January 28-29, 2005
Xavier University
Members in attendance:
Chris Christensen, Northern Kentucky University (after dinner Friday)
Daniel Curtin, Northern Kentucky University
Richard Davitt, University of Louisville
Charles Groetsch, University of Cincinnati (Saturday)
Kevin Kirby, Northern Kentucky University
David Kullman, Miami Univeristy (Friday)
Daniel Otero, Xavier University
Richard Pulskamp, Xavier University
Friday night dinner at 6pm was at The Quarter Bistro in Mariemont. A fine restaurant, one in which we dallied too long over dessert. (We almost got Davitt to admit that this was better than Buck's in Louisville.)
We reconvened at XU somewhat after 8pm. After a lively discussion regarding the redesign of Lake Inferior on the NKU campus as well as the redesign of the college in which computer science will reside there, we got to work on a review of the source materials that were brought to supplement our work.
John von Neumann, Zur Theorie der Gesellschaftspiele (Math. Annalen 100 (1928), 295-320), translated as On the theory of games of strategy by Sonya Bargmann, in Contributions to the Theory of Games, IV (Annals of Mathematics Studies 40), A. W. Tucker and R. D. Luce, eds., Princeton U. Pr., 1950, pp. 13-42. [In this paper, von Neumann first proved the Minimax Theorem and launched the serious mathematical theory of games.]
Emile Borel, La théorie du jeux et les équations intégrales à noyau symétriques (C. R. Math. Acad. Sci. Paris, vol. 173 (1921), 1304-1308) , translated as Theory of games and integral equations with skew symmetric kernels by Leonard J. Savage, Econometrica, vol. 21, no. 1 (Jan 1953), 97-100. [This was one of the papers that started von Neumann thinking about the mathematics of games of strategy.]
Members kindly prepared this bibliography of the supporting materials that were made available:
Dick Davitt:
William Aspray and Arthur Burks, eds., Papers of John von Neumann on Computing and Computer Theory, Charles Babbage Institute Reprint Series for the History of Computing, vol. 12, MIT Press, Cambridge, MA and London and Tomash Publishers, Los Angeles/San Francisco, 1987.
Mohammed Dore, Sukhamoy Chakravarty, and Richard Goodwin, eds., John von Neumann and Modern Economics, Clarendon Press, Oxford, 1989.
Melvin Dresher, The Mathematics of Games of Strategy, Theory and Applications, Dover Publications, Inc., NY 1981.
Steve Heims, John von Neumann and Norbert Weiner, From Mathematics to the Technologies of Life and Death, MIT Press, Cambridge, MA and London, 1980.
Wang Jianhua, The Theory of Games, Tsinghua University Press, Beijing and Clarendon Press, Oxford, 1988.
Norman Macrae, John von Neumann, the Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More, American Mathematical Society, Providence RI, 1999. [This is a literal reprint of this biography first published by Pantheon Books, a division of Random House, Inc., NY]
John von Neumann, Continuous geometries with a transition probability, Memoirs of the American Mathematical Society, November 1981, Volume 34, Number 252 (third of 5 numbers), AMS, Providence, RI, 1981.
-----, and Oskar Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, NJ, 1947.Kevin Kirby:
J.B.G. (Hans) Frenk, G. Kassay, and J. Kolumbán, "Equivalent Results in Minimax Theory" (January 2002). ERIM Report Series Reference No. ERS-2002-08-LIS. <http://ssrn.com/abstract=370953>
Danny Otero:
Emile Borel, "On systems of linear forms of skew symmetric determinant and the general theory of play" ("Sur les systèmes des formes linéaires à determinant symétrique gauche et la théorie générale du jeu," C. R. Math. Acad. Sci. Paris, vol. 184, 1927, pp. 52-53, trans. by Leonard J. Savage), Econometrica, 21 1 (Jan 1953), 116-117.
—————, "On games that involve chance and the skill of the players" ("Sur les jeux où interviennent l’hasard et l’habileté des joueurs," Théorie des Probabilités, 1924, pp. 204-224, trans. by Leonard J. Savage), Econometrica, 21 1 (Jan 1953), 101-115.
J. Dieudonné, “Von Neumann, Johann (or John),” Dictionary of Scientific Biography, C. Scribner's Sons, 1981, vol. 16, 88-92.
R. Duncan Luce and Howard Raiffa, Games and Decisions: introduction and critical survey, Wiley, 1957.
Maurice Fréchet, "Commentary on the three notes of Emile Borel," Econometrica, 21 1 (Jan 1953), 118-124.
—————, "Emile Borel, initiator of the theory of psychological games and its application," Econometrica, 21 1 (Jan 1953), 95-96.
James Glimm, John Impagliazzo, and Isadore Singer, eds., The Legacy of John von Neumann, Proc. Symp. Pure Math. 50, AMS, 1990.
Steve Heims, “Von Neumann, John,” Thinkers of the Twentieth Century, St. James, 1987.
Guillermo Owen, Game Theory, 2nd ed., Academic Press, 1982.
John von Neumann, "Communication on the Borel Notes," Econometrica, 21 1 (Jan 1953), 124-127.
————— and Oskar Morgenstern, Theory of Games and Economic Behavior, 3rd ed., Princeton, 1953.
E. Roy Weintraub, ed., Toward a History of Game Theory, Annual Supplement to vol. 24, History of Political Economy, Duke U. Press, 1992.Dick Pulskamp:
[1713] James de Waldegrave, Excerpt from a letter of Pierre de Montmort to Nicholas Bernoulli, Essai d'Analyse sur les jeux de hazard.
[1907] J. Bertrand, Calcul des Probabilités, Chapter II §33, "On Baccarat."
[1913] E. Zermelo, Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels, Proc. Fifth Internat. Cong. Math., vol. II. [A translation is appended to the paper Zermelo and the early history of game theory by Ulrich Schwabe and Paul Walker. The first theorem of game theory asserts that chess is strictly determined.]
[1921] E. Borel, "La théorie du jeux...."
[1923] —————, "Sur lex jeux ou interviennent l'hasard et l'habilité des joueurs," Association Français pour l'Advancement des Sciences, 79-85.
[1924] —————, "Sur lex jeux ou interviennent l'hasard et l'habilité des joueurs," in Théorie des Probabilités, note iv, 204-224.
[1926] —————, "Un théorème sur les systèmes des formes linéaires à déterminant symètrique gauche," C. R. Math. Acad. Sci. Paris, vol.183, 925-927.
[1926] —————, Correction to an error in "Un théorème sur les systèmes des formes linéaires à déterminant symètrique gauche," C. R. Math. Acad. Sci. Paris, vol.183, 933.
[1927] —————, "Algebre et calcul des probabilités -- Sur les systèmes des formes linéaires à déterminant symètrique gauche et la théorie générale du jeu," C. R. Math. Acad. Sci. Paris, vol.184, 52-54. [Translated into English by Leonard J. Savage as "On systems of linear forms of skew symmetric determinant and the general theory of play." Reference above in Otero's bibliography.]
[1928] John von Neumann, Zur Theorie der Gesellschaftspiele.
[1934] R. A. Fisher, "Randomisation, and an old enigma of card play," Math. Gazette 18, 294-297. [Fisher discovers Waldegrave's solution to the game of Her.]
[1938] Emile Borel, Traité du calcul des probabilités et des applications: Applications des jeux de hasard, vol. IV, fasc. 2.
[1938] Jean Ville, "Note sur la théorie générale ges jeux or interviennent l'habilité des joueurs, " Applications des jeux de hasard, vol. IV, fasc. 2. of Traité du calcul des probabilités et des applications. [Ville gives the first elementary but still partially topological proof of the Minimax Theorem.]
[1944] John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior.
[1946] L. H. Loomis, On a theorem of von Neumann, Proc. Nat. Acad. Sci., vol. 32, no. 8. [The first entirely algebraic proof of the Minimax Theorem.]
We then embarked on a consideration of the life and times of our man, Johnny von Neumann. Largely because of the wealth of biographical information on the man, and because he was such an interesting character, too, we found ourselves spending most of the evening discussing his life. It was well after 9pm when we started looking at our sources. It was deemed appropriate that we consider first the Borel paper that was part of the common readings. The major issue was why Borel had used a two-dimensional distribution for his strategy spaces (rather than a one-dimensional distribution) when he considered the continuous case. Otero distributed copies of the two-page paper of Borel (first entry in his bibliography above) which was actually the one cited in von Neumann's paper.
On Saturday morning, after Panera bagels and cream cheese, orange juice and coffee, we tore into the paper of von Neumann. The timing was perfect as we reached the end of the paper just at noon. Despite the ellipsis-crazy notation, the paper was surprisingly easy to read.
The fall meeting, tentatively scheduled for September 16-17, 2005, at Northern Kentucky University, will be devoted to the work of Hermann Weyl. David Kullman will look for suitable materials, likely to include a selection from Weyl's Symmetry.
Respectfully submitted,
Danny Otero