# Game theory

**gamegame theoristgame theoreticgame-theoreticgame theoreticalgame-theorygamestheory of gamesgame theoristsgame-theoretical**

Game theory is the study of mathematical models of strategic interaction among rational decision-makers.wikipedia

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### Zero-sum game

**zero-sumzero sumNon-zero-sum game**

Originally, it addressed zero-sum games, in which each participant's gains or losses are exactly balanced by those of the other participants. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann.

In game theory and economic theory, a zero-sum game is a mathematical representation of a situation in which each participant's gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants.

### John von Neumann

**von NeumannJ. von NeumannNeumann, John von**

Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann.

He made major contributions to a number of fields, including mathematics (foundations of mathematics, functional analysis, ergodic theory, representation theory, operator algebras, geometry, topology, and numerical analysis), physics (quantum mechanics, hydrodynamics, and quantum statistical mechanics), economics (game theory), computing (Von Neumann architecture, linear programming, self-replicating machines, stochastic computing), and statistics.

### Mathematical economics

**mathematical economisteconomicsmathematical**

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.

Economics became more mathematical as a discipline throughout the first half of the 20th century, but introduction of new and generalized techniques in the period around the Second World War, as in game theory, would greatly broaden the use of mathematical formulations in economics.

### Mathematical model

**mathematical modelingmodelmathematical models**

Game theory is the study of mathematical models of strategic interaction among rational decision-makers.

Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models.

### Jean Tirole

**Tirole, J.Tirole, JeanTirole**

, with the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole, eleven game theorists have won the economics Nobel Prize.

He focuses on industrial organization, game theory, banking and finance, and economics and psychology.

### Minimax

**maximinminimax algorithmminmax**

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.

Minimax (sometimes MinMax, MM or saddle point ) is a decision rule used in artificial intelligence, decision theory, game theory, statistics and philosophy for minimizing the possible loss for a worst case (maximum loss) scenario.

### Oskar Morgenstern

**MorgensternMorgenstern, OskarO. Morgenstern**

His paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players.

In collaboration with mathematician John von Neumann, he founded the mathematical field of game theory and its application to economics (see von Neumann–Morgenstern utility theorem).

### John Maynard Smith

**Maynard SmithJohn Maynard-SmithMaynard Smith, J.**

John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy.

Maynard Smith was instrumental in the application of game theory to evolution with George R. Price, and theorised on other problems such as the evolution of sex and signalling theory.

### Prisoner's dilemma

**iterated prisoner's dilemmaprisoner’s dilemmaprisoners' dilemma**

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.

The prisoner's dilemma is a standard example of a game analyzed in game theory that shows why two completely rational individuals might not cooperate, even if it appears that it is in their best interests to do so.

### Extensive-form game

**extensive form gameextensive formextensive form games**

Game theory experienced a flurry of activity in the 1950s, during which the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed.

An extensive-form game is a specification of a game in game theory, allowing (as the name suggests) for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the (possibly imperfect) information each player has about the other player's moves when they make a decision, and their payoffs for all possible game outcomes.

### Strictly determined game

**Determined gamestrictly determined**

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.

In game theory, a strictly determined game is a two-player zero-sum game that has at least one Nash equilibrium with both players using pure strategies.

### Core (game theory)

**coretheory of the corecore (economics)**

Game theory experienced a flurry of activity in the 1950s, during which the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed.

In game theory, the core is the set of feasible allocations that cannot be improved upon by a subset (a coalition) of the economy's agents.

### Non-cooperative game theory

**non-cooperative gamenon-cooperativenon-cooperative games**

Nash proved that every finite n-player, non-zero-sum (not just 2-player zero-sum) non-cooperative game has what is now known as a Nash equilibrium in mixed strategies.

In game theory, a non-cooperative game is a game with competition between individual players, as opposed to cooperative games, and in which alliances can only operate if self-enforcing (e.g. through credible threats).

### Merrill M. Flood

**Merrill FloodFlood, Merrill M.**

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.

Merrill Meeks Flood (1908 – 1991 ) was an American mathematician, notable for developing, with Melvin Dresher, the basis of the game theoretical Prisoner's dilemma model of cooperation and conflict while being at RAND in 1950 (Albert W. Tucker gave the game its prison-sentence interpretation, and thus the name by which it is known today).

### Duopoly

**duopoliesduopolistic markettwin-duopolistic**

In his 1838 Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth), Antoine Augustin Cournot considered a duopoly and presents a solution that is the Nash equilibrium of the game.

### Brouwer fixed-point theorem

**Brouwer fixed point theoremBrouwer's fixed point theoremBrouwer's fixed-point theorem**

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.

It appears in unlikely fields such as game theory.

### Repeated game

**repeated gamesiterated gamedynamic game**

Game theory experienced a flurry of activity in the 1950s, during which the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed.

In game theory, a repeated game is an extensive form game that consists of a number of repetitions of some base game (called a stage game).

### Shapley value

**Aumann-Shapley valueAumann–ShapleyAumann–Shapley value**

The Shapley value is a solution concept in cooperative game theory.

### Solution concept

**equilibrium conceptequilibrium refinementrefinement**

In 1965, Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium.

In game theory, a solution concept is a formal rule for predicting how a game will be played.

### Evolutionary game theory

**Evolutionary Gamesstrategyevolution**

Schelling worked on dynamic models, early examples of evolutionary game theory.

Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology.

### Evolutionarily stable strategy

**evolutionarily stable strategiesevolutionarily stableevolutionary strategies**

In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy.

It is relevant in game theory, behavioural ecology, and evolutionary psychology.

### John Harsanyi

**John C. HarsanyiHarsanyiJános Harsányi**

In 1994 Nash, Selten and Harsanyi became Economics Nobel Laureates for their contributions to economic game theory.

He is best known for his contributions to the study of game theory and its application to economics, specifically for his developing the highly innovative analysis of games of incomplete information, so-called Bayesian games.

### Fictitious play

In game theory, fictitious play is a learning rule first introduced by George W. Brown.

### Leonid Hurwicz

**Leo HurwiczHurwicz, L.Leonid "Leo" Hurwicz**

In 2007, Leonid Hurwicz, Eric Maskin, and Roger Myerson were awarded the Nobel Prize in Economics "for having laid the foundations of mechanism design theory".

Leonid "Leo" Hurwicz (August 21, 1917 – June 24, 2008) was a Polish-American economist and mathematician, known for his work in game theory and mechanism design.

### Waldegrave problem

**Waldegrave**

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 probability and game theory, the Waldegrave problem refers to a problem first described in the second edition of Montmort`s Essay d'analyse sur les jeux de hazard.