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). 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 1713, a letter attributed to Charles Waldegrave, an active Jacobite and uncle to British diplomat James Waldegrave, analyzed a game called " le her". Huygens later published his gambling calculus as De ratiociniis in ludo aleæ ( On Reasoning in Games of Chance) in 1657. 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. In the 1650s, Pascal and Huygens developed the concept of expectation on reasoning about the structure of games of chance. 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. Wilson.ĭiscussions on the mathematics of games began long before the rise of modern mathematical game theory. 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. Game theory has been widely recognized as an important tool in many fields. 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. The second edition provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty. His paper was followed by Theory of Games and Economic Behavior (1944), co-written with Oskar Morgenstern, which considered cooperative games of several players. 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. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. 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 logical decision making in humans, animals, as well as computers. 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. It has applications in many fields of social science, used extensively in economics as well as in logic, systems science and computer science. Game theory is the study of mathematical models of strategic interactions among rational agents.
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