Historical Development of Game Theory

Discussion on the mathematics of games began long before the rise of modern, mathematical game theory. Cardano wrote on the game of chance on Liber de ludo aleae (Book on Games of Chance), written around 1564 but published posthumously in 1663. In the 1650s, Pascal and Huygens developed the concept of expectation on reasoning about the structure of games of chance, and Huygens published his gambling calculus in De ratiociniis in ludo aleae (On Reasoning in Games of Chance) in 1657.

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

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 was symmetric and provides 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.

Game theory did not really exist as a unique field until John von Neumann published the book On the Theory of Games of Strategy in 1928. Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern. The second edition of this book provided an axiomatic theory of utility, which reincarnated Daniel Bernoulli's old theory of utility (of money) as an independent discipline. Von Neumann's work in game theory culminated in this 1944 book. This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. Subsequent work focused primarily on cooperative game theory, which analyzes optimal strategies for group of individuals, presuming that they can enforce agreements between them about proper strategies.

In 1950, the first mathematical discussion of the prisoner's dilemma appeared, and an experiment was undertaken by notable mathematicians Merrill M. Flood and Melvin Dresher, as part of the RAND Corporation's investigations into game theory. RAND pursued the studies because of possible applications of global nuclear strategy. Around this time, John Nash developed a criterion for mutual consistency of players' strategies known as the Nash equilibrium, applicable to a wider variety of games than the criterion proposed by Von Neumann and Morgenstern. Nash proved that every finite, n-player, non-zero-sum (not just two-player zero-sum) non-competitive game has what is now known as a Nash equilibrium in mixed strategies.

Game theory experienced a flurry of activitiy in the 1950s, during which the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapely value were developed. The 1950s also saw the first application of game theory to philosophy and political science.

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