That the study of gambling was central to the development of probability theory is indicated in the title of David’s history of the subject, Games, Gods, and Gambling (1962). 2 Initially, there was no true dichotomy between the more analytical gaming manuals and mathematical work on probability. Early probabilists studied games of pure chance, using simple and widely known gambling games as a starting point and to introduce readers to thinking in probabilistic terms. For example, Louis Bachelier (1901) used simple games of pure chance to make his novel contributions to probability theory more accessible to readers.
For probabilists used to pondering games, it was a short step from games of pure chance such as roulette or craps to contemplation of baccarat, blackjack or poker, where the probability of winning depends on the strategy chosen by the player. (The relationship between the analysis of specific games, probability theory and game theory was discussed at greater length in the first chapter.) Such a step was taken in the early twentieth century by Borel and his associates, notably Ville, working within the French tradition of probability theory, the tradition of Laplace, Poisson, Cournot, Bertrand and Poincaré. Starting from Bertrand’s analysis of baccarat, Borel (1921, 1924, 1927) found minimax solutions to several two-person zero-sum strategic games and failed to find a counterexample of such a game without a minimax solution (see Chapter 7). This preceded von Neumann’s proof of the minimax theorem, which Borel communicated to the Académie des Sciences (von Neumann 1928a, b; Leonard 1992). Borel papers of 1921, 1924 and 1927 attracted attention when translated by Savage in Econometrica in 1953. Von Neumann and Morgenstern (1944) made no mention of Borel’s early minimax papers, one of which von Neumann had cited in 1928, but acknowledged (1944, 154n.) that their proof of the minimax theorem was based on the first largely elementary proof by Ville (1938), 3 rather than on von Neumann’s intricate topological proof of 1928.
Apart from mention of Ville’s 1938 proof, and from a footnote by von Neumann and Morgenstern (1944, 186-7n.) on a 1938 treatment of poker by Borel and Ville, the literature of game theory has ignored the study of 131