Unstimmigkeiten über die Spielregeln, etwa, ob bei Mensch ärgere Dich nicht die Pflicht besteht, einen gegnerischen Kegel zu schlagen, wenn dies im betreffenden Zug möglich ist, oder ob bei Mau Mau eine gezogene Karte sofort gelegt werden darf, wenn sie passt, werden in der Regel als ernsthafte Störung betrachtet, wenn sie nicht vor dem Spiel geklärt wurden.

Andererseits wird die Spieltheorie auf viele Situationen angewendet, für die dieses Informationserfordernis zu rigide wäre, da mit dem Vorhandensein gewisser Informationen nicht gerechnet werden kann z.

Darum ist es sinnvoll, die klassische Spieltheorie, die mit vollständiger Information arbeitet, um die Möglichkeit unvollständiger Information zu erweitern.

Andererseits ist dieses Feld dadurch begrenzt, weil sich für jedes Spiel mit unvollständiger Information ein Spiel mit vollständiger Information konstruieren lässt, das strategisch äquivalent ist.

Perfekte Information , also die Kenntnis sämtlicher Spieler über sämtliche Züge sämtlicher Spieler, ist eine rigorose Forderung, die in vielen klassischen Spielen nicht erfüllt ist: Sie ist beispielsweise in den meisten Kartenspielen dadurch verletzt, weil zu Spielbeginn der Zug des Zufallsspielers und die Verteilung der Blätter unbekannt ist, da man jeweils nur die eigenen Karten einsehen kann.

Darum wird in spieltheoretischen Modellen meist nicht von perfekter Information ausgegangen. Perfektes Erinnerungsvermögen ist das Wissen jedes Spielers über sämtliche Informationen, die ihm bereits in der Vergangenheit zugänglich waren.

Obwohl diese Annahme zumindest vom Prinzip her auf den ersten Blick immer erfüllt zu sein scheint, gibt es Gegenbeispiele: Spiele werden meist entweder in strategischer Normal- Form oder in extensiver Form beschrieben.

Weiterhin ist noch die Agentennormalform zu nennen. Da es Spiele gibt, denen keine dieser Formen gerecht wird, muss bisweilen auf allgemeinere mathematische oder sprachliche Beschreibungen zurückgegriffen werden.

Die Extensivform bezeichnet in der Spieltheorie eine Darstellungsform von Spielen , die sich auf die Baumdarstellung zur Veranschaulichung der zeitlichen Abfolge von Entscheidungen stützt.

Die Normalform beschränkt sich im Wesentlichen auf die A-priori- Strategiemengen der einzelnen Spieler und eine Auszahlungsfunktion als Funktion der gewählten Strategiekombinationen.

Gerecht wird diese Darstellungsform am ehesten solchen Spielen, bei denen alle Spieler ihre Strategien zeitgleich und ohne Kenntnis der Wahl der anderen Spieler festlegen.

Zur Veranschaulichung verwendet man meist eine Bimatrixform. Wer oder was ist eigentlich ein Spieler in einer gegebenen Situation? Die Agentennormalform beantwortet diese Frage so: Jeder Zug im Verlauf eines Spiels verlangt nach einem Spieler im Sinne eines unabhängigen Entscheiders, da die lokale Interessenlage einer Person oder Institution von Informationsbezirk zu Informationsbezirk divergieren kann.

Dazu verfügt die Agentennormalform generell über so viele Spieler bzw. Agenten, wie es Informationsbezirke persönlicher Spieler gibt.

Sobald ein Spiel definiert ist, kann man sodann das Analyseinstrumentarium der Spieltheorie anwenden, um beispielsweise zu ermitteln, welche die optimalen Strategien für alle Spieler sind und welches Ergebnis das Spiel haben wird, falls diese Strategien zur Anwendung kommen.

Die obige Fragestellung — welche möglichen Ausgänge ein Spiel hat, wenn sich alle Spieler individuell optimal verhalten — kann durch die Ermittlung der Nash-Gleichgewichte eines Spiels beantwortet werden: Die Menge der Nash-Gleichgewichte eines Spiels enthält per Definition diejenigen Strategieprofile, in denen sich ein einzelner Spieler durch Austausch seiner Strategie durch eine andere Strategie bei gegebenen Strategien der anderen Spieler nicht verbessern könnte.

Für andere Fragestellungen gibt es andere Lösungskonzepte. Wichtige sind das Minimax-Gleichgewicht , das wiederholte Streichen dominierter Strategien sowie Teilspielperfektheit und in der kooperativen Spieltheorie der Core, der Nucleolus , die Verhandlungsmenge und die Imputationsmenge.

Während die reine Strategie eines Spielers eine Funktion ist, die jeder Spielstufe, in der die Aktionsmenge des Spielers nicht leer ist, eine Aktion zuordnet, ist eine gemischte Strategie eine Funktion, die jeder Spielstufe, in der die Aktionsmenge des Spielers nichtleer ist, eine Wahrscheinlichkeitsverteilung über der in dieser Spielstufe verfügbaren Aktionsmenge zuordnet.

Damit ist eine reine Strategie der Spezialfall einer gemischten Strategie, in der immer dann, wenn die Aktionsmenge eines Spielers nichtleer ist, die gesamte Wahrscheinlichkeitsmasse auf eine einzige Aktion der Aktionsmenge gelegt wird.

Man kann leicht zeigen, dass jedes Spiel, dessen Aktionsmengen endlich sind, ein Nash-Gleichgewicht in gemischten Strategien haben muss.

In reinen Strategien ist die Existenz eines Nash-Gleichgewichtes hingegen für viele Spiele nicht gewährleistet. Die Analyse von Gleichgewichten in gemischten Strategien wurde wesentlich durch eine Reihe von Beiträgen John Harsanyis in den 70er und 80er Jahren vorangebracht.

Im Folgenden sollen auf der Basis der beschriebenen Spielformen und deren Lösungskonzepte einige Probleme genannt werden, die sich in der spieltheoretischen Behandlung als besonders einflussreich erwiesen haben.

Ein Spiel, das nach einmaliger Durchführung nicht wiederholt wird, wird als sogenanntes One-Shot-Game bezeichnet. Wird ein One-Shot-Game mehrmals hintereinander durchgeführt, wobei sich im Allgemeinen die Gesamtauszahlung für jeden Spieler durch die eventuell aufdiskontierten Auszahlungen jedes einzelnen One-Shot-Games ergibt, so spricht man von einem wiederholten Spiel.

In der Spieltheorie unterscheidet man zudem zwischen endlich wiederholten und unendlich wiederholten Superspielen.

Die Analyse wiederholter Spiele wurde wesentlich von Robert J. Ein Lösungskonzept vieler endlich wiederholter Spiele ist die sogenannte Rückwärtsinduktion , indem zunächst die Lösung des letzten One-Shot-Games ermittelt und darauf basierend die Lösungen der vorangegangenen Spiele bis zum ersten Spiel bestimmt werden.

Kennt ein Spieler selbst nur seinen eigenen Typ, während andere nur diesbezügliche probabilistische Erwartungen hegen, so spricht man von unvollständiger, speziell asymmetrischer Information.

Reputationseffekte treten immer dann auf, wenn ein Spieler für andere als einem bestimmten Typ zugehörig identifiziert werden kann.

Die Spieltheorie unterstellt zunächst nicht nur jedem Spieler Rationalität, sondern auch, dass alle Spieler wissen, dass alle Spieler rational sind etc ….

Man unterstellt also allgemein bekannte Spielregeln, bzw. Von evolutionärer Spieltheorie spricht man meist dann, wenn das Verhalten der Spieler nicht durch rationale Entscheidungskalküle abgeleitet wird, sondern als Ergebnis von kulturellen oder genetischen Evolutionsprozessen begründet wird.

Oft kann man die stabilen Ergebnisse durch statische Stabilitätskonzepte charakterisieren. Evolutionstheoretisch besagt diese Spieltheorie, dass jeweils nur die am besten angepasste Strategie bzw.

Die Spieltheorie untersucht, wie rationale Spieler ein gegebenes Spiel spielen. In der Mechanismus-Designtheorie wird diese Fragestellung jedoch umgekehrt, und es wird versucht, zu einem gewollten Ergebnis ein entsprechendes Spiel zu entwerfen, um den Ausgang bestimmter regelbezogener Prozesse zu bestimmen oder festzulegen.

Dieses Vorgehen kann nicht nur für "reine" Spiele, sondern auch für das Verhalten von Gruppen in Wirtschaft und Gesellschaft genutzt werden.

Die Spieltheorie erlaubt es, soziale Konfliktsituationen, die strategische Spiele genannt werden, facettenreich abzubilden und mathematisch streng zu lösen.

Aufgrund der unrealistischen Modellannahmen wird die empirische Erklärungskraft der Spieltheorie in der Regel in Abrede gestellt.

Kein Mensch wird jemals so rational sein, wie es den Spielern durch die spieltheoretischen Lösungskonzepte unterstellt wird. Dieser Artikel beschreibt die Spieltheorie als Teilgebiet der Mathematik.

Zur Erforschung von Spielen siehe Spielwissenschaft. Allgemeine Teilgebiete der Kybernetik. Sequential games or dynamic games are games where later players have some knowledge about earlier actions.

This need not be perfect information about every action of earlier players; it might be very little knowledge. The difference between simultaneous and sequential games is captured in the different representations discussed above.

Often, normal form is used to represent simultaneous games, while extensive form is used to represent sequential ones. The transformation of extensive to normal form is one way, meaning that multiple extensive form games correspond to the same normal form.

Consequently, notions of equilibrium for simultaneous games are insufficient for reasoning about sequential games; see subgame perfection. An important subset of sequential games consists of games of perfect information.

A game is one of perfect information if all players know the moves previously made by all other players.

Most games studied in game theory are imperfect-information games. Many card games are games of imperfect information, such as poker and bridge.

Games of incomplete information can be reduced, however, to games of imperfect information by introducing " moves by nature ".

Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games.

Examples include chess and go. Games that involve imperfect information may also have a strong combinatorial character, for instance backgammon.

There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve particular problems and answer general questions.

Games of perfect information have been studied in combinatorial game theory , which has developed novel representations, e. These methods address games with higher combinatorial complexity than those usually considered in traditional or "economic" game theory.

A related field of study, drawing from computational complexity theory , is game complexity , which is concerned with estimating the computational difficulty of finding optimal strategies.

Research in artificial intelligence has addressed both perfect and imperfect information games that have very complex combinatorial structures like chess, go, or backgammon for which no provable optimal strategies have been found.

The practical solutions involve computational heuristics, like alpha-beta pruning or use of artificial neural networks trained by reinforcement learning , which make games more tractable in computing practice.

Games, as studied by economists and real-world game players, are generally finished in finitely many moves. Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner or other payoff not known until after all those moves are completed.

The focus of attention is usually not so much on the best way to play such a game, but whether one player has a winning strategy. The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory.

Much of game theory is concerned with finite, discrete games, that have a finite number of players, moves, events, outcomes, etc. Many concepts can be extended, however.

Continuous games allow players to choose a strategy from a continuous strategy set. For instance, Cournot competition is typically modeled with players' strategies being any non-negative quantities, including fractional quantities.

Differential games such as the continuous pursuit and evasion game are continuous games where the evolution of the players' state variables is governed by differential equations.

The problem of finding an optimal strategy in a differential game is closely related to the optimal control theory.

In particular, there are two types of strategies: A particular case of differential games are the games with a random time horizon.

Therefore, the players maximize the mathematical expectation of the cost function. It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval.

Evolutionary game theory studies players who adjust their strategies over time according to rules that are not necessarily rational or farsighted.

Such rules may feature imitation, optimization or survival of the fittest. In biology, such models can represent biological evolution , in which offspring adopt their parents' strategies and parents who play more successful strategies i.

In the social sciences, such models typically represent strategic adjustment by players who play a game many times within their lifetime and, consciously or unconsciously, occasionally adjust their strategies.

Individual decision problems with stochastic outcomes are sometimes considered "one-player games". These situations are not considered game theoretical by some authors.

Although these fields may have different motivators, the mathematics involved are substantially the same, e. Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" " moves by nature ".

For some problems, different approaches to modeling stochastic outcomes may lead to different solutions. For example, the difference in approach between MDPs and the minimax solution is that the latter considers the worst-case over a set of adversarial moves, rather than reasoning in expectation about these moves given a fixed probability distribution.

The minimax approach may be advantageous where stochastic models of uncertainty are not available, but may also be overestimating extremely unlikely but costly events, dramatically swaying the strategy in such scenarios if it is assumed that an adversary can force such an event to happen.

General models that include all elements of stochastic outcomes, adversaries, and partial or noisy observability of moves by other players have also been studied.

The " gold standard " is considered to be partially observable stochastic game POSG , but few realistic problems are computationally feasible in POSG representation.

These are games the play of which is the development of the rules for another game, the target or subject game. Metagames seek to maximize the utility value of the rule set developed.

The theory of metagames is related to mechanism design theory. The term metagame analysis is also used to refer to a practical approach developed by Nigel Howard.

Subsequent developments have led to the formulation of confrontation analysis. These are games prevailing over all forms of society.

Pooling games are repeated plays with changing payoff table in general over an experienced path and their equilibrium strategies usually take a form of evolutionary social convention and economic convention.

Pooling game theory emerges to formally recognize the interaction between optimal choice in one play and the emergence of forthcoming payoff table update path, identify the invariance existence and robustness, and predict variance over time.

The theory is based upon topological transformation classification of payoff table update over time to predict variance and invariance, and is also within the jurisdiction of the computational law of reachable optimality for ordered system.

Mean field game theory is the study of strategic decision making in very large populations of small interacting agents. This class of problems was considered in the economics literature by Boyan Jovanovic and Robert W.

Rosenthal , in the engineering literature by Peter E. The games studied in game theory are well-defined mathematical objects. To be fully defined, a game must specify the following elements: These equilibrium strategies determine an equilibrium to the game—a stable state in which either one outcome occurs or a set of outcomes occur with known probability.

Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.

The extensive form can be used to formalize games with a time sequencing of moves. Games here are played on trees as pictured here.

Here each vertex or node represents a point of choice for a player. The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player.

The payoffs are specified at the bottom of the tree. The extensive form can be viewed as a multi-player generalization of a decision tree. It involves working backward up the game tree to determine what a rational player would do at the last vertex of the tree, what the player with the previous move would do given that the player with the last move is rational, and so on until the first vertex of the tree is reached.

The game pictured consists of two players. The way this particular game is structured i. Next in the sequence, Player 2 , who has now seen Player 1 ' s move, chooses to play either A or R.

Suppose that Player 1 chooses U and then Player 2 chooses A: Player 1 then gets a payoff of "eight" which in real-world terms can be interpreted in many ways, the simplest of which is in terms of money but could mean things such as eight days of vacation or eight countries conquered or even eight more opportunities to play the same game against other players and Player 2 gets a payoff of "two".

The extensive form can also capture simultaneous-move games and games with imperfect information. To represent it, either a dotted line connects different vertices to represent them as being part of the same information set i.

See example in the imperfect information section. The normal or strategic form game is usually represented by a matrix which shows the players, strategies, and payoffs see the example to the right.

More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions.

In the accompanying example there are two players; one chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns.

The payoffs are provided in the interior. The first number is the payoff received by the row player Player 1 in our example ; the second is the payoff for the column player Player 2 in our example.

Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff of 4, and Player 2 gets 3.

When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other.

If players have some information about the choices of other players, the game is usually presented in extensive form. Every extensive-form game has an equivalent normal-form game, however the transformation to normal form may result in an exponential blowup in the size of the representation, making it computationally impractical.

In games that possess removable utility, separate rewards are not given; rather, the characteristic function decides the payoff of each unity.

The idea is that the unity that is 'empty', so to speak, does not receive a reward at all. The balanced payoff of C is a basic function.

Although there are differing examples that help determine coalitional amounts from normal games, not all appear that in their function form can be derived from such.

Formally, a characteristic function is seen as: N,v , where N represents the group of people and v: Such characteristic functions have expanded to describe games where there is no removable utility.

As a method of applied mathematics , game theory has been used to study a wide variety of human and animal behaviors. It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers.

The first use of game-theoretic analysis was by Antoine Augustin Cournot in with his solution of the Cournot duopoly. The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well.

Although pre-twentieth century naturalists such as Charles Darwin made game-theoretic kinds of statements, the use of game-theoretic analysis in biology began with Ronald Fisher 's studies of animal behavior during the s.

This work predates the name "game theory", but it shares many important features with this field. The developments in economics were later applied to biology largely by John Maynard Smith in his book Evolution and the Theory of Games.

In addition to being used to describe, predict, and explain behavior, game theory has also been used to develop theories of ethical or normative behavior and to prescribe such behavior.

Game-theoretic arguments of this type can be found as far back as Plato. The primary use of game theory is to describe and model how human populations behave.

This particular view of game theory has been criticized. It is argued that the assumptions made by game theorists are often violated when applied to real-world situations.

Game theorists usually assume players act rationally, but in practice, human behavior often deviates from this model.

Game theorists respond by comparing their assumptions to those used in physics. Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicists.

There is an ongoing debate regarding the importance of these experiments and whether the analysis of the experiments fully captures all aspects of the relevant situation.

Price , have turned to evolutionary game theory in order to resolve these issues. These models presume either no rationality or bounded rationality on the part of players.

Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning for example, fictitious play dynamics.

Some scholars, like Leonard Savage , [ citation needed ] see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave.

Since a strategy, corresponding to a Nash equilibrium of a game constitutes one's best response to the actions of the other players — provided they are in the same Nash equilibrium — playing a strategy that is part of a Nash equilibrium seems appropriate.

This normative use of game theory has also come under criticism. Game theory is a major method used in mathematical economics and business for modeling competing behaviors of interacting agents.

This research usually focuses on particular sets of strategies known as "solution concepts" or "equilibria". A common assumption is that players act rationally.

In non-cooperative games, the most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies.

If all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing.

The payoffs of the game are generally taken to represent the utility of individual players. A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of a particular economic situation.

One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type.

Naturally one might wonder to what use this information should be put. Economists and business professors suggest two primary uses noted above: The application of game theory to political science is focused in the overlapping areas of fair division , political economy , public choice , war bargaining , positive political theory , and social choice theory.

In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians.

Early examples of game theory applied to political science are provided by Anthony Downs. In his book An Economic Theory of Democracy , [52] he applies the Hotelling firm location model to the political process.

In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space.

Downs first shows how the political candidates will converge to the ideology preferred by the median voter if voters are fully informed, but then argues that voters choose to remain rationally ignorant which allows for candidate divergence.

Game Theory was applied in to the Cuban missile crisis during the presidency of John F. It has also been proposed that game theory explains the stability of any form of political government.

Taking the simplest case of a monarchy, for example, the king, being only one person, does not and cannot maintain his authority by personally exercising physical control over all or even any significant number of his subjects.

Sovereign control is instead explained by the recognition by each citizen that all other citizens expect each other to view the king or other established government as the person whose orders will be followed.

Coordinating communication among citizens to replace the sovereign is effectively barred, since conspiracy to replace the sovereign is generally punishable as a crime.

Thus, in a process that can be modeled by variants of the prisoner's dilemma , during periods of stability no citizen will find it rational to move to replace the sovereign, even if all the citizens know they would be better off if they were all to act collectively.

A game-theoretic explanation for democratic peace is that public and open debate in democracies sends clear and reliable information regarding their intentions to other states.

In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy.

On the other hand, game theory predicts that two countries may still go to war even if their leaders are cognizant of the costs of fighting. War may result from asymmetric information; two countries may have incentives to mis-represent the amount of military resources they have on hand, rendering them unable to settle disputes agreeably without resorting to fighting.

Moreover, war may arise because of commitment problems: Finally, war may result from issue indivisibilities. Game theory could also help predict a nation's responses when there is a new rule or law to be applied to that nation.

One example would be Peter John Wood's research when he looked into what nations could do to help reduce climate change.

Wood thought this could be accomplished by making treaties with other nations to reduce greenhouse gas emissions.

However, he concluded that this idea could not work because it would create a prisoner's dilemma to the nations. Unlike those in economics, the payoffs for games in biology are often interpreted as corresponding to fitness.

In addition, the focus has been less on equilibria that correspond to a notion of rationality and more on ones that would be maintained by evolutionary forces.

Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium , every ESS is a Nash equilibrium.

In biology, game theory has been used as a model to understand many different phenomena. It was first used to explain the evolution and stability of the approximate 1: Fisher suggested that the 1: Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication.

For example, the mobbing behavior of many species, in which a large number of prey animals attack a larger predator, seems to be an example of spontaneous emergent organization.

Ants have also been shown to exhibit feed-forward behavior akin to fashion see Paul Ormerod 's Butterfly Economics.

Biologists have used the game of chicken to analyze fighting behavior and territoriality. According to Maynard Smith, in the preface to Evolution and the Theory of Games , "paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behaviour for which it was originally designed".

Evolutionary game theory has been used to explain many seemingly incongruous phenomena in nature. One such phenomenon is known as biological altruism.

This is a situation in which an organism appears to act in a way that benefits other organisms and is detrimental to itself.

This is distinct from traditional notions of altruism because such actions are not conscious, but appear to be evolutionary adaptations to increase overall fitness.

Examples can be found in species ranging from vampire bats that regurgitate blood they have obtained from a night's hunting and give it to group members who have failed to feed, to worker bees that care for the queen bee for their entire lives and never mate, to vervet monkeys that warn group members of a predator's approach, even when it endangers that individual's chance of survival.

Evolutionary game theory explains this altruism with the idea of kin selection. Altruists discriminate between the individuals they help and favor relatives.

The more closely related two organisms are causes the incidences of altruism to increase because they share many of the same alleles.

This means that the altruistic individual, by ensuring that the alleles of its close relative are passed on through survival of its offspring, can forgo the option of having offspring itself because the same number of alleles are passed on.

Ensuring that enough of a sibling's offspring survive to adulthood precludes the necessity of the altruistic individual producing offspring.

Similarly if it is considered that information other than that of a genetic nature e. Game theory has come to play an increasingly important role in logic and in computer science.

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Such characteristic functions have expanded to describe games where there is no removable utility. As a method of applied mathematics , game theory has been used to study a wide variety of human and animal behaviors.

It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers.

The first use of game-theoretic analysis was by Antoine Augustin Cournot in with his solution of the Cournot duopoly.

The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well.

Although pre-twentieth century naturalists such as Charles Darwin made game-theoretic kinds of statements, the use of game-theoretic analysis in biology began with Ronald Fisher 's studies of animal behavior during the s.

This work predates the name "game theory", but it shares many important features with this field. The developments in economics were later applied to biology largely by John Maynard Smith in his book Evolution and the Theory of Games.

In addition to being used to describe, predict, and explain behavior, game theory has also been used to develop theories of ethical or normative behavior and to prescribe such behavior.

Game-theoretic arguments of this type can be found as far back as Plato. The primary use of game theory is to describe and model how human populations behave.

This particular view of game theory has been criticized. It is argued that the assumptions made by game theorists are often violated when applied to real-world situations.

Game theorists usually assume players act rationally, but in practice, human behavior often deviates from this model. Game theorists respond by comparing their assumptions to those used in physics.

Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicists.

There is an ongoing debate regarding the importance of these experiments and whether the analysis of the experiments fully captures all aspects of the relevant situation.

Price , have turned to evolutionary game theory in order to resolve these issues. These models presume either no rationality or bounded rationality on the part of players.

Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning for example, fictitious play dynamics.

Some scholars, like Leonard Savage , [ citation needed ] see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave.

Since a strategy, corresponding to a Nash equilibrium of a game constitutes one's best response to the actions of the other players — provided they are in the same Nash equilibrium — playing a strategy that is part of a Nash equilibrium seems appropriate.

This normative use of game theory has also come under criticism. Game theory is a major method used in mathematical economics and business for modeling competing behaviors of interacting agents.

This research usually focuses on particular sets of strategies known as "solution concepts" or "equilibria". A common assumption is that players act rationally.

In non-cooperative games, the most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies.

If all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing.

The payoffs of the game are generally taken to represent the utility of individual players. A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of a particular economic situation.

One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type.

Naturally one might wonder to what use this information should be put. Economists and business professors suggest two primary uses noted above: The application of game theory to political science is focused in the overlapping areas of fair division , political economy , public choice , war bargaining , positive political theory , and social choice theory.

In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians.

Early examples of game theory applied to political science are provided by Anthony Downs. In his book An Economic Theory of Democracy , [52] he applies the Hotelling firm location model to the political process.

In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. Downs first shows how the political candidates will converge to the ideology preferred by the median voter if voters are fully informed, but then argues that voters choose to remain rationally ignorant which allows for candidate divergence.

Game Theory was applied in to the Cuban missile crisis during the presidency of John F. It has also been proposed that game theory explains the stability of any form of political government.

Taking the simplest case of a monarchy, for example, the king, being only one person, does not and cannot maintain his authority by personally exercising physical control over all or even any significant number of his subjects.

Sovereign control is instead explained by the recognition by each citizen that all other citizens expect each other to view the king or other established government as the person whose orders will be followed.

Coordinating communication among citizens to replace the sovereign is effectively barred, since conspiracy to replace the sovereign is generally punishable as a crime.

Thus, in a process that can be modeled by variants of the prisoner's dilemma , during periods of stability no citizen will find it rational to move to replace the sovereign, even if all the citizens know they would be better off if they were all to act collectively.

A game-theoretic explanation for democratic peace is that public and open debate in democracies sends clear and reliable information regarding their intentions to other states.

In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept.

Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy.

On the other hand, game theory predicts that two countries may still go to war even if their leaders are cognizant of the costs of fighting.

War may result from asymmetric information; two countries may have incentives to mis-represent the amount of military resources they have on hand, rendering them unable to settle disputes agreeably without resorting to fighting.

Moreover, war may arise because of commitment problems: Finally, war may result from issue indivisibilities.

Game theory could also help predict a nation's responses when there is a new rule or law to be applied to that nation.

One example would be Peter John Wood's research when he looked into what nations could do to help reduce climate change. Wood thought this could be accomplished by making treaties with other nations to reduce greenhouse gas emissions.

However, he concluded that this idea could not work because it would create a prisoner's dilemma to the nations. Unlike those in economics, the payoffs for games in biology are often interpreted as corresponding to fitness.

In addition, the focus has been less on equilibria that correspond to a notion of rationality and more on ones that would be maintained by evolutionary forces.

Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium , every ESS is a Nash equilibrium.

In biology, game theory has been used as a model to understand many different phenomena. It was first used to explain the evolution and stability of the approximate 1: Fisher suggested that the 1: Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication.

For example, the mobbing behavior of many species, in which a large number of prey animals attack a larger predator, seems to be an example of spontaneous emergent organization.

Ants have also been shown to exhibit feed-forward behavior akin to fashion see Paul Ormerod 's Butterfly Economics. Biologists have used the game of chicken to analyze fighting behavior and territoriality.

According to Maynard Smith, in the preface to Evolution and the Theory of Games , "paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behaviour for which it was originally designed".

Evolutionary game theory has been used to explain many seemingly incongruous phenomena in nature. One such phenomenon is known as biological altruism.

This is a situation in which an organism appears to act in a way that benefits other organisms and is detrimental to itself.

This is distinct from traditional notions of altruism because such actions are not conscious, but appear to be evolutionary adaptations to increase overall fitness.

Examples can be found in species ranging from vampire bats that regurgitate blood they have obtained from a night's hunting and give it to group members who have failed to feed, to worker bees that care for the queen bee for their entire lives and never mate, to vervet monkeys that warn group members of a predator's approach, even when it endangers that individual's chance of survival.

Evolutionary game theory explains this altruism with the idea of kin selection. Altruists discriminate between the individuals they help and favor relatives.

The more closely related two organisms are causes the incidences of altruism to increase because they share many of the same alleles.

This means that the altruistic individual, by ensuring that the alleles of its close relative are passed on through survival of its offspring, can forgo the option of having offspring itself because the same number of alleles are passed on.

Ensuring that enough of a sibling's offspring survive to adulthood precludes the necessity of the altruistic individual producing offspring.

Similarly if it is considered that information other than that of a genetic nature e. Game theory has come to play an increasingly important role in logic and in computer science.

Several logical theories have a basis in game semantics. In addition, computer scientists have used games to model interactive computations.

Also, game theory provides a theoretical basis to the field of multi-agent systems. Separately, game theory has played a role in online algorithms ; in particular, the k-server problem , which has in the past been referred to as games with moving costs and request-answer games.

The emergence of the internet has motivated the development of algorithms for finding equilibria in games, markets, computational auctions, peer-to-peer systems, and security and information markets.

Algorithmic game theory [63] and within it algorithmic mechanism design [64] combine computational algorithm design and analysis of complex systems with economic theory.

Game theory has been put to several uses in philosophy. Responding to two papers by W. In so doing, he provided the first analysis of common knowledge and employed it in analyzing play in coordination games.

In addition, he first suggested that one can understand meaning in terms of signaling games. This later suggestion has been pursued by several philosophers since Lewis.

Game theory has also challenged philosophers to think in terms of interactive epistemology: Philosophers who have worked in this area include Bicchieri , , [69] [70] Skyrms , [71] and Stalnaker Since games like the prisoner's dilemma present an apparent conflict between morality and self-interest, explaining why cooperation is required by self-interest is an important component of this project.

This general strategy is a component of the general social contract view in political philosophy for examples, see Gauthier and Kavka Other authors have attempted to use evolutionary game theory in order to explain the emergence of human attitudes about morality and corresponding animal behaviors.

These authors look at several games including the prisoner's dilemma, stag hunt , and the Nash bargaining game as providing an explanation for the emergence of attitudes about morality see, e.

From Wikipedia, the free encyclopedia. The study of mathematical models of strategic interaction between rational decision-makers.

This article is about the mathematical study of optimizing agents. For the mathematical study of sequential games, see Combinatorial game theory.

For the study of playing games for entertainment, see Game studies. For other uses, see Game theory disambiguation. A supply and demand diagram, illustrating the effects of an increase in demand.

History of economics Schools of economics Mainstream economics Heterodox economics Economic methodology Economic theory Political economy Microeconomics Macroeconomics International economics Applied economics Mathematical economics Econometrics.

Economic systems Economic growth Market National accounting Experimental economics Computational economics Game theory Operations research. Cooperative game and Non-cooperative game.

Simultaneous game and Sequential game. Prior knowledge of opponent's move? Extensive-form game Extensive game. Strategy game Strategic game. List of games in game theory.

Applied ethics Chainstore paradox Collective intentionality Combinatorial game theory Confrontation analysis Glossary of game theory Intra-household bargaining Kingmaker scenario Parrondo's paradox Precautionary principle Quantum game theory Quantum refereed game Rationality Risk management Reverse game theory Self-confirming equilibrium Zermelo's theorem Tragedy of the commons Law and economics.

Analysis of Conflict, Harvard University Press, p. Archived 11 April at the Wayback Machine. An Outline of the History of Economic Thought' 2nd ed.

Game theory applications in network design. Toward a History of Game Theory. From Wikipedia, the free encyclopedia. For other uses, see Metagaming disambiguation.

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