Game theory

Game theory is a branch of mathematics that studies strategic interactions among rational decision-makers. It has applications in a variety of fields, including economics, political science, biology, computer science, and psychology. Game theory aims to understand the behavior of these agents in situations where the outcome for each participant depends not only on their own actions but also on the actions of others.

Overview

Game theory models situations as games, with rules that specify the possible actions for each player and the outcomes that result from the combination of these actions. Players in a game are assumed to be rational, meaning they will strive to maximize their own payoff, given their knowledge of the outcomes and the actions of other players.

Key Concepts

Nash Equilibrium

A central concept in game theory is the Nash Equilibrium, named after mathematician John Nash. It is a situation in a game where no player can benefit by changing their strategy while the other players keep theirs unchanged. This concept is crucial for predicting the outcome of strategic interactions.

Zero-Sum and Non-Zero-Sum Games

Games can be classified as zero-sum, where one player's gain is exactly balanced by the losses of the other players, or non-zero-sum, where the total payoff to all players can vary.

Cooperative and Non-Cooperative Games

In cooperative games, players can form coalitions and make binding agreements, while in non-cooperative games, agreements are not enforceable, and each player acts independently.

Applications

Game theory has been applied to a wide range of real-world situations, including business negotiations, auctions, voting systems, public goods provision, and warfare strategies. In economics, it is used to model markets, competition, and bargaining. In biology, it helps explain the evolution of cooperation and conflict among organisms.

Mathematical Formulation

The mathematical analysis of games involves defining the game's components, such as the set of players, the set of actions available to each player, the payoffs for each combination of actions, and the information available to each player at the time of decision-making.

Limitations

While game theory provides valuable insights, it has limitations. It assumes rational behavior and common knowledge among players, which may not always be realistic. Additionally, the complexity of some games makes finding the Nash Equilibrium challenging.

See Also

• Decision theory
• Evolutionary game theory
• Mechanism design
• Von Neumann–Morgenstern utility theorem

References

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  John von Neumann at Los Alamos

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  John F. Nash

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  Prisoner's Dilemma with Outside Option

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  Game Theory

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  Ultimatum Game Extensive Form

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  Centipede Game

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  Cournot Duopoly Nash Equilibrium