Sources page biographical material

4.A.5. GENERAL THEORY
Charles Babbage. The Philosophy of Analysis -- unpublished collection of MSS in the BM as Add. MS 37202, c1820. ??NX. Ff. 134-144 are: Essay 10 Part 5. See 4.B.1 for more details. At the top of f. 134.r, he has added a note: "This is probably my earliest Note on Games of Skill. I do not recollect the date. 3 March 1865". He then describes Tit Tat To and makes some simple analysis, but he never uses a name for it.

Charles Babbage. Notebooks -- unpublished collection of MSS in the BM as Add. MS 37205. ??NX. See 4.B.1 for more details. On f. 304, he starts on analysis of games. Ff. 310 383 are almost entirely devoted to Tit-Tat-To, with some general discussions. F. 321.r, 10 Sep 1860, is the beginning of a summary of his work on games of skill in general. F. 324-333, Oct 1844, studies "General laws for all games of Skill between two players" and draws flow charts showing the basic recursive analysis of a game tree (ff. 325.v & 325.r). On f. 332, he counts the number of positions in Tit Tat To as 9! + 8! + ... + 1! = 409,113. F. 333 has an idea of the tree structure of a game.

John M. Dubbey. The Mathematical Work of Charles Babbage. CUP, 1978, pp. 96 97 & 125 130. See 4.B.1 for more details. He discusses the above Babbage material. On p. 127, Dubbey has: "The basic problem is one that appears not to have been previously considered in the history of mathematics." Dubbey, on p. 129, says: "This analysis ... must count as the first recorded stochastic process in the history of mathematics." However, it is really a deterministic two-person game.

E. Zermelo. Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels. Proc. 5th ICM (1912), CUP, 1913, vol. II, 501 504. Gives general idea of first and second person games.

Ahrens. A&N. 1918. P. 154, note. Says that each particular Dots and Boxes board, with rational play, has a definite outcome.

W. Rivier. Archives des Sciences Physiques et Naturelles (Nov/Dec 1921). ??NYS -- cited by Rivier (1935) who says that the later article is a new and simpler version of this one.

H. Steinhaus. Difinicje potrzebne do teorji gry i pościgu (Definitions for a theory of games and pursuit). Myśl Akademicka (Lwów) 1:1 (Dec 1925) 13 14 (in Polish). Translated, with an introduction by Kuhn and a letter from Steinhaus in: Naval Research Logistics Quarterly 7 (1960) 105 108.

Dénès König. Über eine Schlussweise aus dem Endlichen ins Unendliche. Mitteilungen der Universitä Szeged 3 (1927) 121-130. ??NYS -- cited by Rivier (1935). Kalmár cites it to the same Acta as his article.

László Kalmár. Zur Theorie der abstracten Spiele. Acta Litt. Sci. Regia Univ. Hungaricae Francisco Josephine (Szeged) 4 (1927) 62 85. Says there is a gap in Zermelo which has been mended by König. Lengthy approach, but clearly gets the idea of first and second person games.

Max Euwe. Proc. Koninklijke Akadamie van Wetenschappen te Amsterdam 32:5 (1929). ??NYS -- cited by Rivier (1935).

Emanuel Lasker. Brettspiele der Völker. Rätsel  und mathematische Spiele. A. Scherl, Berlin, 1931, pp. 170 203. Studies the one pile game (100, 5) and the sum of two one pile games: (100, 5) + (50, 3). Discusses Nimm, "an old Chinese game according to Ahrens" and says the solver is unknown. Gives Lasker's Nim -- one can take any amount from a pile or split it in two -- and several other variants. Notes that 2nd person + 2nd person is 2nd person while 2nd person + 1st person is 1st person. Gives the idea of equivalent positions. Studies three (and more) person games, assuming the pay offs are all different. Studies some probabilistic games. Jörg Bewersdorff [email of 6 Jun 1999] observes that Lasker's analysis of his Nim got very close to the idea of the Sprague-Grundy number. See: Jörg Bewersdorff; Glück, Logik und Bluff Mathematik im Spiel -- Methoden, Ergebnisse und Grenzen; Vieweg, 1998, Section 2.5 Lasker-Nim: Gewinn auf verborgenem Weg, pp. 118-124.

W. Rivier. Une theorie mathématique des jeux de combinaisions. Comptes-Rendus du Premier Congrès International de Récréation Mathématique, Bruxelles, 1935. Sphinx, Bruxelles, 1935, pp. 106 113. A revised and simplified version of his 1921 article. He cites and briefly discusses Zermelo, König and Euwe. He seems to be classifying games as first player or second player.

René de Possel. Sur la Théorie Mathématique des Jeux de Hasard et de Réflexion. Actualités Scientifiques et Industrielles 436. Hermann, Paris, 1936. Gives the theory of Nim and also the misère version. Shows that any combinatorial game is a win, loss or draw and describes the nature of first and second person positions. He then goes on to consider games with chance and/or bluffing, based on von Neumann's 1927 paper.

R. Sprague. Über mathematische Kampfspiele. Tôhoku Math. J. 41 (1935/36) 438 444.

P. M. Grundy. Mathematics and games. Eureka 2 (1939) 6 8. Reprinted, ibid. 27 (1964) 9 11. These two papers develop the Sprague-Grundy Number of a game.

D. W. Davies. A theory of chess and noughts and crosses. Penguin Science News 16 (Jun 1950) 40-64. Sketches general ideas of tree structure, Sprague-Grundy number, rational play, etc.

H. Steinhaus. Games, an informal talk. AMM 72 (1965) 457 468. Discusses Zermelo and says he wasn't aware of Zermelo in 1925. Gives Mycielski's formulation and proof via de Morgan's laws. Goes into pursuit and infinite games and their relation to the Axiom of Choice.

H. Steinhaus. (Proof that a game without ties has a strategy.) In: M. Kac; Hugo Steinhaus -- a reminiscence and a tribute; AMM 81 (1974) 572 581. Repeats idea of his 1965 talk.