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## Game theory
Game theory is closely related to economics in that it seeks to find rational strategies in situations where the outcome depends not only on one's own strategy and "market conditions", but upon the strategies chosen by other players with possibly different or overlapping goals. It also finds wider application in fields such as political science and military strategy.
The results can be applied to simple games of entertainment or to more significant aspects of life and society. An example of the application of game theory to real life is the prisoner's dilemma as popularized by mathematician Albert W. Tucker; it has many implications for the nature of human cooperation. Biologists have used game theory to understand and predict certain outcomes of evolution, such as the concept of evolutionarily stable strategy introduced by John Maynard Smith in his essay Other branches of mathematics, in particular probability, statistics and linear programming, are commonly used in conjunction with game theory to analyse games.
## Types of games and examples
A convenient way to represent a game is given by its payoff matrix. Consider for example the two-player zero-sum game with the following matrix:
This game is played as follows: the first player chooses one of the two actions 1 or 2, and the second player, unaware of the first player's choice, chooses one of the three actions A, B or C. Once these choices have been made, the payoff is allocated according to the table; for instance, if the first player chose action 2 and the second player chose action B, then the first player gains 20 points and the second player loses 20 points. Both players know the payoff matrix and attempt to maximize the number of their points. What should they do? Player 1 could reason as follows: "with action 2, I could lose up to 20 points and can win only 20, while with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, player 2 would choose action C (negative numbers in the table are good for him). If both players take these actions, the first player will win 20 points. But how about if player 2 anticipates the first player's reasoning and choice of action 1, and deviously goes for action B, so as to win 10 points? Or if the first player in turn anticipates this devious trick and goes for action 2, so as to win 20 points after all? The fundamental and surprising insight by John von Neumann was that probability provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. The probabilities are computed so as to maximize the expected point gain independent of the opponent's strategy; this leads to a linear programming problem with a unique solution for each player. This method can compute provably optimal strategies for all two-player zero-sum games. For the example given above, it turns out that the first player should chose action 1 with probability 57% and action 2 with 43%, while the second player should assign the probabilities 0%, 57% and 43% to the three actions A, B and C. Player one will then win 2.85 points on average per game.
## Risk aversion
For the above example to work, the participants in the game have to be assumed to be One example of risk aversion can be seen on Game Shows. For example, if a person has a 1 in 3 chance of winning $50,000, or can take a sure $10,000, many people will take the sure $10,000. ## Games and numbersJohn Conway developed a notation for certain games and defined several operations on those games, originally in order to study Go endgames. In a surprising connection, he found that a certain subclass of these games can be used as numbers, leading to the very general class of surreal numbers. ## History
Though touched on by earlier mathematical results, modern Around 1950, John Nash developed a definition of an "optimum" strategy for multi player games where no such optimum was previously defined, known as Nash equilibrium. This concept was further refined by Reinhard Selten. These men were awarded The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel in 1994 for their work on game theory, along with John Harsanyi who developed the analysis of games of incomplete information. Conway's number-game connection was found in the early 1970s. See also Mathematical game; Artificial intelligence; Newcomb's paradox; Game classification. ## External links and references- Paul Walker, An Outline of the History of Game Theory.
- Oskar Morgenstern, John von Neumann:
*The Theory of Games and Economic Behavior, 3rd ed.*, Princeton University Press 1953 - Alvin Roth:
*Game Theory and Experimental Economics page*, http://www.economics.harvard.edu/~aroth/alroth.html Comprehensive list of links to game theory information on the Web - Mike Shor:
*Game Theory .net*, http://www.gametheory.net Lecture notes, interactive illustrations and other information. - Maynard Smith:
*Evolution and the Theory of Games*, Cambridge University Press 1982 - Don Ross: Review Of Game Theory.
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