Game theory

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 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 1980ernalso in biology. Occasionally also except-mathematical kinds of the theoretical treatment of the play are called game theory; about Homo compares ludens, Spielpädagogik and Ludologie.

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history

historical starting pointthe game theory is the analysis of society plays by John von Neumann (Hungarian: Neumann Janos) 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 andeconomic behavior " (Theory OF Games and Economic Behavior), which it 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 six Wirtschaftsnobelpreise were assigned: 1994 at John Forbes Nash Jr., John Harsanyi and pure hard rare, 1996 on William Vickrey and 2005at Robert Aumann and Thomas Schelling. Also the Nobelpreise for the study of limited Rationalität at harsh ore Simon 1978 and Daniel Kahneman 2002 stands in a close relationship to play-theoretical questions.

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methodology of the game theory

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The game theory model

the most diverse situations model interaction as play 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 whichAction options each player in the individual stages of the sequence has. (With) of plays: In the play Cournot - duo pole are the players the companies and their respective action option is their offer quantity. In the play Bertrand - duo pole are the players again the duo polists, their action options are but here the asking prices. 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, during in biology the players mostly genesor species are.

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

One can recognize two important aspects in the 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: Techniques of the combinatorial optimization and artificial intelligence) plays, like chess, determined SameGame, Awari, solve Go or e.g. to prove that that, which begins always wins with correct strategy (that e.g. is. the case for four wins, Qubic and five into oneRow) or e.g. that, that the 2. Course has, always at least an undecided to obtain can (example mill).

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approaches

as soon as a play one is defined, can then the analysis equipment of the game theory use, 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, each game situation, in that the action quantity of this player nonemptyis, an element from the action quantity assigns. One calls a quantity, in which for each player exactly one strategy is contained, strategy profile.)

around questions play-theoretically to analyze, so-called approaches is 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 oneContains those strategy profiles of play 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 that Min max equilibrium, repeated capers of dominanter 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.

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mixing vs. pure strategies

during the pure strategyplayers function is, 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 each play stage, in which the action quantity of the player is nonempty, a probability distribution over thatin this play stage assigns available action quantity. 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 easily it shows 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 became substantially througha set of contributions John Harsanyis in and the 80's 70's gotten going.

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representations of a play

of plays are described either in strategic (normal) form or in extensive form. The extensive form comes particularly with sequential plays toEmployment, while the strategic form with most single-step plays such as prisoner's dilemma, Ultimatumspiel, Chicken Game among other things one uses. In principle however each play in both forms can be described. An explicitly multi-level play in the strategic form is represented,thus one speaks of pseudostrategic form, while the representation of a single-step play in extensive form is called pseudoextensive form.

One can illustrate plays in strategic form with a disbursement matrix. In a two-dimensional disbursement matrix the lines correspond to the actionsor strategies of the first player (“line player”) and the columns the actions or strategies of the second player (“column player”). The fields of the matrix contain then information about it, which disbursements contain the two players, if the appropriate line and column are played.

Underaction player understands one action player in game situation, whereas one under strategy function understands, which to each game situation, in which a player assigns has a nonempty action quantity, an action from this quantity. IfPlay from only one game situation exists, then strategy and action collapse obviously. Since disbursement stencils are used particularly for the representation by single-step plays, there is mostly actions, for the lines and columns of the matrix. If the playis however not multi-level and the lines and columns of the matrix for strategies for actions, has one it with a pseudostrategic play representation to do.

One can illustrate plays in extensive form with arranged graphs. Each knot is, ofthat go off arrows, one point, in which a player can transact an action, while arrows the going off this knot represent the actions standing to the selection. If two knots A and B by an arrow x from A to Bit is connected then this means that by the selection of the action x by the player, who had a nonempty action quantity in the game situation A the point B was reached, where possibly another or the same player again an actionto select must. A knot, which no arrow goes off, marks a game situation, in which for no more player an action quantity stands to the selection - therefore the play ends. At this knot then mostly stand the disbursements, which receive all players,if this play exit is reached. One calls such a graph play tree and this kind of the representation a tree representation. A player function is one there on the quantity of the knots of the play tree defined function, which assigns a player to each knot, thatto pull may. A strategy of a player can be regarded then as a function, which is defined on the knots, to which this player is assigned by the player function. It forms everyone this knot on an element in this knot of the availableAction quantity off, is thus an illustration of this knot quantity into the set union of all action quantities of this player. A strategy profile can be regarded to that as one on all knots of the play tree defined function, which assigns an element of that activity space to each knot, thathas players at this knot, assigned by the player function this knot. A strategy profile is then an illustration of the quantity of the knots into the set union of all activity spaces of all players.

With many plays in extensive form is assumed playersin certain situations do not know, which history had the play so far: They do not know clearly in which knots them are. If one wants to clarify this in a play tree graphically, then one draws a connected line around the quantity for thatKnot, from which the player does not know, in which it is. He knows only that he is in one of this knots. The range encircled by this line is called information area.

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unique ones vs. repeated plays

A play, which is not repeated after unique execution, becomes as so-called.One SHOT Game designates. If a One SHOT Game is accomplished several times one behind the other, whereby generally the total disbursement for each player arises as a result of (possibly up-discounted) the disbursements of each individual One SHOT Games, then speaksone of a repeated play. If a repeated play has infinitely many repetitions, then one calls it superplay.

The analysis of repeated plays became substantially of Robert J. Aumann gotten going.

Important element of many repeated plays is the so-called Backward Breakdown, which result can, if the players know, how many rounding the repeated play has.

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cooperative ones vs. Nichtkooperative game theory

the distinction between cooperative and not-cooperative game theory is much blurred. Often the following definition is used: Can the playersbinding contracts lock, then one speaks of cooperative game theory. However all behaviors (thus also a possible co-operation between players) are self enforcing, i.e. they result from the egoistic behavior of the players, without binding contracts can be locked, thusone speaks of not-cooperative game theory.

Nearly all prominent examples, which admits also laymen is, as for example the prisoner's dilemma, the play Battle OF Sexes etc., come of to the not-cooperative game theory. The not-cooperative game theory displaces the cooperative game theory for some decades inincreasing measure, in particular in the teachings of universities. Nowadays many text books to the game theory and it appear give at universities many meetings with the title game theory, in which the cooperative game theory no more is not only treated or at the edge.Although the Nobelpreisträger Robert J. Aumann and John Forbes Nash Jr. both crucial contributions to the cooperative game theory carried out, the price by the Nobelpreiskommitee expressly for their contributions to the not-cooperative game theory were assigned.

Nothing the defiance becomes in the current research further thosecooperative game theory examines, and a majority new play-theoretical scientific article are to be assigned to the cooperative game theory.

The further great importance of the cooperative game theory in the research is to be read off also from the fact that in the scientific discussion very present research fields like the negotiation theory and the Matching theory to a large part with the means of the cooperative game theory to be analyzed.

See also Nash Program.

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game theory in the Mechanism Design

the game theory is among other things also used, around the exit of certain rule-referred processesto determine or specify. This happens in the course of the solutions for a Mechanism Design problem. This procedure can be used not only for “pure” plays, but also for the behavior by groups in economics and society.

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Examples

famous problems of the game theory:

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see also

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to literature

  • Robert Axelrod: The evolution OF Cooperation, 1985, ISBN 0465021212 dt. The evolution of co-operation ISBN 3486539957
  • Christian Rieck:Game theory - an introduction. 5. Edition 2005, ISBN 3-924043-91-4. Very good introduction, didactically descriptive written. Explained also background to the concepts.
  • Avinash K. Dixit, Barry J. Nalebuff: Game theory for a risers. Strategic know-how for winners. 1991, ISBN 3-7910-1239-8 (American original title: Thinking Strategically. The Competitive Edge in Business, Politics, and Everyday would run, ISBN 0393974219.) - easily readable introduction to the game theory
  • Jörg Bewersdorff: Luck, logic and bluff: Mathematics in the play - methods, results and borders. 2003, ISBN 3528269979 - which states the game theory over correct plays (relatively elementarily, many historical details, no economics)
  • Manfred own, Ruthild Winkler: The play, 1976, ISBN 3492021514
  • Morton D. Davis, Dietmar red ago mouth: Game theory for Nichtmathematiker, 1999, ISBN 348656448X
  • Drew Fudenberg, Jean Tirole:Game Theory. 1991, ISBN 0-262-06141-4 - Zurzeit the standard text book of the game theory - at least for economists. All bases comprehensively, very precisely, strongly mathematically formalizes, for the entrance less suitably.
  • Shaun P. Hargreaves Heap, Yanis Varoufakis: GameTheory - A Critical text, 2004, ISBN 0415250951 (the book describes not only the theories, but also her meaning)
  • walter Schlee, introduction to the game theory, 2004, ISBN 3528032146 (“strictly mathematically”)
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