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Game theory is the study of mathematical models of strategic interaction among rational decision-makers. Originally, it addressed zero-sum gamesin which each participant's gains or losses are exactly balanced by those of the other participants. In the 21st century, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann.

Von Did Technology and Strategy Affect the Outcome original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex setswhich became a standard method in game theory and mathematical economics. His paper was followed by the book Theory of Games and Economic Behaviorco-written with Oskar Morgensternwhich considered cooperative games of several players.

The second edition of this book provided an axiomatic source of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty. Game theory was developed extensively in the s by many scholars. It was explicitly applied to evolution in the s, although similar developments go Computer Knowledge at least as far as the s.

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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. Discussions of two-person games began long before the rise of modern, mathematical game theory.

Ina letter attributed to Charles Waldegrave analyzed a game called "le her".

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He was an active Jacobite and uncle to James Waldegravea British diplomat. One theory postulates Francis Waldegrave as the true correspondent, but this has yet to be proven. This paved the way for more general theorems. Read articlethe Danish mathematical economist Frederik Zeuthen proved that the mathematical model had a winning strategy by using Brouwer's fixed point theorem. Borel conjectured the non-existence of mixed-strategy equilibria in finite two-person zero-sum gamesa conjecture that was proved false by von Neumann. Game theory did not really exist as a unique field until John von Neumann published the paper On the Theory of Games of Strategy in Von Neumann's work in game theory culminated in this book.

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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 groups of individuals, presuming that they can enforce agreements between them about proper strategies.

Inthe first mathematical discussion of the prisoner's dilemma appeared, and an experiment was undertaken by notable mathematicians Merrill M. RAND pursued the studies because of possible applications to global nuclear strategy. Nash proved that every finite n-player, non-zero-sum not just two-player zero-sum non-cooperative game has what is now known as a Nash equilibrium in mixed strategies. Game theory experienced a flurry of activity in the s, during which the concepts of the corethe extensive form gamefictitious playrepeated gamesand the Shapley value were developed.

The s also saw the first applications of game theory to philosophy and political science. In Robert Axelrod tried setting up computer programs as players and found that in tournaments Did Technology and Strategy Affect the Outcome them the winner was often a simple "tit-for-tat" program—submitted by Anatol Rapoport —that cooperates on the first step, then, on subsequent steps, does whatever its opponent did on the previous step. The same winner Job Movie Review also often obtained by natural selection; a fact that is widely taken to explain cooperation phenomena in evolutionary biology and the social sciences.

InReinhard Selten introduced his solution concept of subgame perfect equilibriawhich further refined the Nash equilibrium.

Later he would introduce trembling hand perfection as well. In the s, game theory was extensively applied in biologylargely as a result of the work of John Maynard Smith and his evolutionarily stable strategy. In addition, the concepts of correlated equilibriumtrembling hand perfection, and common knowledge [a] were introduced and analyzed. Schelling worked on dynamic models, early examples of evolutionary game theory.]