op. cit.

27
. Richard Duffin, interview, 10.94.

28
. Halmos, op. cit.

29
. Ulam, “John von Neumann,” op. cit., pp. 35–39.

30
. Interviews with Donald Spencer, 11.18.95; David Gale, 9.20.95; and Harold Kuhn, 9.23.95.

31
. Poundstone, op. cit.

32
. Herman H. Goldstine, “A Brief History of the Computer,”
A Century of Mathematics in America,
Part I
,
op. cit.

33
. John von Neumann, as quoted in ibid.

8: The Theory of Games

1
. John von Neumann and Oskar Morgenstern,
The Theory of Games and Economic Behavior
(Princeton: Princeton University Press, 1944, 1947, 1953).

2
. Both von Neumann and Morgenstern came to the seminar. Albert W. Tucker, interview, 10.94. See also Martin Shubik, “Game Theory and Princeton, 1940–1955: A Personal Reminiscence,” Cowles Foundation Preliminary Paper, undated, p. 3; David Gale, interview, 9.20.95; and Harold Kuhn, interview, 9.20.95.

3
. A. W. Tucker, “Combinatorial Problems Related to Mathematical Aspects of Logistics: Final Summary Report” (U.S. Department of the Navy, Office of Naval Research, Logistics Branch, February 28, 1957), p. 1.

4
. Melvin Hausner, interview, 2.6.96.

5
. Interviews with David Yarmush, 2.6.96, and John Mayberry, 4.15.96.

6
. David Gale, interview.

7
. Kuhn, interview.

8
. Ibid.; Hausner, interview.

9
. Robert J. Leonard, “From Parlor Games to Social Science,” op. cit.

10
. See, for example, H. W. Kuhn and A. W. Tucker, “John von Neumann’s Work in the Theory of Games and Mathematical Economics,”
Bulletin of the American Mathematical Society
(May 1958).

11
. Leonard, “From Parlor Games to Social Science,” op. cit.

12
. Ibid.

13
. Ibid.

14
. Dorothy Morgenstern Thomas, interview, 1.25.96. Morgenstern kept a portrait of the kaiser hanging in his home.

15
. Letter from George Mowbry to author, 4.5.95.

16
. Leonard, “From Parlor Games to Social Science,” op. cit.

17
. As quoted in ibid.

18
. Ibid.

19
. Ibid.

20
. Ibid.

21
. Ibid.

22
. Ibid.

23
. Ibid.

24
. Ibid.

25
. A. W. Tucker, who knew both men well, said, “If he hadn’t been forced to write a book, it wouldn’t have gotten written,” interview, 10.94. Von Neumann was interested in economics before he met Morgenstern.

26
. Leonard, “From Parlor Games to Social Science,” op. cit.

27
. Ibid.

28
. Von Neumann and Morgenstern, op. cit., p. 6.

29
. Leonid Hurwicz, “The Theory of Economic Behavior,”
The American Economic Review
(1945), pp. 909–25.

30
. Von Neumann and Morgenstern, op. cit., p. 7.

31
. Ibid., p. 3.

32
. Ibid.

33
. Ibid., p. 4.

34
. Ibid., p. 7.

35
. Ibid., p. 2.

36
. Ibid.

37
. Ibid., p. 6.

38
.
New York Times,
3.46.

39
. See, for example, Herbert Simon,
The American Journal of Sociology,
no. 50 (1945), pp. 558–60. Hurwicz, op. cit.; Jacob Marschak, “Neumann’s and Morgenstern’s New Approach to Static Economics,”
Journal of Political Economy,
no. 54 (1946), pp. 97–115; John McDonald, “A Theory of Strategy,”
Fortune
(June 1949), pp. 100–110.

40
. Leonard, “From Parlor Games to Social Science,” op. cit.

41
. Ibid.

42
. Ibid.

43
. Shubik, “Game Theory and Princeton,” op. cit., p. 2.

44
. Von Neumann and Morgenstern, op. cit. See also Eatwell, Milgate, and Newman, op. cit.

45
. Von Neumann and Morgenstern, op. cit.

46
. Ibid.

47
. See, for example, John C. Harsanyi, “Nobel Seminar,” in
Les Prix Nobel 1994.

48
. Von Neumann and Morgenstern, op. cit.

49
. Ibid.

50
. Ibid.

51
. Harsanyi, op. cit.

9: The Bargaining Problem

1
. John Forbes Nash, Jr., “The Bargaining Problem,”
Econometrica,
vol. 18 (1950), pp. 155–62.

2
. Nash’s bargaining solution was “virtually unanticipated in the literature,” according to Roger B. Myerson, “John Nash’s Contribution to Economics,”
Games and Economic Behavior,
no. 14 (1996), p. 291. See also Ariel Rubinstein, “John Nash: The Master of Economic Modeling,”
The Scandinavian Journal of Economics,
vol. 97, no. 1 (1995), pp. 11–12; John C. Harsanyi, “Bargaining,” in Eatwell, Milgate, and Newman, op. cit., pp. 56–60; Andrew Schotter, interview, 10.25.96; Ariel Rubinstein, interview, 11.25.96; James W. Friedman, professor of economics, University of North Carolina, interview, 10.2.96.

3
. “This is the classical problem of exchange and, more specifically, of bilateral monopoly as treated by Cournot, Bowley, Tintner, Fellner and others,” Nash, “The Bargaining Problem,” p. 155. As Harold Kuhn points out, Nash’s delineation of the history of the problem was undoubtedly supplied by Oskar Morgenstern, “It is now clear that Nash had not read those writers,” Harold Kuhn, “Nobel Seminar,”
Les Prix Nobel 1994.
For a delightful short history of exchange, including the references to pharaohs and kings, see Robert L. Heilbroner,
The Worldly Philosophers,
6th edition (New York: Touchstone, 1992), p. 27.

4
. John C. Harsanyi, “Approaches to the Bargaining Problem Before and After the Theory of Games: A Critical Discussion of Zeuthen’s, Hick’s and Nash’s Theories,”
Econometrica,
vol. 24 (1956), pp. 144–57.

5
. In his now-classic reformulation of the Nash bargaining model, Ariel Rubinstein traces the bargaining problem to Edgeworth, “Mathematical Psychics: An Essay on the Application of MatKematics to the Moral Sciences” (London: C. Kegan Paul, 1881), reprinted in
Mathematical Psychics and Other Essays
(Mountain Center, Calif.: James & Gordon, 1995). Martin Shubik writes, “Even as a graduate student I was struck by the contrast between cooperative game theory, the seeds of which I regarded as already present in Edgeworth and noncooperative theory which was present in Cournot,” Martin Shubik,
Collected Works,
forthcoming, p. 6. For lively accounts of Edgeworth’s life and contributions, see Heilbroner, op. cit., pp. 174–76, and John Maynard Keynes, “Obituary of Francis Isidro Edgeworth, March 26, 1926,” reprinted in Edgeworth, op. cit.

6
. Fleilbroner, op. cit., p. 173.

7
. Ibid., p. 174.

8
. Edgeworth, op. cit.

9
. Ibid.

10
. Ibid.

11
. Harsanyi, op. cit.

12
. John von Neumann and Oskar Morgenstern,
The Theory of Games and Economic Behavior,
op. cit., p. 9. “It may also be regarded as a nonzero-sum two-person game,” Nash, “The Bargaining Problem,” op. cit., p. 155; “even though von Neumann and Morgenstern’s theory of games was an essential step toward a strong bargaining theory, their own analysis of two-person bargaining games did not go significantly beyond the weak bargaining theory of neoclassical economics,” Harsanyi, “Bargaining,” op. cit., pp. 56–57.

13
. See, for example, Robert J. Leonard, “From Parlor Games to Social Science,” op. cit., for a history of the axiomatic approach, and a superb interpretive discussion of “axiomatics” in Robert J. Aumann, “Game Theory,” in John Eatwell, Murray Milgate, and Peter Newman,
The New Palgrave,
op. cit., pp. 26–28.

14
. Von Neumann and Morgenstern used the axiomatic method to derive their theory of expected or von Neumann-Morgenstern utilities in the second, 1947, edition of
The Theory of Games and Economic Behavior.
The first application to a problem in social sciences, I believe, was Kenneth J. Arrow’s Ph.D. thesis
Social Choice and Individual Values
(New York: John Wiley & Sons, 1951). Lloyd S. Shapley’s “A Value of N-Person Games,”
Contributions to the Theory of Games II
(Princeton: Princeton University Press, 1953), pp. 307–17, is another stellar example.

15
. John Nash, “The Bargaining Problem,” op. cit., p. 155.

16
. John Nash,
Les Prix Nobel 1994,
op. cit., pp. 276–77.

17
. The sketch of Bart Hoselitz is based on an interview with his friend Sherman Robinson, professor of economics, University of Chicago, 7.95, and questionnaires, letters, and a curriculum vitae from Carnegie-Mellon University archives.

18
. This bit of history about international trade theory after World War II was supplied by Kenneth Rogoff, professor of economics, Princeton University, interview.

19
. John Nash,
Les Prix Nobel 1994,
op. cit., pp. 176–77.

20
. Nash told Myerson that he was inspired by a problem posed by Hoselitz. Roger Myerson, professor of economics, Northwestern University’, interview, 8.7.97.

21
. Myerson, e-mail, 8.11.97.

22
. Letter from John Nash to Martin Shubik, undated (written in 1950 or 1951).