Game theory
Game theory is the study of mathematical models of strategic interactions among rational agents.[1] It has applications in many fields of social science, used extensively in economics as well as in logic, systems science and computer science.[2] Traditional game theory addressed two-person zero-sum games, in which a participant's gains or losses are exactly balanced by the losses and gains of the other participant. In the 21st century, game theory applies to a wider range of behavioral relations, and it is now an umbrella term for the science of rational decision making in humans, animals, as well as computers.
This article is about the mathematical study of optimizing agents. For the mathematical study of sequential games, see Combinatorial game theory. For the study of playing games for entertainment, see Game studies. For the YouTube series, see MatPat. For other uses, see Game theory (disambiguation).
Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. Von Neumann's original proof used the Brouwer 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 Theory of Games and Economic Behavior (1944), co-written with Oskar Morgenstern, which considered cooperative games of several players.[3] The second edition provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.[4]
Game theory was developed extensively in the 1950s, and was explicitly applied to evolution in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. John Maynard Smith was awarded the Crafoord Prize for his application of evolutionary game theory in 1999, and fifteen game theorists have won the Nobel Prize in economics as of 2020, including most recently Paul Milgrom and Robert B. Wilson.
History[edit]
Precursors[edit]
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.[5] 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".[6][7] 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.[8]
In 1938, the Danish mathematical economist Frederik Zeuthen proved that the mathematical model had a winning strategy by using Brouwer's fixed point theorem.[9] 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.
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