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Mathematical origins

Discussions on the mathematics of games began long before the rise of modern mathematical game theory. Cardano 's work Liber de ludo aleae (Book on Games of Chance), which was written around 1564 but published posthumously in 1663, sketches some basic ideas on games of chance. In the 1650s, Pascal and Huygens developed the concept of expectation on reasoning about the structure of games of chance. Pascal argued for equal division when chances are equal while Huygens extended the argument by considering strategies for a player who can make any bet with any opponent so long as its terms are equal. Huygens later published his gambling calculus as De ratiociniis in ludo aleæ (On Reasoning in Games of Chance) in 1657.

In 1713, a letter attributed to Charles Waldegrave, an active Jacobite and uncle to British diplomat James Waldegrave, analyzed a game called " le her ". Waldegrave provided a minimax mixed strategy solution to a two-person version of the card game, and the problem is now known as Waldegrave problem. In 1838, Antoine Augustin Cournot considered a duopoly and presented a solution that is the Nash equilibrium of the game in his Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth).

In 1913, Ernst Zermelo published Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels (On an Application of Set Theory to the Theory of the Game of Chess), which proved that the optimal chess strategy is strictly determined. This paved the way for more general theorems.

In 1938, the Danish mathematical economist Frederik Zeuthen proved that the mathematical model had a winning strategy by using Brouwer's fixed point theorem. In his 1938 book Applications aux Jeux de Hasard and earlier notes, Émile Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix is symmetric and provided a solution to a non-trivial infinite game (known in English as Blotto game). Borel conjectured the non-existence of mixed-strategy equilibria in finite two-person zero-sum games, a conjecture that was proved false by von Neumann.