# Game theory

**Game theory**is a branch ofapplied mathematics that is used in thesocial science s (most notablyeconomics ),biology ,engineering ,political science ,computer science (mainly forartificial intelligence ), andphilosophy . Game theory attempts to mathematically capture behavior in "strategic situations", in which an individual's success in making choices depends on the choices of others. While initially developed to analyze competitions in which one individual does better at another's expense (zero sum game s), it has been expanded to treat a wide class of interactions, which are classified according to several criteria. Today, "game theory is a sort of umbrella or 'unified field' theory for the rational side of social science, where 'social' is interpreted broadly, to include human as well as non-human players (computers, animals, plants)" harv|Aumann|1987.Traditional applications of game theory attempt to find

equilibria in these games—sets of strategies in which individuals are unlikely to change their behavior. Manyequilibrium concept s have been developed (most famously theNash equilibrium ) in an attempt to capture this idea. These equilibrium concepts are motivated differently depending on the field of application, although they often overlap or coincide. This methodology is not without criticism, and debates continue over the appropriateness of particular equilibrium concepts, the appropriateness of equilibria altogether, and the usefulness of mathematical models more generally.Although some developments occurred before it, the field of game theory came into being with the 1944 book "

Theory of Games and Economic Behavior " byJohn von Neumann andOskar Morgenstern . This theory was developed extensively in the 1950s by many scholars. Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. Eight game theorists have wonNobel prize s in economics, andJohn Maynard Smith was awarded theCrafoord Prize for his application of game theory to biology.**Representation of games**The games studied in game theory are well-defined mathematical objects. A game consists of a set of players, a set of moves (or strategies) available to those players, and a specification of payoffs for each combination of strategies. Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.

**Extensive form**The extensive form can be used to formalize games with some important order. Games here are often presented as trees (as pictured to the left). 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.

In the game pictured here, there are two players. "Player 1" moves first and chooses either "F" or "U". "Player 2" sees "Player 1"'s move and then chooses "A" or "R". Suppose that "Player 1" chooses "U" and then "Player 2" chooses "A", then "Player 1" gets 8 and "Player 2" gets 2.

The extensive form can also capture simultaneous-move games and games with incomplete information. To represent it, either a dotted line connects different vertices to represent them as being part of the same information set (i.e., the players do not know at which point they are), or a closed line is drawn around them.

**Normal form**Payoff matrix

Name = Normal form or payoff matrix of a 2-player, 2-strategy game

2L = Player 2 chooses "Left"

2R = Player 2 chooses "Right"

1U = Player 1 chooses "Up"

1D = Player 1 chooses "Down"

UL =**4**,**3**

UR =**–1**,**–1**

DL =**0**,**0**

DR =**3**,**4**

Float = right | Width = 330The 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.

**Characteristic function form**In

cooperative game s withtransferable utility no individual payoffs are given. Instead, the characteristic function determines the payoff of each coalition. The standard assumption is that the empty coalition obtains a payoff of 0.The origin of this form is to be found in the seminal book of von Neumann and Morgenstern who, when studying coalitional

normal form game s, assumed that when a coalition $C$ forms, it plays against the complementary coalition ($Nsetminus\; C$) as if they were playing a 2-player game. The equilibrium payoff of $C$ is "characteristic". Now there are different models to derive coalitional values from normal form games, but not all games in characteristic function form can be derived from normal form games.Formally, a characteristic function form game (also known as a TU-game) is given as a pair $(N,v)$, where $N$ denotes a set of players and $v:2^Nlongrightarrowmathbb\{R\}$ is a characteristic function.

The characteristic function form has been generalised to games without the assumption of

transferable utility .**Partition function form**The characteristic function form ignores the possible externalities of coalition formation. In the partition function form the payoff of a coalition depends not only on its members, but also on the way the rest of the players are partitioned harv|Thrall|Lucas|1963.

**Application and challenges**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 use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well.Game theoretic analysis was initially used to study animal behavior by

Ronald Fisher in the 1930s (although evenCharles Darwin makes a few informal game theoretic statements). 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 byJohn Maynard Smith in his book "Evolution and the Theory of Games ".In addition to being used to predict and explain behavior, game theory has also been used to attempt to develop theories of ethical or normative behavior. In economics and

philosophy , scholars have applied game theory to help in the understanding of good or proper behavior. Game theoretic arguments of this type can be found as far back asPlato . [*cite web |url=http://plato.stanford.edu/archives/spr2008/entries/game-theory/ |title=Game Theory |accessdate=2008-08-21 |last=Ross |first=Don |date= |work=The Stanford Encyclopedia of Philosophy (Spring 2008 Edition) |publisher=Edward N. Zalta (ed.)*]**Political science**The application of game theory to

political science is focused in the overlapping areas offair division ,political economy ,public choice ,positive political theory , andsocial 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.For early examples of game theory applied to political science, see the work of

Anthony Downs . In his bookAn Economic Theory of Democracy harvard citations|last1=Downs|year=1957, he applies a hotelling firm location model to the political process. In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. The theorist shows how the political candidates will converge to the ideology preferred by the median voter. For more recent examples, see the books bySteven Brams ,George Tsebelis ,Gene M. Grossman andElhanan Helpman , orDavid Austen-Smith andJeffrey S. Banks .A game-theoretic explanation for democratic peace is that public and open debate in democracies send 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 nondemocracy harvard citations| last1=Levy | last2=Razin| year=2003.

**Economics and business**Economists have long used game theory to analyze a wide array of economic phenomena, including

auction s,bargaining , duopolies,fair division , oligopolies,social network formation, andvoting system s. This research usually focuses on particular sets of strategies known as equilibria in games. These "solution concepts" are usually based on what is required by norms of rationality. In non-cooperative games, the most famous of these is theNash equilibrium . A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. So, 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. Often in modeling situations the payoffs represent money, which presumably corresponds to an individual's utility. This assumption, however, can be faulty.A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of some 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 should this information be put. Economists and business professors suggest two primary uses.

**Descriptive**The first known use is to inform us about how actual human populations behave. Some scholars believe that by finding the equilibria of games they can predict how actual human populations will behave when confronted with situations analogous to the game being studied. This particular view of game theory has come under recent criticism. First, it is criticized because the assumptions made by game theorists are often violated. Game theorists may assume players always act in a way to directly maximize their wins (the

Homo economicus model), but in practice, humans behaviour is often contrary to this model. Explanations of this phenomenon are many;irrationality , new models ofdeliberation , or even differentmotives (like that ofaltruism ). 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 byphysicist s. However, additional criticism of this use of game theory has been levied because some experiments have demonstrated that individuals do not play equilibrium strategies. For instance, in thecentipede game ,guess 2/3 of the average game, and thedictator game , people regularly do not play Nash equilibria. There is an ongoing debate regarding the importance of these experiments. [*Experimental work in game theory goes by many names,*]experimental economics ,behavioral economics , andbehavioural game theory are several. For a recent discussion on this field see harvtxt|Camerer|2003.Alternatively, some authors claim that Nash equilibria do not provide predictions for human populations, but rather provide an explanation for why populations that play Nash equilibria remain in that state. However, the question of how populations reach those points remains open.

Some game theorists have turned to

evolutionary game theory in order to resolve these worries. These models presume either no rationality orbounded rationality on the part of players. Despite the name, evolutionary game theory does not necessarily presumenatural 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).**Prescriptive or normative analysis**Payoff matrix | Name = The Prisoner's Dilemma

2L = Cooperate | 2R = Defect

1U = Cooperate | UL = -1, -1 | UR = -10, 0

1D = Defect | DL = 0, -10 | DR = -5, -5On the other hand, some scholars 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

Nash equilibrium of a game constitutes one'sbest response to the actions of the other players, playing a strategy that is part of a Nash equilibrium seems appropriate. However, this use for game theory has also come under criticism. First, in some cases it is appropriate to play a non-equilibrium strategy if one expects others to play non-equilibrium strategies as well. For an example, seeGuess 2/3 of the average .Second, the

Prisoner's dilemma presents another potential counterexample. In the Prisoner's Dilemma, each player pursuing his own self-interest leads both players to be worse off than had they not pursued their own self-interests.**Biology**Payoff matrix | Name = The hawk-dove game

2L = Hawk | 2R = Dove

1U = Hawk | UL = v−c, v−c | UR = 2v, 0

1D = Dove | DL = 0, 2v | DR = v, vUnlike 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, but rather on ones that would be maintained byevolution ary forces. The best known equilibrium in biology is known as theEvolutionarily stable strategy or (ESS), and was first introduced in harv|Maynard Smith|1982. Although its initial motivation did not involve any of the mental requirements of theNash equilibrium , every ESS is a Nash equilibrium.In biology, game theory has been used to understand many different phenomena. It was first used to explain the evolution (and stability) of the approximate 1:1

sex ratio s. harv|Fisher|1930 suggested that the 1:1 sex ratios are a result of evolutionary forces acting on individuals who could be seen as trying to maximize their number of grandchildren.Additionally, biologists have used

evolutionary game theory and the ESS to explain the emergence ofanimal communication harv|Harper|Maynard Smith|2003. The analysis ofsignaling games and other communication games has provided some insight into the evolution of communication among animals. For example, theMobbing 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.Biologists have used the

hawk-dove game (also known as chicken) to analyze fighting behavior and territoriality.Maynard Smith, in the preface to Evolution of the Theory of Games writes, " [p] aradoxically, 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 phenomena 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. [

*http://www.seop.leeds.ac.uk/entries/altruism-biological/ Biological Altruism (Stanford Encyclopedia of Philosophy)*] ] All of these actions increase the overall fitness of a group, but occur at a cost to the individual.Evolutionary game theory explains this altruism with the idea of

kin selection . Altruists discriminate between the individuals they help and favor relatives. Hamilton’s rule explains the evolutionary reasoning behind this selection with the equation cRecent applications of biological game theory to humans has garnered some criticism because evolutionary analysis cannot provide a value-neutral evaluation of a given cultural situation. The valuations of whether an action is good or bad constitute a normative judgment of whether an action is altruistic. Altruism also has a different socially constructed meaning in the context of human society because altruistic actions within culture are not all instinctually driven and do not always result in increased fitness for a group. [

*http://plato.stanford.edu/entries/game-evolutionary/ Evolutionary Game Theory (Stanford Encyclopedia of Philosophy)*] ]**Computer science and logic**Game theory has come to play an increasingly important role in

logic and incomputer science . Several logical theories have a basis ingame semantics . In addition, computer scientists have used games to modelinteractive computation s. Also, game theory provides a theoretical basis to the field ofmulti-agent system s.Separately, game theory has played a role in

online algorithm s. In particular, thek-server problem , which has in the past been referred to as "games with moving costs" and "request-answer games" harvard citations| last1=Ben David | last2=Borodin | last3=Karp | last4=Tardos| last5=Wigderson | year=1994.Yao's principle is a game-theoretic technique for proving lower bounds on thecomputational complexity ofrandomized algorithm s, and especially of online algorithms.**Radio networks**Game theory has recently become a useful tool for modeling and studying interactions between

cognitive radios envisioned to operate in future communications systems. Such terminals will have the capability to adapt to the context they operate in, through possibly power and rate control as well as channel selection. Software agents embedded in these terminals will potentially be selfish, meaning they will only try to maximize the throughput/connectivity of the terminal they function for, as opposed to maximizing the welfare (total capacity) of the system they operate in. Thus, the potential interactions among them can be modeled through non-cooperative games. The researchers in this field often strive to determine the stable operating points of systems composed of such selfish terminals, and try to come up with a minimum set of rules (etiquette) so as to make sure that the optimality loss compared to a cooperative - centrally controlled setting- is kept at a minimum. [*M. Felegyhazi and J.P. Hubaux, "Game Theory in Wireless Networks: A Tutorial"*]**Philosophy**Payoff matrix | Name = Stag hunt

2L = Stag | 2R = Hare

1U = Stag | UL = 3, 3 | UR = 0, 2

1D = Hare | DL = 2, 0 | DR = 2, 2 Game theory has been put to several uses inphilosophy . Responding to two papers by harvard citations|txt=yes|first=W.V.O.|last=Quine|author1-link=Willard Van Orman Quine|year=1960|year2=1967, Harvtxt|Lewis|1969 used game theory to develop a philosophical account of convention. In so doing, he provided the first analysis of common knowledge and employed it in analyzing play incoordination game s. In addition, he first suggested that one can understand meaning in terms ofsignaling games . This later suggestion has been pursued by several philosophers since Lewis (Harvtxt|Skyrms|1996, harvard citations|txt=yes|last1=Grim | last2=Kokalis | last3=Alai-Tafti | last4=Kilb | last5=St Denis | year=2004). Inethics , some authors have attempted to pursue the project, begun byThomas Hobbes , of deriving morality from self-interest. Since games like thePrisoner'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 generalsocial contract view inpolitical philosophy (for examples, see Harvtxt|Gauthier|1986 and Harvtxt|Kavka |1986. [*For a more detailed discussion of the use of Game Theory in ethics see the Stanford Encyclopedia of Philosophy's entry [*]*http://plato.stanford.edu/entries/game-ethics/ game theory and ethics*] .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 thePrisoner's dilemma ,Stag hunt , and theNash bargaining game as providing an explanation for the emergence of attitudes about morality (see, e.g., harvard citations|txt=yes|last=Skyrms|year=1996|year2=2004 and harvard citations|txt=yes|last1=Sober|last2=Wilson|year=1999).Some assumptions used in some parts of game theory have been challenged in philosophy;

psychological egoism states that rationality reduces to self-interest—a claim debated among philosophers. ("seePsychological egoism#Criticism ")**Types of games****Cooperative or non-cooperative**A game is "cooperative" if the players are able to form binding commitments. For instance the legal system requires them to adhere to their promises. In noncooperative games this is not possible.

Often it is assumed that "communication" among players is allowed in cooperative games, but not in noncooperative ones. This classification on two binary criteria has been rejected harv|Harsanyi|1974.

Of the two types of games, noncooperative games are able to model situations to the finest details, producing accurate results. Cooperative games focus on the game at large. Considerable efforts have been made to link the two approaches. The so-called Nash-programme has already established many of the cooperative solutions as noncooperative equilibria.

"Hybrid" games contain cooperative and non-cooperative elements. For instance, coalitions of players are formed in a

cooperative game , but these play in a non-cooperative fashion.**ymmetric and asymmetric**Payoff matrix | Name = An asymmetric game

2L = E | 2R = F

1U = E | UL = 1, 2 | UR = 0, 0

1D = F | DL = 0, 0 | DR = 1, 2 A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric. Many of the commonly studied 2×2 games are symmetric. The standard representations of chicken, theprisoner's dilemma , and thestag hunt are all symmetric games. Some scholars would consider certain asymmetric games as examples of these games as well. However, the most common payoffs for each of these games are symmetric.Most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the

ultimatum game and similarly thedictator game have different strategies for each player. It is possible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the game pictured to the right is asymmetric despite having identical strategy sets for both players.**Zero sum and non-zero sum**Payoff matrix | Name = A zero-sum game

2L = A | 2R = B

1U = A | UL = –1, 1 | UR = 3, –3

1D = B | DL = 0, 0 | DR = –2, 2Zero sum games are a special case of constant sum games, in which choices by players can neither increase nor decrease the available resources. In

zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the equal expense of others).Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose. Other zero sum games includematching pennies and most classical board games including Go andchess .Many games studied by game theorists (including the famous

prisoner's dilemma ) are non-zero-sum games, because some outcomes have net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.Constant sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential gains from trade. It is possible to transform any game into a (possibly asymmetric) zero-sum game by adding an additional dummy player (often called "the board"), whose losses compensate the players' net winnings.

**imultaneous and sequential**Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players' actions (making them "effectively" simultaneous). 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. For instance, a player may know that an earlier player did not perform one particular action, while he does not know which of the other available actions the first player actually performed.The difference between simultaneous and sequential games is captured in the different representations discussed above. Often, normal form is used to represent simultaneous games, and extensive form is used to represent sequential ones; although this isn't a strict rule in a technical sense.

**Perfect information and imperfect information**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. Thus, only sequential games can be games of perfect information, since in simultaneous games not every player knows the actions of the others. Most games studied in game theory are imperfect information games, although there are some interesting examples of perfect information games, including the

ultimatum game andcentipede game . Perfect information games include alsochess , go,mancala , andarimaa .Perfect information is often confused with

complete information , which is a similar concept. Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions.**Infinitely long games**Games, as studied by economists and real-world game players, are generally finished in a finite number of moves. Pure mathematicians are not so constrained, and set theorists in particular study games that last for an infinite number of 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 what is the best way to play such a game, but simply on whether one or the other player has a winning strategy. (It can be proven, using the

axiom of choice , that there are games—even with perfect information, and where the only outcomes are "win" or "lose"—for which "neither" player has a winning strategy.) The existence of such strategies, for cleverly designed games, has important consequences indescriptive set theory .**Discrete and continuous games**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 game s 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 game s such as the continuous pursuit and evasion game are continuous games.**Metagames**These are games the play of which is the development of the rules for another game, the target or subject game.

Metagame s seek to maximize the utility value of the rule set developed. The theory of metagames is related tomechanism design theory.**History**The first known discussion of game theory occurred in a letter written by James Waldegrave in 1713. In this letter, Waldegrave provides a

minimax mixed strategy solution to a two-person version of the card gamele Her . It was not until the publication ofAntoine Augustin Cournot 's "Recherches sur les principes mathématiques de la théorie des richesses" ("Researches into the Mathematical Principles of the Theory of Wealth") in 1838 that a general game theoretic analysis was pursued. In this work Cournot considers aduopoly and presents a solution that is a restricted version of theNash equilibrium .Although Cournot's analysis is more general than Waldegrave's, game theory did not really exist as a unique field until

John von Neumann published a series of papers in 1928. While the French mathematicianÉmile Borel did some earlier work on games, Von Neumann can rightfully be credited as the inventor of game theory. Von Neumann was a brilliant mathematician whose work was far-reaching from set theory to his calculations that were key to development of both the Atom and Hydrogen bombs and finally to his work developing computers. Von Neumann's work in game theory culminated in the 1944 book "Theory of Games and Economic Behavior " by von Neumann andOskar Morgenstern . This profound work contains the method for finding mutually consistent solutions for two-person zero-sum games. During this time period, work on game theory was primarily focused oncooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.In 1950, the first discussion of the

prisoner's dilemma appeared, and an experiment was undertaken on this game at theRAND corporation . Around this same time, John Nash developed a criterion for mutual consistency of players' strategies, known asNash equilibrium , applicable to a wider variety of games than the criterion proposed by von Neumann and Morgenstern. This equilibrium is sufficiently general, allowing for the analysis ofnon-cooperative game s in addition to cooperative ones.Game theory experienced a flurry of activity in the 1950s, during which time the concepts of the core, the

extensive form game ,fictitious play ,repeated game s, and theShapley value were developed. In addition, the first applications of Game theory tophilosophy andpolitical science occurred during this time.In 1965,

Reinhard Selten introduced hissolution concept of subgame perfect equilibria, which further refined theNash equilibrium (later he would introducetrembling hand perfection as well). In 1967,John Harsanyi developed the concepts ofcomplete information andBayesian game s. Nash, Selten and Harsanyi became Economics Nobel Laureates in 1994 for their contributions to economic game theory.In the 1970s, game theory was extensively applied in

biology , largely as a result of the work ofJohn Maynard Smith and hisevolutionarily stable strategy . In addition, the concepts ofcorrelated equilibrium , trembling hand perfection, and common knowledge [*Although common knowledge was first discussed by the philosopher David Lewis in his dissertation (and later book) "Convention" in the late 1960s, it was not widely considered by economists until*] were introduced and analysed.Robert Aumann 's work in the 1970s.In 2005, game theorists

Thomas Schelling andRobert Aumann followed Nash, Selten and Harsanyi as Nobel Laureates. Schelling worked on dynamic models, early examples ofevolutionary game theory . Aumann contributed more to the equilibrium school, introducing an equilibrium coarsening, correlated equilibrium, and developing an extensive formal analysis of the assumption of common knowledge and of its consequences.In 2007,

Roger Myerson , together withLeonid Hurwicz andEric Maskin , was awarded of the Nobel Prize in Economics "for having laid the foundations ofmechanism design theory." Among his contributions, is also the notion ofproper equilibrium , and an important graduate text: "Game Theory, Analysis of Conflict" harv|Myerson|1997.**ee also***

Analytic narrative

*Game

*Glossary of game theory

*List of games in game theory

*"The Trap", in whichAdam Curtis examines the rise of game theory during theCold War **Notes****References****Textbooks and general references***.

*. Suitable for undergraduate and business students.

*. Suitable for upper-level undergraduates.

*. Acclaimed reference text, [

*http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=8204 public description*] .*. Suitable for advanced undergraduates. :*Published in Europe as Citation | title=A Primer in Game Theory | publisher=Harvester Wheatsheaf | isbn=978-0-7450-1159-2 | location=London.

*

*. Presents game theory in formal way suitable for graduate level.

*. [

*http://www.gametheorists.com Snippets from interviews*] .*

*. Suitable for a general audience.

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*. Undergraduate textbook.

*. A general history of game theory and game theoreticians.

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*. A modern introduction at the graduate level.

* Praised primer and popular introduction for everybody, never out of print.

**Historically important texts*** Aumann, R.J. and Shapley, L.S. (1974), "Values of Non-Atomic Games", Princeton University Press

*

*

*:*reprinted edition: Citation | title=The Genetical Theory of Natural Selection: A Complete Variorum Edition | publisher=

Oxford University Press | isbn=978-0-19-850440-5 | year=1999*:*reprinted edition: Citation | title=Games and decisions: introduction and critical survey | publisher=

Dover Publications | location=New York | isbn=978-0-486-65943-5 | year=1989*

*

* Shapley, L.S. (1953), A Value for n-person Games, In: Contributions to the Theory of Games volume II, H.W. Kuhn and A.W. Tucker (eds.)

* Shapley, L.S. (1953), Stochastic Games, Proceedings of National Academy of Science Vol. 39, pp. 1095-1100.

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**Other print references***

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*, ISBN 978-0-631-23257-5 (2002 edition)

*. A layman's introduction.

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*Citation

last =Thrall

first =Robert M.

author-link = Robert M. Thrall

last2 = Lucas

first2 =William F.

author2-link =William F. Lucas

title =$n$-person games in partition function form

journal =Naval Research Logistics Quarterly

volume =10

issue =4

pages =281–298

date =1963

year =1963**Websites*** Paul Walker: [

*http://www.econ.canterbury.ac.nz/personal_pages/paul_walker/gt/hist.htm History of Game Theory Page*] .

* David Levine: [*http://dklevine.com Game Theory. Papers, Lecture Notes and much more stuff.*]

* Alvin Roth: [*http://www.economics.harvard.edu/~aroth/alroth.html Game Theory and Experimental Economics page*] - Comprehensive list of links to game theory information on the Web

* Adam Kalai: [*http://wiki.cc.gatech.edu/theory/index.php/CS_8803_-_Game_Theory_and_Computer_Science._Spring_2008 Game Theory and Computer Science*] - Lecture notes on Game Theory and Computer Science

* Mike Shor: [*http://www.gametheory.net Game Theory .net*] - Lecture notes, interactive illustrations and other information.

* Jim Ratliff's [*http://virtualperfection.com/gametheory/ Graduate Course in Game Theory*] (lecture notes).

* Valentin Robu's [*http://homepages.cwi.nl/~robu/aamas/aamas_demo.html software tool*] for simulation of bilateral negotiation (bargaining)

* Don Ross: [*http://plato.stanford.edu/entries/game-theory/ Review Of Game Theory*] in the "Stanford Encyclopedia of Philosophy".

* Bruno Verbeek and Christopher Morris: [*http://plato.stanford.edu/entries/game-ethics/ Game Theory and Ethics*]

* Chris Yiu's [*http://www.yiu.co.uk/gametheory.php Game Theory Lounge*]

* Elmer G. Wiens: [*http://www.egwald.ca/operationsresearch/gameintroduction.php Game Theory*] - Introduction, worked examples, play online two-person zero-sum games.

* Marek M. Kaminski: [*http://webfiles.uci.edu/mkaminsk/www/courses.html Game Theory and Politics*] - syllabuses and lecture notes for game theory and political science.

* [*http://www.socialcapitalgateway.org/eng-gametheory.htm Web sites on game theory and social interactions*]

* Kesten Green's [*http://conflictforecasting.com Conflict Forecasting*] - See Papers for evidence on the accuracy of forecasts from game theory and other methods.

* McKelvey, Richard D., McLennan, Andrew M., and Turocy, Theodore L. (2007) " [*http://gambit.sourceforge.net Gambit: Software Tools for Game Theory*] ".

*Wikimedia Foundation.
2010.*

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**game theory**— ➔ theory * * * game theory UK US noun [U] ECONOMICS ► an economic theory that tries to calculate what choices people will make, when those choices depend on the choices of others: »Some of the most interesting ideas are coming from game theory,… … Financial and business terms**game theory**— n. A branch of mathematics that deals with strategies for maximizing gains or minimizing losses in competitive situations having defined constraints and involving random factors. Note: Game theory is used for modelling and analysis of various… … The Collaborative International Dictionary of English**game theory**— n. a method of using mathematical analysis to select the best available strategy in order to minimize one s maximum losses or maximize one s minimum winnings in a game, war, business competition, etc … English World dictionary**Game Theory**— [engl.], Spieltheorie … Universal-Lexikon**game theory**— a mathematical theory that deals with strategies for maximizing gains and minimizing losses within prescribed constraints, as the rules of a card game: widely applied in the solution of various decision making problems, as those of military… … Universalium**game theory**— The general theory of the rational behaviour of two or more people in circumstances where their interests are, at least in part, conflicting. In The Theory of Games and Economic Behaviour(1947), John Von Neumann and Oskar Morgenstern attempted to … Dictionary of sociology**game theory**— noun (economics) a theory of competition stated in terms of gains and losses among opposing players • Syn: ↑theory of games • Topics: ↑economics, ↑economic science, ↑political economy • Members of this Topic: ↑ … Useful english dictionary**game theory**— A mathematical theory, developed by J. von Neumann (1903 57) and O. Morgenstern (1902 77) in 1944, concerned with predicting the outcome of games of strategy (rather than games of chance) in which the participants have incomplete information… … Big dictionary of business and management**game theory**— The systematic analysis in competitive conditions of strategies for the optimal selection of alternative courses of action. In game theory, unlike *decision theory, the outcomes of an individual’s decisions depend on the actions of other… … Auditor's dictionary**Game Theory**— A model of optimality taking into consideration not only benefits less costs, but also the interaction between participants. Game theory attempts to look at the relationships between participants in a particular model and predict their optimal… … Investment dictionary