The game theory (English game theory) is a subsection of mathematics, which is occupied with the modelling and investigation of society plays, of interaction systems society-play-similar in the broadest sense as well as with in these assigned play strategies. The game theory is thereby less a coherent theory than more a sentence of analysis instruments. The game theory applies particularly in operation the Research, in the economic science, in the political sciences, in the sociology, in the psychology and since the 1980ern also in biology. Occasionally also except-mathematical kinds of the theoretical treatment of the play are called game theory; about Homo compares ludens, and Ludologie.

History

Historical starting point of the game theory is the analysis of society plays by John von Neumann (Hungarian: Neumann in the year 1928. Fast John von Neumann recognized the applicability of the beginning developed by him for the analysis of economic questions, so that 1944 in the book "game theory and economic behavior" (Theory OF Games and Economic Behavior), which he together with Oskar morning star wrote, already a Verquickung between the mathematical theory and economiceconomics application took place. The appearance of this book is regarded generally as starting point of the modern game theory. Worth mentioning that there were play-theoretical analyses already before and parallel to John von Neumann, in particular by Bernoulli, Bertrand, Cournot, is Edgeworth, Zeuthen and of Stackelberg. These play-theoretical analyses were however always answers to specific questions, without a more general theory would have been developed for the analysis of strategic interaction from it.

For play-theoretical work so far five Wirtschaftsnobelpreise were assigned: 1994 at John Forbes Nash Jr., John Harsanyi and pure hard rare, 1996 at William Vickrey and 2005 at Robert Aumann and Thomas Schelling. Also the Nobelpreise for the study of limited at Herbert Simon 1978 and Daniel Kahneman 2002 stands in a close relationship to play-theoretical questions.

Methodology of the game theory

Interaction as play model

The game theory models the most diverse situations as a play. The term play is to be taken quite literally: In the mathematical-formal description one specifies, there are which players, the play has which sequential operational sequence and each player in the individual stages of the sequence has which action options. (With) of plays: In the play Cournot Duopol are the players the companies and their respective action option are their offer quantity. In the play Bertrand Duopol are the players again the duo polists, their action options are the asking prices however here. In the play prisoner's dilemma the players are the two prisoners and their action quantities are state and are silent. In applications of the political science the players are often parties or lobby federations, while in biology the players are mostly genes or species.

To the description of a play belongs besides a disbursement function: This function assigns a disbursement vector to each possible play exit, i.e. one specifies by it, a player makes which profit, if a certain play exit occurs. With applications of the economic science the disbursement is to be mostly understood as monetary size, with politics-scientific applications can it however votes concern, while with biological applications mostly the disbursement consists of reproduction ability or survivability.

One can recognize two important aspects in the game theory:

• Formalizing: An important step is to formalize a play in the sense of the game theory. The game theory developed for this a rich language. See under: Play representation
• Solution: Dependent on the context one can try a forecast of the play exit in a further step. See for this: Approaches (game theory).

An important technology when finding equilibrium in the game theory is regarding fixed points.

In computer science one tries, by search strategies and heuristics (general: ) Plays, like chess, determined techniques of of the combinatorial one optimization and artificial intelligence SameGame, Awari, solve or e.g. prove Go that that, which begins always wins with correct strategy (that is e.g. the case for four wins, Qubic and five into a row) or e.g. that, that the 2. Course has, always at least an undecided to obtain can (example mill).

Approaches

As soon as a play is defined, one can use then the analysis equipment of the game theory, in order to determine for example, which are the optimal strategies for all players and which result will have the play, if these strategies are used. (Under the strategy of a player one understands a function, which assigns an element from the action quantity to each game situation, in which the action quantity of this player is nonempty. One calls a quantity, in which for each player exactly one strategy is contained, strategy profile.)

In order to analyze questions play-theoretically, so-called approaches are used. The by far most prominent such approach, the Nash equilibrium, comes from John Forbes Nash Jr. (1951). The above question - a play has which possible exits, if all players behave individually optimally - can by the determination of the Nash equilibrium of a play be answered: The quantity of the Nashgleichgewichte of a play contains those strategy profiles by definition, in which an individual player could not improve by exchange of his strategy by another strategy with given strategies of the other players.

For other questions there are other approaches. Important ones are the min max equilibrium, repeated capers of dominated strategies as well as partial play perfection and in the cooperative game theory the core, the nucleolus, the Shapley VALUE, the negotiation quantity and the Imputationsmenge.

Mixed one vs. pure strategies

While the pure is strategy of a player a function, which assigns to each play stage, in which the action quantity of the player is nonempty, an action, is a mixed strategy a function, which assigns a probability distribution over the action quantity to each available in this play stage play stage, in which the action quantity of the player is nonempty. Thus a pure strategy is the special case of a mixed strategy, in that whenever the action quantity is nonempty player, which is put entire probability mass on only one action of the action quantity. One can show easily that each play, whose action quantities are finite must have an Nash equilibrium in mixed strategies. In pure strategies the existence of an Nash equilibrium is not however ensured for many plays. The analysis strategies mixed by equilibrium in was gotten going substantially by a set of contributions John Harsanyis in and the 80's 70's.

Articles in category "Game theory"

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Related Websites

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• Game theory - Wikipedia, the free encyclopedia
  Game theory is a branch of applied mathematics that studies strategic situations ... The games studied by game theory are well-defined mathematical objects. ...


• Game Theory .net
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• Game Theory: An Introductory Sketch
  by Roger McCain at Drexel University.


• History of Game Theory
  An outline of the history of game theory by Paul Walker.