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{{distinguish|Game studies}}
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{{about|the mathematical study of optimizing agents}}
{{refimprove|date=October 2011}}
{{Economics sidebar}}
{{JEL code|C7}}
'''Game theory''' is a study of strategic [[decision making]]. More formally, it is "the study of [[mathematical model]]s of conflict and cooperation between intelligent rational decision-makers".<ref>[[Roger B. Myerson]] (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p. [http://books.google.com/books?id=E8WQFRCsNr0C&printsec=find&pg=PA1 1]. Chapter-preview links, pp. [http://books.google.com/books?id=E8WQFRCsNr0C&printsec=find&pg=PR7 vii–xi].
</ref> An alternative term suggested "as a more descriptive name for the discipline" is ''interactive [[decision theory]]''.<ref>[[Robert Aumann|R. J. Aumann]] ([[The New Palgrave: A Dictionary of Economics|[1987] ]] 2008). "game theory," Introduction, ''[[The New Palgrave Dictionary of Economics]]'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_G000007&q=game%20theory&topicid=&result_number=3 Abstract.]</ref> Game theory is mainly used in economics, political science, and psychology, as well as logic and biology. The subject first addressed [[zero-sum game]]s, such that one person's gains exactly equal net losses of the other participant(s). Today, however, game theory applies to a wide range of behavioral relations, and has developed into an [[umbrella term]] for the logical side of decision science, to include both human and non-humans, like computers.
 
Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by [[John von Neumann]]. Von Neumann's original proof used [[Brouwer's fixed-point theorem]] on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book ''[[Theory of Games and Economic Behavior]]'', with [[Oskar Morgenstern]], which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.
 
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 won the [[Nobel Memorial Prize in Economic Sciences]], and [[John Maynard Smith]] was awarded the [[Crafoord Prize]] for his application of game theory to biology.
 
==Representation of games==
{{see also|List of games in game theory}}
The games studied in game theory are well-defined mathematical objects. To be fully defined, a game must specify the following elements: the [[Player (game)|''players'' of the game]], the ''information'' and ''actions'' available to each player at each decision point, and the [[Utility|''payoffs'']] for each outcome. (Rasmusen refers to these four "essential elements" by the acronym PAPI.)<ref name = r7/> A game theorist typically uses these elements, along with a [[solution concept]] of their choosing, to deduce a set of equilibrium [[Strategy (game theory)|strategies]] for each player such that, when these strategies are employed, no player can profit by unilaterally deviating from their strategy. These equilibrium strategies determine an [[Economic equilibrium|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.
 
===Extensive form===
{{main|Extensive form game}}
[[Image:Ultimatum Game Extensive Form.svg|thumb|left|An extensive form game]]
The extensive form can be used to formalize games with a time sequencing of moves. Games here are played on [[tree (graph theory)|trees]] (as pictured to the left). Here each [[Graph (mathematics)|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]]. {{harv|Fudenberg|Tirole|1991|p=67}}
 
In the game pictured to the left, 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 imperfect 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. (See example in the [[#Perfect information and imperfect information|imperfect information section]].)
 
===Normal form===
<!--{| align=right border="1" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 1em; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-size: 95%;"
|+ align=bottom |''A normal form game''
|-
|
! scope="col" style="color: #900;width: 90px;"|''Player 2 chooses left''
! scope="col" style="color: #900;width: 90px;"|''Player 2 chooses right''
|-
! scope="col" style="color: #009;width: 90px;"|''Player 1 chooses top''
|align=center| <span style="color: #009">4</span>, <span style="color: #900">3</span>
|align=center| <span style="color: #009">-1</span>, <span style="color: #900">-1</span>
|-
! scope="col" style="color: #009;width: 100px;"|''Player 1 chooses bottom''
|align=center| <span style="color: #009">0</span>, <span style="color: #900">0</span>
|align=center| <span style="color: #009">3</span>, <span style="color: #900">4</span>
|}-->
{{Payoff matrix
| Name = Normal form or payoff matrix of a 2-player, 2-strategy game
| 2L = <span style="color:#900;">Player 2<br />chooses ''Left''</span>
| 2R = <span style="color:#900;">Player 2<br />chooses ''Right''</span>
| 1U = <span style="color:#009;">Player 1<br />chooses ''Up''</span>
| 1D = <span style="color:#009;">Player 1<br />chooses&nbsp;''Down''</span>
| UL = <span style="color:#009;">'''4'''</span>, <span style="color:#900;">'''3'''</span>
| UR = <span style="color:#009;">'''–1'''</span>, <span style="color:#900;">'''–1'''</span>
| DL = <span style="color:#009;">'''0'''</span>, <span style="color:#900;">'''0'''</span>
| DR = <span style="color:#009;">'''3'''</span>, <span style="color:#900;">'''4'''</span>
| Float = right | Width = 330}}
 
{{main|Normal-form game}}
The normal (or strategic form) game is usually represented by a [[Matrix (mathematics)|matrix]] which shows the players, strategies, and pay-offs (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. {{harv|Leyton-Brown|Shoham|2008|p=35}}
 
===Characteristic function form===
{{main|Cooperative game}}
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 origin of this form is to be found in John von Neumann and Oskar Morgenstern's book; when looking at these instances, they guessed that when a union <math>\mathbf{C}</math> appears, it works against the fraction
<math>
\left(\frac{\mathbf{N}}{\mathbf{C}}\right)
</math>
as if two individuals were playing a normal game. 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 <math>v:2^N \to \mathbf{R}</math> is a normal utility.
 
Such characteristic functions have expanded to describe games where there is no removable utility.
 
==General and applied uses==
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 1838 with his solution of the [[Cournot competition|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.
 
Game-theoretic analysis, in biology, was initially used to study animal behavior by [[Ronald Fisher]] in the 1930s (although even [[Charles 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 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 [[Decision theory#Normative and descriptive decision theory|prescribe]] such behavior.<ref name = "Camerer2003">[[Colin F. Camerer]] (2003). ''Behavioral Game Theory: Experiments in Strategic Interaction'', pp. 5–7 (scroll to at [http://press.princeton.edu/chapters/i7517.html 1.1 What Is Game Theory Good For?]).</ref> 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 as [[Plato]].<ref>{{cite web |url=http://plato.stanford.edu/archives/spr2008/entries/game-theory/ |title=Game Theory |accessdate=2008-08-21 |last=Ross |first=Don |work=The Stanford Encyclopedia of Philosophy (Spring 2008 Edition) |publisher=Edward N. Zalta (ed.)}}</ref>
 
===Description and modeling===
[[Image:Centipede game.svg|thumb|300px|right|A four-stage [[centipede game]]]]
The first known use is to describe and [[Conceptual model#Economic models|model]] how human populations behave. Some{{Who|date=July 2012}} 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, human behavior often deviates from this model. Explanations of this phenomenon are many; [[irrationality]], new models of [[deliberation]], or even different motives (like that of [[altruism]]). 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 [[Idealization|ideal]] akin to the models used by [[physicist]]s. However, in the [[centipede game]], [[guess 2/3 of the average]] game, and the [[dictator game]], people regularly do not play Nash equilibria. These experiments have demonstrated that individuals do not play equilibrium strategies. There is an ongoing debate regarding the importance of these experiments.<ref>Experimental work in game theory goes by many names, [[experimental economics]], [[behavioral economics]], and [[behavioural game theory]] are several. For a recent discussion, see Colin F. Camerer (2003). ''Behavioral Game Theory: Experiments in Strategic Interaction'' ([http://press.princeton.edu/titles/7517.html description] and [http://press.princeton.edu/chapters/i7517.html Introduction], pp.&nbsp;1–25).</ref>
 
Alternatively, some{{Who|date=July 2012}} 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{{Who|date=July 2012}} game theorists 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).
 
===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, -5 }}
 
On the other hand, some{{Who|date=July 2012}} 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 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. However, the rationality of such a decision has been proved only for special cases. This normative use of 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, see [[guess 2/3 of the average]].
 
Second, the [[prisoner's dilemma]] presents another potential counterexample. In the prisoner's dilemma, each player pursuing their own self-interest leads both players to be worse off than had they not pursued their own self-interests.
 
===Economics and business===
{{incomplete|date=November 2010}}
Game theory is a major method used in [[mathematical economics]] and business for [[economic model|modeling]] competing behaviors of interacting [[agent (economics)|agents]].<ref name=r1/> Applications include a wide array of economic phenomena and approaches, such as [[auction]]s, [[bargaining]], [[Mergers & Acquisitions|mergers & acquisitions]] pricing,<ref name=GT-A-E-00/> [[fair division]], [[duopoly|duopolies]], [[oligopoly|oligopolies]], [[social network]] formation, [[agent-based computational economics]],<ref name=r2/> [[general equilibrium]], [[mechanism design]],<ref name=r3/> and [[voting system]]s,<ref name=r4/> and across such broad areas as [[experimental economics]],<ref name=r5/> [[Behavioral game theory|behavioral economics]],<ref name=r6/> [[information economics]],<ref name=r7/> [[industrial organization]],<ref name=r8/> and [[political economy]].<ref name=r9/><ref name=r10/>
 
This research usually focuses on particular sets of strategies known as [[solution concept|equilibria]] in games. These "solution concepts" are usually based on what is required by norms of [[perfect rationality|rationality]]. 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. 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.<ref name=GT-F-R-09>[http://www.insead.edu/facultyresearch/research/doc.cfm?did=46503 Game-theoretic model to examine the two tradeoffs in the acquisition of information for a careful balancing act] Research paper [[INSEAD]]</ref><ref name=GT-F-R-10>[http://www.europeanfinancialreview.com/?p=4645 Options Games: Balancing the trade-off between flexibility and commitment]. Europeanfinancialreview.com (2012-02-15). Retrieved on 2013-01-03.</ref>
 
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 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): ''descriptive'' and ''[[Decision theory#Normative and descriptive decision theory|prescriptive]]''.<ref name = "Camerer2003"/>
 
===Political science===
The application of game theory to [[political science]] is focused in the overlapping areas of [[fair division]], [[political economy]], [[public choice]], [[war's inefficiency puzzle|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.
 
For early examples of game theory applied to political science, see the work of [[Anthony Downs]]. In his book ''[[An Economic Theory of Democracy]]'' {{harvard citations|last1=Downs|year=1957}}, he applies the [[Hotelling's law|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.
 
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.<ref>Morrison, Andrew Stumpff, Eminent Legal Philosophers, or Yes, Law is the Command of the Sovereign, at 8-12 (2013). Available at SSRN: http://ssrn.com/abstract=2371076</ref>
 
A game-theoretic explanation for [[democratic peace theory|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 non-democracy {{harvard citations| last1=Levy | last2=Razin| year=2003}}.
 
Game theory could also help predict 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 (2013) 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 green house gas emissions. However, he concluded that this idea could not work because it would create a prisoner's dilemma to the nations.<ref>Wood, Peter John. (2011, February). "Climate change and game theory," ''[[Ecological Economics Review]]'' '''1219''': 153-70.</ref>
 
===Biology===
{{Payoff matrix | Name = The hawk-dove game
| 2L = Hawk | 2R = Dove |
1U = Hawk | UL = 20, 20 | UR = 80, 40 |
1D = Dove | DL = 40, 80 | DR = 60, 60 }}
 
The payoffs for games in [[biology]], unlike those in economics, are often interpreted as corresponding to [[fitness (biology)|fitness]]. In addition, the focus has been less on [[solution concept|equilibria]] that correspond to a notion of rationality and more on ones that would be maintained by [[evolution]]ary forces. The best known equilibrium in biology is known as the ''[[evolutionarily stable strategy]]'' (or ESS), and was first introduced in {{harv|Smith|Price|1973}}. 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 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 of [[animal communication]] {{harv|Harper|Maynard Smith|2003}}. The analysis of [[signaling games]] and [[cheap talk|other communication games]] has provided insight into the evolution of communication among animals. 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 [[chicken (game)|game of chicken]] to analyze fighting behavior and territoriality.<ref>{{cite doi|10.1016/0022-5193(74)90110-6}}</ref>
 
Maynard Smith, in the preface to ''Evolution and the Theory of Games'', writes, "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.<ref name="stan-egt">[http://plato.stanford.edu/entries/game-evolutionary/ Evolutionary Game Theory (Stanford Encyclopedia of Philosophy)]. Plato.stanford.edu. Retrieved on 2013-01-03.</ref>
 
One such phenomenon is known as [[Altruism in animals|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.<ref name = "qudzyh">[http://www.seop.leeds.ac.uk/entries/altruism-biological/ Biological Altruism (Stanford Encyclopedia of Philosophy)]. Seop.leeds.ac.uk. Retrieved on 2013-01-03.</ref> 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 c<b*r where the cost (c) to the altruist must be less than the benefit (b) to the recipient multiplied by the coefficient of relatedness (r). 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. Helping a sibling for example (in diploid animals), has a coefficient of ½, because (on average) an individual shares ½ of the alleles in its sibling's offspring. Ensuring that enough of a sibling’s offspring survive to adulthood precludes the necessity of the altruistic individual producing offspring.<ref name = "qudzyh"/> The coefficient values depend heavily on the scope of the playing field; for example if the choice of whom to favor includes all genetic living things, not just all relatives, we assume the discrepancy between all humans only accounts for approximately 1% of the diversity in the playing field, a co-efficient that was ½ in the smaller field becomes 0.995. Similarly if it is considered that information other than that of a genetic nature (e.g. epigenetics, religion, science, etc.) persisted through time the playing field becomes larger still, and the discrepancies smaller.
 
===Computer science and logic===
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 computation]]s. Also, game theory provides a theoretical basis to the field of [[multi-agent system]]s.
 
Separately, game theory has played a role in [[online algorithm]]s. In particular, the [[k-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 the [[computational complexity]] of [[randomized algorithm]]s, and especially of online algorithms.
 
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]]<ref name="algorithmic-game"/> and within it [[algorithmic mechanism design]]<ref name=r11/> combine computational [[algorithm design]] and analysis of [[complex system]]s with economic theory.<ref name=r12/>
 
===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 in [[philosophy]]. 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 (norm)|convention]]. In so doing, he provided the first analysis of [[Common knowledge (logic)|common knowledge]] and employed it in analyzing play in [[coordination game]]s. In addition, he first suggested that one can understand [[Meaning (semiotics)|meaning]] in terms of [[signaling 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}}). Following {{Harvtxt|Lewis|1969}} game-theoretic account of conventions, Edna Ullmann-Margalit (1977) and [[Cristina Bicchieri|Bicchieri]] (2006) have developed theories of [[social norms]] that define them as Nash equilibria that result from transforming a mixed-motive game into a coordination game.<ref>{{Citation |first=E. |last=Ullmann-Margalit |title=The Emergence of Norms |publisher=Oxford University Press |year=1977 |isbn=0198244118 }}</ref><ref>{{Citation |first=C. |last=Bicchieri |title=The Grammar of Society: the Nature and Dynamics of Social Norms |publisher=Cambridge University Press |year=2006 |isbn=0521573726 }}</ref>
 
Game theory has also challenged philosophers to think in terms of interactive [[epistemology]]: what it means for a collective to have common beliefs or knowledge, and what are the consequences of this knowledge for the social outcomes resulting from agents' interactions. Philosophers who have worked in this area include Bicchieri (1989, 1993),<ref>{{Citation |title=Self-Refuting Theories of Strategic Interaction: A Paradox of Common Knowledge |journal=Erkenntnis |volume=30 |issue=1–2 |year=1989 |pages=69–85 |doi=10.1007/BF00184816 |last1=Bicchieri |first1=Cristina }} See also {{Citation |title=Rationality and Coordination |publisher=Cambridge University Press |year=1993 |isbn=0521381231 }}</ref> [[Brian Skyrms|Skyrms]] (1990),<ref>{{Citation |title=The Dynamics of Rational Deliberation |publisher=Harvard University Press |year=1990 |isbn=067421885X }}</ref> and [[Robert Stalnaker|Stalnaker]] (1999).<ref>{{Citation |chapter=Knowledge, Belief, and Counterfactual Reasoning in Games |editor-first=Cristina |editor-last=Bicchieri |editor2-first=Richard |editor2-last=Jeffrey |editor3-first=Brian |editor3-last=Skyrms |title=The Logic of Strategy |location=New York |publisher=Oxford University Press |year=1999 |isbn=0195117158 |pages= }}</ref>
 
In [[ethics]], some{{Who|date=July 2012}} authors have attempted to pursue the project, begun by [[Thomas Hobbes]], of deriving morality from self-interest. 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 {{Harvtxt|Gauthier|1986}} and {{Harvtxt|Kavka |1986}}).<ref>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].</ref>
 
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.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; for example, [[psychological egoism]] states that rationality reduces to self-interest—a claim debated among philosophers. (''see [[Psychological egoism#Criticisms]]'')
 
==Types of games==
 
===Cooperative or non-cooperative===
{{main|Cooperative game|Non-cooperative game}}
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. However, this classification on two binary criteria has been questioned, and sometimes 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 (Nash program is the research agenda for investigating on the one hand axiomatic bargaining solutions and on the other hand the equilibrium outcomes of strategic bargaining procedures)<ref>Harold Houba, Wilko Bolt. ''Credible Threats in Negotiations. A Game-theoretic Approach.'' Chapter 4. The Nash Program. ISBN 978-1-4020-7183-6.</ref> 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.
 
===Symmetric 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 }}
{{main|Symmetric game}}
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 [[Game of chicken|chicken]], the [[prisoner's dilemma]], and the [[stag hunt]] are all symmetric games. Some{{Who|date=July 2012}} 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 the [[dictator 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, 2 }}
{{main|Zero–sum game}}
 
Zero-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 include [[matching pennies]] and most classical board games including [[go (board game)|Go]] and [[chess]].
 
Many games studied by game theorists (including the infamous [[prisoner's dilemma]]) are non-zero-sum games, because the [[Outcome (game theory)|outcome]] has 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 a dummy player (often called "the board"), whose losses compensate the players' net winnings.
 
===Simultaneous and sequential===
{{main|Sequential game| Simultaneous game}}
 
[[Simultaneous game]]s 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 game]]s (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 game|normal form]] is used to represent simultaneous games, and [[extensive form game|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]].
 
In short, the differences between sequential and simultaneous games are as follows:
 
{| class="wikitable"
|-
! !! Sequential !! Simultaneous
|-
| Normally denoted by: || Decision Trees || Payoff Matrices
|-
| Prior knowledge of opponent's move: || Yes || No
|-
| Time Axis: || Yes || No
|-
| Also known as: || Extensive Game || Strategic Game
|}
 
===Perfect information and imperfect information===
{{main|Perfect information}}
[[Image:PD with outside option.svg|thumb|250px|right|A game of imperfect information (the dotted line represents ignorance on the part of player 2, formally called an [[Information set (game theory)|information set]])]]
 
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 because players in simultaneous games do not know the actions of the other players. Most games studied in game theory are imperfect-information games. Interesting examples of perfect-information games include the [[ultimatum game]] and [[centipede game]]. Recreational games of perfect information games include [[chess]], [[go (board game)|go]], and [[mancala]]. Many card games are games of imperfect information, for instance [[poker]] or [[contract bridge]].
 
Perfect information is often confused with [[complete information]], which is a similar concept. Complete information requires that every player know the strategies and payoffs available to the other players but not necessarily the actions taken. Games of incomplete information can be reduced, however, to games of imperfect information by introducing "[[move by nature|moves by nature]]" {{harv|Leyton-Brown|Shoham|2008|p=60}}.
 
===Combinatorial games===
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 or incomplete 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.<ref name="Bewersdorff2005"/>
 
Games of perfect information have been studied in [[combinatorial game theory]], which has developed novel representations, e.g. [[surreal numbers]], as well as [[combinatorics|combinatorial]] and [[abstract algebra|algebraic]] (and [[strategy stealing argument|sometimes non-constructive]]) proof methods to [[solved game|solve games]] of certain types, including "loopy" games that may result in infinitely long sequences of moves. These methods address games with higher combinatorial complexity than those usually considered in traditional (or "economic") game theory.<ref>{{Citation
| last1 = Albert | first1 = Michael H. | author1-link = Michael H. Albert
| last2 = Nowakowski | first2 = Richard J.
| last3 = Wolfe | first3 = David
| isbn = 978-1-56881-277-9
| publisher = A K Peters Ltd
| title = Lessons in Play: In Introduction to Combinatorial Game Theory
| year = 2007
| pages = 3–4
}}</ref><ref>{{Citation
| last = Beck | first = József | author-link = József Beck
| isbn = 978-0-521-46100-9
| publisher = Cambridge University Press
| title = Combinatorial games: tic-tac-toe theory
| year = 2008
| pages = 1–3}}</ref> A typical game that has been solved this way is [[Hex (board game)|hex]]. 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.<ref>{{Citation|author1=Robert A. Hearn|author2=Erik D. Demaine|title=Games, Puzzles, and Computation|year=2009|publisher=A K Peters, Ltd.|isbn=978-1-56881-322-6}}</ref>
 
Research in [[artificial intelligence]] has addressed both perfect and imperfect (or incomplete) 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 network]]s trained by [[reinforcement learning]], which make games more tractable in computing practice.<ref name="Bewersdorff2005">{{Citation|author=[[Jörg Bewersdorff]]|title=Luck, logic, and white lies: the mathematics of games|year=2005|publisher=A K Peters, Ltd.|isbn=978-1-56881-210-6|pages=ix–xii and chapter 31}}</ref><ref name="Jones2008">{{Citation|author=M. Tim Jones|title=Artificial Intelligence: A Systems Approach|year=2008|publisher=Jones & Bartlett Learning|isbn=978-0-7637-7337-3|pages=106–118}}</ref>
 
===Infinitely long games===
{{main|Determinacy}}
 
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 theory|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 what is the best way to play such a game, but simply on whether one or the other player has a [[Determinacy#Basic notions|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 in [[descriptive 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 games===
[[Differential game]]s such as the continuous [[pursuit-evasion|pursuit and evasion game]] are continuous games where the evolution of the players' state variables is governed by [[differential equation]]s. 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: the open-loop strategies are found using the [[Pontryagin's Minimum Principle|Pontryagin maximum principle]] while the closed-loop strategies are found using [[Hamilton–Jacobi–Bellman equation|Bellman's Dynamic Programming]] method.
 
A particular case of differential games are the games with random time horizon.<ref name="PM66">Petrosjan, L.A. and Murzov, N.V. (1966). ''Game-theoretic problems of mechanics''. Litovsk. Mat. Sb. 6, 423–433 (in Russian).</ref> In such games, the terminal time is a random variable with a given [[probability distribution]] function. 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.
 
===Many-player and population games===
Games with an arbitrary, but finite, number of players are often called n-person games {{harv|Luce & Raiffa|1957}}. [[Evolutionary game theory]] considers games involving a [[population]] of decision makers, where the frequency with which a particular decision is made can change over time in response to the decisions made by all individuals in the population. In biology, this is intended to model (biological) [[evolution]], where genetically programmed organisms pass along some of their strategy programming to their offspring. In economics, the same theory is intended to capture population changes because people play the game many times within their lifetime, and consciously (and perhaps rationally) switch strategies {{harv|Webb|2007}}.
 
===Stochastic outcomes (and relation to other fields)===
Individual decision problems with stochastic outcomes are sometimes considered "one-player games". These situations are not considered game theoretical by some authors.{{By whom|date=March 2011}} They may be modeled using similar tools within the related disciplines of [[decision theory]], [[operations research]], and areas of [[artificial intelligence]], particularly [[AI planning]] (with uncertainty) and [[multi-agent system]]. Although these fields may have different motivators, the mathematics involved are substantially the same, e.g. using [[Markov decision process]]es (MDP).{{Citation needed|date=March 2011}}
 
Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves", also known as "[[move by nature|moves by nature]]" {{harv|Osborne & Rubinstein|1994}}. This player is not typically considered a third player in what is otherwise a two-player game, but merely serves to provide a roll of the dice where required by the game.
 
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|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.<ref name=McMahan>Hugh Brendan McMahan (2006), ''[http://www.cs.cmu.edu/~mcmahan/research/mcmahan_thesis.pdf Robust Planning in Domains with Stochastic Outcomes, Adversaries, and Partial Observability]'', CMU-CS-06-166, pp. 3–4</ref> (See [[black swan theory]] for more discussion on this kind of modeling issue, particularly as it relates to predicting and limiting losses in investment banking.)
 
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.<ref name=McMahan/>
 
===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 to [[mechanism design]] theory.
 
The term [[metagame analysis]] is also used to refer to a practical approach developed by Nigel Howard {{Harvard citation|Howard|1971|}} whereby a situation is framed as a strategic game in which stakeholders try to realise their objectives by means of the options available to them. Subsequent developments have led to the formulation of [[confrontation analysis]].
 
==History==
[[File:JohnvonNeumann-LosAlamos.gif|thumb|[[John von Neumann]]]]
Early discussions of examples of two-person games occurred long before the rise of modern, mathematical game theory. The first known discussion of game theory occurred in a letter written by [[James Waldegrave, 1st Earl Waldegrave|James Waldegrave]] in 1713.<ref name=GT-PD-01>{{Citation|url=http://www.jehps.net/Decembre2007/Bellhouse.pdf |title=The Problem of Waldegrave |author=Bellhouse, David |journal=Journal Électronique d'Histoire des Probabilités et de la Statistique |year=2007 |volume=3 |issue=2}}</ref> In this letter, Waldegrave provides a [[minimax]] [[mixed strategy]] solution to a two-person version of the card game [[le Her]]. [[James Madison]] made what we now recognize as a game-theoretic analysis of the ways states can be expected to behave under different systems of taxation.<ref>James Madison, Vices of the Political System of the United States, April 1787. [http://www.constitution.org/jm/17870400_vices.htm Link]</ref><ref>Jack Rakove, "James Madison and the Constitution", ''History Now'', Issue 13 September 2007. [http://www.historynow.org/09_2007/historian2.html Link]</ref> In his 1838 ''Recherches sur les principes mathématiques de la théorie des richesses'' (''Researches into the Mathematical Principles of the Theory of Wealth''), [[Antoine Augustin Cournot]] considered a [[duopoly]] and presents a solution that is a restricted version of the [[Nash equilibrium]].
 
The Danish mathematician Zeuthen proved that the mathematical model had a winning strategy by using [[Brouwer fixed point theorem|Brouwer's fixed point theorem]]. In his 1938 book ''Applications aux Jeux de Hasard'' and earlier notes, [[Émile Borel]] proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix was symmetric. Borel conjectured that non-existence of mixed-strategy equilibria in two-person zero-sum games would occur, a conjecture that was proved false.
 
Game theory did not really exist as a unique field until [[John von Neumann]] published a paper<!-- only one referenced in Aumann (1987), so no plural here. --> in 1928.<ref>{{Citation |first=J. v. |last=Neumann |year=1928 |title=Zur Theorie der Gesellschaftsspiele |journal=[[Mathematische Annalen]] |volume=100 |issue=1 |pages=295–320 |doi=10.1007/BF01448847 }} English translation: {{Citation |chapter=On the Theory of Games of Strategy |editor1-first=A. W. |editor1-last=Tucker |editor2-first=R. D. |editor2-last=Luce |year=1959 |title=Contributions to the Theory of Games |volume=4 |pages=13–42 |chapterurl=http://books.google.com/books?hl=en&lr=&id=9lSVFzsTGWsC&oi=fnd&pg=PA13&dq==P_RGaKOVtC&sig=J-QB_GglFSVWw9KfXjut62E6AmM#v=onepage&q&f=false }}</ref> Von Neumann's original proof used [[Brouwer's fixed-point theorem]] on continuous mappings into compact convex sets, which became a standard method in game theory and [[mathematical economics]]. His paper was followed by his 1944 book ''[[Theory of Games and Economic Behavior]]''. The second edition of this book provided an axiomatic theory of utility, which reincarnated Daniel Bernoulli's old theory of utility (of the money) as an independent discipline. Von Neumann's work in game theory culminated in this 1944 book. This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. During the following time period, work on game theory was primarily focused on [[cooperative game]] theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.<ref>{{Citation |last=Leonard |first=Robert |title=Von Neumann, Morgenstern, and the Creation of Game Theory |location=New York |publisher=Cambridge University Press |year=2010 |isbn=9780521562669 }}</ref>
 
In 1950, the first mathematical discussion of the [[prisoner's dilemma]] appeared, and an experiment was undertaken by notable mathematicians [[Merrill M. Flood]] and [[Melvin Dresher]], as part of the [[RAND corporation]]'s investigations into game theory. Rand pursued the studies because of possible applications to global [[nuclear strategy]].<ref name=GT-PD-01>[http://plato.stanford.edu/entries/prisoner-dilemma/ http://plato.stanford.edu/entries/prisoner-dilemma/]. Plato.stanford.edu. Retrieved on 2013-01-03.</ref> Around this same time, [[John Forbes Nash|John Nash]] developed a criterion for mutual consistency of players' strategies, known as Nash equilibrium, applicable to a wider variety of games than the criterion proposed by von Neumann and Morgenstern. This equilibrium is sufficiently general to allow for the analysis of [[non-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 (economics)|core]], the [[extensive form game]], [[fictitious play]], [[repeated game]]s, and the [[Shapley value]] were developed. In addition, the first applications of game theory to [[philosophy]] and [[political science]] occurred during this time.
 
In 1965, [[Reinhard Selten]] introduced his [[solution concept]] of [[subgame perfect equilibrium|subgame perfect equilibria]], which further refined the Nash equilibrium (later he would introduce [[trembling hand perfection]] as well). In 1967, [[John Harsanyi]] developed the concepts of [[complete information]] and [[Bayesian game]]s. Nash, Selten and Harsanyi became [[Nobel Memorial Prize in Economic Sciences|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 of [[John Maynard Smith]] and his [[evolutionarily stable strategy]]. In addition, the concepts of [[correlated equilibrium]], trembling hand perfection, and [[Common knowledge (logic)|common knowledge]]<ref>Although common knowledge was first discussed by the philosopher [[David Kellogg Lewis|David Lewis]] in his dissertation (and later book) ''Convention'' in the late 1960s, it was not widely considered by economists until [[Robert Aumann]]'s work in the 1970s.</ref> were introduced and analyzed.
 
In 2005, game theorists [[Thomas Schelling]] and [[Robert Aumann]] followed Nash, Selten and Harsanyi as Nobel Laureates. Schelling worked on dynamic models, early examples of [[evolutionary 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, [[Leonid Hurwicz]], together with [[Eric Maskin]] and [[Roger Myerson]], was awarded the Nobel Prize in Economics "for having laid the foundations of [[mechanism design]] theory." Myerson's contributions include the notion of [[proper equilibrium]], and an important graduate text: ''Game Theory, Analysis of Conflict'' {{harv|Myerson|1997}}. Hurwicz introduced and formalized the concept of [[incentive compatibility]].
 
==Popular culture==
The life story of game theorist and mathematician [[John Forbes Nash, Jr.|John Nash]] was turned into a biopic, ''[[A Beautiful Mind (film)|A Beautiful Mind]]'' starring [[Russell Crowe]],<ref>Simon Singh [http://www.nytimes.com/books/98/06/14/reviews/980614.14singht.html "Between Genius and Madness."] New York Times (June 14, 1998)</ref> based on the book by [[Sylvia Nasar]].<ref>Sylvia Nasar, ''[[A Beautiful Mind (book)|A Beautiful Mind]]'', (1998). Simon & Schuster. ISBN 0-684-81906-6</ref>
 
"Games-theory" and "theory of games" are mentioned in the [[military science fiction]] novel ''Starship Troopers'' by [[Robert A. Heinlein]].<ref>{{Citation|last=Heinlein|first=Robert A.|title=Starship Troopers|year=1959}}</ref> In the 1997 film of the same name the character Carl Jenkins refers to his assignment, military intelligence, as "games and theory."
 
One of the main gameplay decision-making mechanics of the video game ''[[Zero Escape: Virtue's Last Reward]]'' is based on game theory. Some of the characters even reference the prisoner's dilemma.
 
The film ''[[Dr. Strangelove]]'' satirizes game theoretic ideas about [[deterrence theory]]. For example, nuclear deterrence depends on the threat to retaliate catastrophically if a nuclear attack is detected. A game theorist might argue that such threats can fail to be ''credible'', in the sense that they can lead to [[Subgame perfect equilibrium|subgame imperfect equilibria]]. The movie takes this idea one step further, with the Russians irrevocably committing to a catastrophic nuclear response without making the threat public.
 
==See also==
* [[AI effect]]
* [[Applications of artificial intelligence]]
* [[Chainstore paradox]]
* [[Collective Intentionality]]
* [[Combinatorial game theory]]
* [[Confrontation analysis]]
* [[Glossary of game theory]]
* [[Intra-household bargaining]]
* [[Parrondo's paradox]]
* [[Quantum game theory]]
* [[Rationality]]
* [[Mechanism design|Reverse game theory]]
* [[Self-confirming equilibrium]]
 
; Lists
* [[List of emerging technologies]]
* [[List of games in game theory]]
* [[Outline of artificial intelligence]]
 
==Notes==
{{reflist|30em|refs=
<ref name=r1>• At [[JEL classification codes#Mathematical and quantitative methods JEL: C Subcategories|JEL:C7]] of the ''[[Journal of Economic Literature]]'' classification codes.<br />&nbsp;&nbsp; • [[Robert Aumann|R.J. Aumann]] (2008). "game theory," ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_G000007&edition=current&q=game%20theory&topicid=&result_number=4 Abstract.]<br />&nbsp;&nbsp; • [[Martin Shubik]] (1981). "Game Theory Models and Methods in Political Economy," in [[Kenneth Arrow]] and Michael Intriligator, ed., ''Handbook of Mathematical Economics'', v. 1, pp. 285–330 {{DOI|10.1016/S1573-4382(81)01011-4}}.<br />&nbsp;&nbsp; • [[Carl Shapiro (professor)|Carl Shapiro]] (1989). "The Theory of Business Strategy," ''RAND Journal of Economics'', 20(1), pp. 125–137 {{JSTOR|2555656}}.</ref>
 
<ref name=r2>• [[Leigh Tesfatsion]] (2006). "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, ''Handbook of Computational Economics'', v. 2, pp. 831-880 {{DOI|10.1016/S1574-0021(05)02016-2}}.<br />&nbsp;&nbsp; • [[Joseph Y. Halpern]] (2008). "computer science and game theory," ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_C000566&edition=current&q=&topicid=&result_number=1 Abstract].</ref>
 
<ref name=r3>• From ''[[The New Palgrave Dictionary of Economics]]'' (2008), 2nd Edition:<br />&nbsp;&nbsp;&nbsp;&nbsp; [[Roger B. Myerson]]. "mechanism design." [http://www.dictionaryofeconomics.com/article?id=pde2008_M000132&edition=current&q=mechanism%20design&topicid=&result_number=3 Abstract].<br />&nbsp;&nbsp;&nbsp;&nbsp; _____. "revelation principle." [http://www.dictionaryofeconomics.com/article?id=pde2008_R000137&edition=current&q=moral&topicid=&result_number=1 Abstract].<br />&nbsp;&nbsp; • Tuomas Sandholm. "computing in mechanism design." [http://www.dictionaryofeconomics.com/article?id=pde2008_C000563&edition=&field=keyword&q=algorithmic%20mechanism%20design&topicid=&result_number=1 Abstract.]<br />&nbsp;&nbsp; • [[Noam Nisan]] and Amir Ronen (2001). "Algorithmic Mechanism Design," ''Games and Economic Behavior'', 35(1–2), pp. [http://www.cs.cmu.edu/~sandholm/cs15-892F09/Algorithmic%20mechanism%20design.pdf 166–196].<br />&nbsp;&nbsp; • [[Noam Nisan]] ''et al''., ed. (2007). ''Algorithmic Game Theory'', Cambridge University Press. [http://www.cup.cam.ac.uk/asia/catalogue/catalogue.asp?isbn=9780521872829 Description].</ref>
 
<ref name=r4>R. Aumann and S. Hart, ed., 1994. ''Handbook of Game Theory with Economic Applications'', v. 2, outline links, ch. 30: [http://www.sciencedirect.com/science/article/pii/S1574000505800621 "Voting Procedures"] & ch. 31: [http://www.sciencedirect.com/science/article/pii/S1574000505800633 "Social Choice."]</ref>
 
<ref name=r5>• [[Vernon L. Smith]], 1992. "Game Theory and Experimental Economics: Beginnings and Early Influences," in E. R. Weintraub, ed., ''Towards a History of Game Theory'', pp. [http://books.google.com/books?hl=en&lr=&id=9CHY2Gozh1MC&oi=fnd&pg=PA241 241–282].<br />&nbsp;&nbsp; • _____, 2001. "Experimental Economics," ''[[International Encyclopedia of the Social & Behavioral Sciences]]'', pp. 5100–5108. [http://www.sciencedirect.com/science/article/pii/B0080430767022324 Abstract] per sect. 1.1 & 2.1.<br />&nbsp;&nbsp; • [[Charles R. Plott]] and Vernon L. Smith, ed., 2008. ''Handbook of Experimental Economics Results'', v. 1, Elsevier, Part 4, Games, [http://www.sciencedirect.com/science/handbooks/15740722 ch. 45–66].<br />&nbsp;&nbsp; • Vincent P. Crawford (1997). "Theory and Experiment in the Analysis of Strategic Interaction," in ''Advances in Economics and Econometrics: Theory and Applications'', pp. [http://weber.ucsd.edu/~vcrawfor/CrawfordThExp97.pdf 206–242]. Cambridge. Reprinted in Colin F. Camerer ''et al''., ed. (2003). ''Advances in Behavioral Economics'', Princeton. 1986–2003 papers. [http://press.princeton.edu/titles/8437.html Description], [http://books.google.com/books?id=sA4jJOjwCW4C&printsec=find&pg=PR7 preview], Princeton, ch. 12.<br />&nbsp;&nbsp; • Martin Shubik, 2002. "Game Theory and Experimental Gaming," in R. Aumann and S. Hart, ed., ''Handbook of Game Theory with Economic Applications'', Elsevier, v. 3, pp. 2327–2351. {{DOI|10.1016/S1574-0005(02)03025-4}}.</ref>
 
<ref name=r6>From ''The New Palgrave Dictionary of Economics'' (2008), 2nd Edition: <br />&nbsp;&nbsp; • [[Faruk Gul]]. "behavioural economics and game theory." [http://www.dictionaryofeconomics.com/article?id=pde2008_G000210&q=Behavioral%20economics%20&topicid=&result_number=2 Abstract.]<br />&nbsp;&nbsp; • [[Colin F. Camerer]]. "behavioral game theory." [http://www.dictionaryofeconomics.com/article?id=pde2008_B000302&q=Behavioral%20economics%20&topicid=&result_number=13 Abstract.]<br />&nbsp;&nbsp; • _____ (1997). "Progress in Behavioral Game Theory," ''Journal of Economic Perspectives'', 11(4), p. 172, pp. [http://authors.library.caltech.edu/22122/1/2138470%5B1%5D.pdf 167–188].<br />&nbsp;&nbsp; • _____ (2003). ''Behavioral Game Theory'', Princeton. [http://press.princeton.edu/chapters/i7517.html Description], [http://books.google.com/books?id=cr_Xg7cRvdcC&printsec=find&pg=PR7 preview] ([ctrl]+), and ch. 1 [http://press.princeton.edu/chapters/i7517.pdf link].<br />&nbsp;&nbsp; • _____, [[George Loewenstein]], and [[Matthew Rabin]], ed. (2003). ''Advances in Behavioral Economics'', Princeton. 1986–2003 papers. [http://press.princeton.edu/titles/8437.html Description], [http://books.google.com/books?id=sA4jJOjwCW4C&printsec=find&pg=PR7 contents], and [http://books.google.com/books?id=sA4jJOjwCW4C&printsec=find&pg=PR7 preview].<br />&nbsp;&nbsp; • [[Drew Fudenberg]] (2006). "Advancing Beyond ''Advances in Behavioral Economics''," ''Journal of Economic Literature'', 44(3), pp. 694–711 {{JSTOR|30032349}}.</ref>
 
<ref name=r7>• Eric Rasmusen (2007). ''Games and Information'', 4th ed. [http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP001009.html Description] and [http://books.google.com/books?id=5XEMuJwnBmUC&printsec=fnd&pg=PR5 chapter-preview.]<br />&nbsp;&nbsp; • [[David M. Kreps]] (1990). ''Game Theory and Economic Modelling''. [http://econpapers.repec.org/bookchap/oxpobooks/9780198283812.htm Description.]<br />&nbsp;&nbsp; • R. Aumann and S. Hart, ed. (1992, 2002). ''Handbook of Game Theory with Economic Applications'' v. 1, [http://www.sciencedirect.com/science/handbooks/15740005/1 ch. 3–6] and v. 3, [http://www.sciencedirect.com/science/article/pii/S1574000502030060 ch. 43].</ref>
 
<ref name=r8>• [[Jean Tirole]] (1988). ''The Theory of Industrial Organization'', MIT Press. [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=8224 Description] and chapter-preview links, pp. [http://books.google.com/books?id=HIjsF0XONF8C&printsec=find&pg=PR7 vii–ix], "General Organization," pp. [http://books.google.com/books?id=HIjsF0XONF8C&dq=find&pg=PA5 5–6], and "Non-Cooperative Game Theory: A User's Guide Manual,' " ch. 11, pp. [http://books.google.com/books?id=HIjsF0XONF8C&dq=find&pg=PA423 423–59].<br />&nbsp;&nbsp; • Kyle Bagwell and Asher Wolinsky (2002). "Game theory and Industrial Organization," ch. 49, ''Handbook of Game Theory with Economic Applications'',
v. 3, pp. [http://www.sciencedirect.com/science/article/pii/S1574000502030126 1851–1895].<br />&nbsp;&nbsp; • Martin Shubik (1959). ''Strategy and Market Structure: Competition, Oligopoly, and the Theory of Games'', Wiley. [http://devirevues.demo.inist.fr/handle/2042/29380 Description] and review [http://www.jstor.org/pss/40434883 extract].<br />&nbsp;&nbsp; • _____ with Richard Levitan (1980). ''Market Structure and Behavior'', Harvard University Press. Review [http://www.jstor.org/pss/2232276 extract].</ref>
 
<ref name=r9>• Martin Shubik (1981). "Game Theory Models and Methods in Political Economy," in ''Handbook of Mathematical Economics'', v. 1, pp. 285–330 {{DOI|10.1016/S1573-4382(81)01011-4}}.<br />&nbsp;&nbsp; •_____ (1987). ''A Game-Theoretic Approach to Political Economy''. MIT Press. [http://mitpress.mit.edu/catalog/item/default.asp?tid=5086&ttype=2 Description].</ref>
<ref name=r10>• Martin Shubik (1978). "Game Theory: Economic Applications," in W. Kruskal and J.M. Tanur, ed., ''International Encyclopedia of Statistics'', v. 2, pp.&nbsp;372–78.<br />&nbsp;&nbsp; • Robert Aumann and [[Sergiu Hart]], ed. ''Handbook of Game Theory with Economic Applications'' (scrollable to chapter-outline or abstract links):
:1992. [http://www.sciencedirect.com/science/handbooks/15740005/1 v. 1]; 1994. [http://www.sciencedirect.com/science/handbooks/15740005/2 v. 2]; 2002. [http://www.sciencedirect.com/science/handbooks/15740005/3 v. 3.]</ref>
 
<ref name=GT-A-E-00>N. Agarwal and P. Zeephongsekul. [http://www.mssanz.org.au/modsim2011/D6/agarwal.pdf Psychological Pricing in Mergers & Acquisitions using Game Theory], School of Mathematics and Geospatial Sciences, RMIT University, Melbourne</ref>
 
<ref name="algorithmic-game">Noam Nisan ''et al''., ed. (2007). ''Algorithmic Game Theory'', Cambridge University Press. [http://www.cup.cam.ac.uk/asia/catalogue/catalogue.asp?isbn=9780521872829 Description].</ref>
 
<ref name=r11>{{Citation |first=Noam |last=Nisan |first2=Amir |last2=Ronen |year=2001 |title=Algorithmic Mechanism Design |journal=Games and Economic Behavior |volume=35 |issue=1–2 |url=http://www.cs.cmu.edu/~sandholm/cs15-892F09/Algorithmic%20mechanism%20design.pdf |pages=166–196 |doi=10.1006/game.1999.0790 }}</ref>
 
<ref name=r12>• [[Joseph Y. Halpern]] (2008). "computer science and game theory," ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_C000566&edition=current&q=&topicid=&result_number=1 Abstract].<br />&nbsp;&nbsp; • {{Citation |first=Yoav |last=Shoham |year=2008 |title=Computer Science and Game Theory |journal=Communications of the ACM |volume=51 |issue=8 |pages=75–79 |url=http://www.robotics.stanford.edu/~shoham/www%20papers/CSGT-CACM-Shoham.pdf |doi=10.1145/1378704.1378721 }}<br />&nbsp;&nbsp; • {{Citation |first=Amy |last=Littman |authorlink2=Michael L. Littman |first2=Michael L. |year=2007 |title=Introduction to the Special Issue on Learning and Computational Game Theory |journal=Machine Learning |volume=67 |issue=1–2 |pages=3–6 |doi=10.1007/s10994-007-0770-1 |last2=Littman }}</ref>
}}
 
==References and further reading==
{{wikibooks|Introduction to Game Theory}}
{{Wikiversity|Game Theory}}
{{Wiktionary|game theory}}
{{commons category}}
 
===Textbooks and general references===
* {{Citation | first1=Robert J.|last1= Aumann|author1-link=Robert Aumann|year=1987 |title=game theory,|series=''[[The New Palgrave: A Dictionary of Economics]]''|volume=2|pages=460–82}}.
* ''[[The New Palgrave Dictionary of Economics]]'' (2008). 2nd Edition:
:"game theory" by Robert J. Aumann. [http://www.dictionaryofeconomics.com/article?id=pde2008_G000007&q=game%20theory&topicid=&result_number=3 Abstract.]
: "game theory in economics, origins of," by Robert Leonard. [http://www.dictionaryofeconomics.com/article?id=pde2008_G000193&goto=a&topicid=B2&result_number=10 Abstract.]
: "behavioural economics and game theory" by [[Faruk Gul]]. [http://www.dictionaryofeconomics.com/article?id=pde2008_G000210&q=Behavioral%20economics%20&topicid=&result_number=2 Abstract.]
* {{Citation | last1=Camerer | first1=Colin|author1-link=Colin Camerer | title=Behavioral Game Theory: Experiments in Strategic Interaction | publisher=Russell Sage Foundation | isbn=978-0-691-09039-9 | year=2003}} [http://press.princeton.edu/titles/7517.html Description] and [http://press.princeton.edu/chapters/i7517.html Introduction], pp.&nbsp;1–25.
* {{Citation |last1=Dixit |first1=Avinash |authorlink1=Avinash Dixit  |last2= Nalebuff |first2=Barry |authorlink2=Barry Nalebuff  |editor=[[David R. Henderson]] (ed.) |encyclopedia=[[Concise Encyclopedia of Economics]] |title=Game Theory |url=http://www.econlib.org/library/Enc/GameTheory.html |year=2008 |edition= 2nd |publisher=[[Library of Economics and Liberty]] |location=Indianapolis |isbn=978-0865976658 |oclc=237794267}}
* {{Citation | last1=Dutta | first1=Prajit K. | title=Strategies and games: theory and practice | publisher=[[MIT Press]] | isbn=978-0-262-04169-0 | year=1999}}. Suitable for undergraduate and business students.
* {{Citation | last1=Fernandez | first1=L F. | last2=Bierman | first2=H S. | title=Game theory with economic applications | publisher=[[Addison-Wesley]] | isbn=978-0-201-84758-1 | year=1998}}. Suitable for upper-level undergraduates.
* {{Citation | last2=Tirole | first2=Jean | author2-link=Jean Tirole | last1=Fudenberg | first1=Drew | title=Game theory | publisher=MIT Press | isbn=978-0-262-06141-4 | year=1991}}. Acclaimed reference text. [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=8204 Description.]
* {{Citation | last1=Gibbons | first1=Robert D. | title=Game theory for applied economists | publisher=[[Princeton University Press]] | isbn=978-0-691-00395-5 | year=1992}}. 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 | author=Robert Gibbons | year=2001}}.
* {{Citation | last1=Gintis | first1=Herbert | title=Game theory evolving: a problem-centered introduction to modeling strategic behavior | publisher=Princeton University Press | isbn=978-0-691-00943-8 | year=2000}}
* {{Citation | last1=Green | first1=Jerry R. | last2=Mas-Colell | first2=Andreu | author2-link = Andreu Mas-Colell| last3=Whinston | first3=Michael D. | title=Microeconomic theory | publisher=[[Oxford University Press]] | isbn=978-0-19-507340-9 | year=1995}}. Presents game theory in formal way suitable for graduate level.
* {{Citation | editor1-last=Hansen | editor1-first=Pelle G. | editor2-last=Hendricks | editor2-first=Vincent F. | title=Game Theory: 5 Questions | publisher=Automatic Press / VIP | location=New York, London | isbn=978-87-991013-4-4 | author=edited by Vincent F. Hendricks, Pelle G. Hansen. | year=2007}}. [http://www.gametheorists.com Snippets from interviews].
* {{Citation
| last = Howard
| first = Nigel
| title = Paradoxes of Rationality: Games, Metagames, and Political Behavior
| publisher = The MIT Press
| year = 1971
| location = Cambridge, Massachusetts
| isbn = 978-0-262-58237-7}}
* {{Citation | last1=Isaacs | first1=Rufus | author1-link=Rufus Isaacs (game theorist) | title=Differential Games: A Mathematical Theory With Applications to Warfare and Pursuit, Control and Optimization | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-40682-4 | year=1999}}
* {{Citation
| last = Julmi
| first = Christian
| title = Introduction to Game Theory
| publisher = BookBooN
| year = 2012
| location = Copenhagen
| isbn = 978-87-403-0280-6
| url = http://bookboon.com/en/business-ebooks/strategy/introduction-to-game-theory
}}
* {{Citation | last2=Shoham | first2=Yoav | last1=Leyton-Brown | first1=Kevin | title=Essentials of Game Theory: A Concise, Multidisciplinary Introduction | publisher=Morgan & Claypool Publishers | isbn=978-1-59829-593-1 | url=http://www.gtessentials.org | year=2008 | location=San Rafael, CA}}. An 88-page mathematical introduction; [http://www.morganclaypool.com/doi/abs/10.2200/S00108ED1V01Y200802AIM003 free online] at many universities.
* {{Citation | last1=Miller | first1=James H. | title=Game theory at work: how to use game theory to outthink and outmaneuver your competition | publisher=[[McGraw-Hill]] | location=New York | isbn=978-0-07-140020-6 | year=2003}}. Suitable for a general audience.
* {{Citation | last1=Myerson | first1=Roger B. | author1-link=Roger Myerson | title=Game theory: analysis of conflict | publisher=[[Harvard University Press]] | isbn=978-0-674-34116-6 | year=1991}}
* {{Citation | last1=Osborne | first1=Martin J. | title=An introduction to game theory | publisher=Oxford University Press | isbn=978-0-19-512895-6 | year=2004}}. Undergraduate textbook.
* {{Citation | last1=Papayoanou | first1=Paul | title=Game Theory for Business | publisher=Probabilistic Publishing | isbn=978-0-9647938-7-3 | year=2010}}. Primer for business men and women.
* {{Citation | last2=Zenkevich | first2=Nikolay | last1=Petrosyan | first1=Leon | title=Game Theory (Series on Optimization, 3) | publisher=World Scientific Publishers | isbn=978-981-02-2396-0 | year=1996}}
* {{Citation | last2=Rubinstein | first2=Ariel | author2-link=Ariel Rubinstein | last1=Osborne | first1=Martin J. | title=A course in game theory | publisher=MIT Press | isbn=978-0-262-65040-3 | year=1994}}. A modern introduction at the graduate level.
* {{Citation | last1=Poundstone | first1=William | title=Prisoner's Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb | publisher=Anchor | isbn=978-0-385-41580-4 | year=1992}}. A general history of game theory and game theoreticians.
* {{Citation | last1=Rasmusen | first1=Eric | title=Games and Information: An Introduction to Game Theory | url=http://www.rasmusen.org/GI/index.html | publisher=Wiley-Blackwell | edition=4th | isbn=978-1-4051-3666-2 | year=2006}}
* {{Citation | last1=Shoham | first1=Yoav | last2=Leyton-Brown | first2=Kevin | title=Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations | publisher=[[Cambridge University Press]] | isbn=978-0-521-89943-7 | url=http://www.masfoundations.org | year=2009 | location=New York}}. A comprehensive reference from a computational perspective; [http://www.masfoundations.org/download.html downloadable free online].
* {{Citation | last1=Williams | first1=John Davis | title=The Compleat Strategyst: Being a Primer on the Theory of Games of Strategy | url=http://www.rand.org/pubs/commercial_books/2007/RAND_CB113-1.pdf|format=PDF|publisher=RAND Corp. | location=Santa Monica | year=1954 | isbn=978-0-8330-4222-4}} Praised primer and popular introduction for everybody, never out of print.
* Roger McCain's Game Theory: [http://faculty.lebow.drexel.edu/McCainR//top/eco/game/game.html A Nontechnical Introduction to the Analysis of Strategy] (Revised Edition)
* Christopher Griffin (2010) ''[http://www.personal.psu.edu/cxg286/Math486.pdf Game Theory: Penn State Math 486 Lecture Notes]'', pp.&nbsp;169, CC-BY-NC-SA license, suitable introduction for undergraduates
* {{Citation | last1=Webb | first1=James N. | year=2007 | title=Game theory: decisions, interaction and evolution | series=Springer undergraduate mathematics series | publisher=Springer | isbn=1-84628-423-6 }} Consistent treatment of game types usually claimed by different applied fields, e.g. [[Markov decision process]]es.
* Joseph E. Harrington (2008) ''Games, strategies, and decision making'', Worth, ISBN 0-7167-6630-2. Textbook suitable for undergraduates in applied fields; numerous examples, fewer formalisms in concept presentation.
 
===Historically important texts===
* [[Robert Aumann|Aumann, R.J.]] and [[Lloyd Shapley|Shapley, L.S.]] (1974), ''Values of Non-Atomic Games'', Princeton University Press
* {{Citation | last1=Cournot | first1=A. Augustin | author1-link=Antoine Augustin Cournot | title=Recherches sur les principles mathematiques de la théorie des richesses | publisher=M. Rivière & C.ie | location=Paris | year=1838 | journal=Libraire des sciences politiques et sociales}}
* {{Citation | last1=Edgeworth | first1=Francis Y. | author1-link=Francis Ysidro Edgeworth | title=Mathematical Psychics | publisher=Kegan Paul | location=London | year=1881}}
* {{Citation
|title=Theory of Voting
|first=Robin
|last=Farquharson
|authorlink = Robin Farquharson
|publisher=Blackwell (Yale U.P. in the U.S.)
|year=1969
|isbn=0-631-12460-8
}}
<!-- * {{Citation | last1=Fisher | first1=Ronald | author1-link=Ronald Fisher | title=[[The Genetical Theory of Natural Selection]] | publisher=Clarendon Press | location=Oxford | year=1930}}
:*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 | author=R.A. Fisher ; edited with a foreword and notes by J. H. Bennett.}} -->
* {{Citation | last1=Luce | first1=R. Duncan | author1-link=R. Duncan Luce | last2=Raiffa | first2=Howard | author2-link=Howard Raiffa | title=Games and decisions: introduction and critical survey | publisher=[[John Wiley & Sons|Wiley]] | location=New York | year=1957}}
:*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 | author=R. Duncan Luce ; Howard Raiffa}}
* {{Citation | last1=Maynard Smith | first1=John | author1-link=John Maynard Smith | title=[[Evolution and the Theory of Games|Evolution and the theory of games]] | publisher=[[Cambridge University Press]] | isbn=978-0-521-28884-2 | year=1982}}
* {{Citation | last1=Maynard Smith | first1=John | author1-link=John Maynard Smith | first2=George R. | last2=Price | author2-link=George R. Price | title=The logic of animal conflict | year=1973 | journal=Nature | volume=246 | issue=5427 | pages=15–18 | doi=10.1038/246015a0 | bibcode=1973Natur.246...15S}}
* {{Citation | last1=Nash | first1=John | author1-link=John Forbes Nash | title= Equilibrium points in n-person games| year=1950 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | volume=36 | issue=1 | pages=48–49 | doi=10.1073/pnas.36.1.48 | pmc=1063129 | pmid=16588946|bibcode = 1950PNAS...36...48N }}
* 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.&nbsp;1095–1100.
* {{Citation | last1=von Neumann | first1=John | title=Zur Theorie der Gesellschaftsspiele | year=1928 | journal=[[Mathematische Annalen]] | volume=100 | issue=1 | pages= [http://www.springerlink.com/content/q07530916862223p/ p. 295]–320. | format= }} English translation: "On the Theory of Games of Strategy," in A. W. Tucker and R. D. Luce, ed. (1959), ''Contributions to the Theory of Games'', v. 4, p. [http://books.google.com/books?hl=en&lr=&id=9lSVFzsTGWsC&oi=fnd&pg=PA42 42.] Princeton University Press.
* {{Citation | last1=von Neumann | first1=John | author1-link=John von Neumann | last2=Morgenstern | first2=Oskar | author2-link=Oskar Morgenstern | title=[[Theory of games and economic behavior]] | publisher=[[Princeton University Press]] | year=1944}}
* {{Citation | last1=Zermelo | first1=Ernst | author1-link=Ernst Zermelo | title=Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels | year=1913 | journal=Proceedings of the Fifth International Congress of Mathematicians | volume=2 | pages=501–4}}
 
===Other print references===
* {{Citation | last1=Ben David | first1=S. | last2=Borodin | first2=Allan |author2-link=Allan Borodin| last3=Karp | first3=Richard |author3-link=Richard Karp| last4=Tardos| first4=G. | last5=Wigderson | first5= A. |author5-link=Avi_Wigderson| title=On the Power of Randomization in On-line Algorithms |url=http://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/BORODIN/paper.pdf |format=PDF| journal=Algorithmica | volume=11 |pages=2–14 | year=1994 | doi=10.1007/BF01294260 | issue=1 }}
* {{Citation | last1=Bicchieri | first1=Cristina |author1-link=Cristina Bicchieri| title=Rationality and Coordination | publisher=[[Cambridge University Press]] | isbn=0-521-57444-7 | year=1993, 2nd. edition, 1997}}
* {{Citation | last1=Downs | first1=Anthony | author1-link=Anthony Downs | title=An Economic theory of Democracy | titlelink=An Economic Theory of Democracy| publisher=Harper | publication-place=New York|year=1957}}
* {{Citation | last1=Gauthier | first1=David | author1-link=David Gauthier | title=Morals by agreement | publisher=[[Oxford University Press]] | isbn=978-0-19-824992-4 | year=1986}}
* [[Allan Gibbard]], "Manipulation of voting schemes: a general result", ''Econometrica'', Vol. 41, No. 4 (1973), pp.&nbsp;587–601.
* {{Citation | last1=Grim | first1=Patrick | last2=Kokalis | first2=Trina | last3=Alai-Tafti | first3=Ali | last4=Kilb | first4=Nicholas | last5=St Denis | first5=Paul | title=Making meaning happen | year=2004 | journal=Journal of Experimental & Theoretical Artificial Intelligence | volume=16 | issue=4 | pages=209–243 | doi=10.1080/09528130412331294715}}
* {{Citation | last1=Harper | first1=David | last2=Maynard Smith | first2=John | author2-link=John Maynard Smith | title=Animal signals | publisher=Oxford University Press | isbn=978-0-19-852685-8 | year=2003}}
* {{Citation | last1=Harsanyi | first1=John C. | author1-link=John Harsanyi | title=An equilibrium point interpretation of stable sets | year=1974 | journal=Management Science | volume=20 | issue=11 | pages=1472–1495 | doi=10.1287/mnsc.20.11.1472}}
* {{Citation | last1=Levy | first1 = Gilat | last2=Razin | first2=Ronny | title =It Takes Two: An Explanation of the Democratic Peace | url =http://papers.ssrn.com/sol3/papers.cfm?abstract_id=433844 | year=2003 | journal=Working Paper}}
* {{Citation | last1=Lewis | first1=David | author1-link=David Kellogg Lewis | title=Convention: A Philosophical Study | year=1969}}, ISBN 978-0-631-23257-5 (2002 edition)
* {{Citation | last1=McDonald | first1=John | title=Strategy in Poker, Business & War | publisher=[[W. W. Norton]] | isbn=0-393-31457-X | year=1950 – 1996}}. A layman's introduction.
* {{Citation | last1=Quine | first1=W.v.O | author1-link=Willard Van Orman Quine | title=Philosophica Essays for A.N. Whitehead | publisher=Russel and Russel Publishers | isbn=978-0-8462-0970-6 | year=1967 | chapter=Truth by Convention}}
* {{Citation | last1=Quine | first1=W.v.O | author1-link=Willard Van Orman Quine | title=Carnap and Logical Truth | year=1960 | journal=Synthese | volume=12 | issue=4 | pages=350–374 | doi=10.1007/BF00485423}}
* Mark A. Satterthwaite, "Strategy-proofness and Arrow's Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions", ''Journal of Economic Theory'' 10 (April 1975), 187–217.
* {{Citation | last1=Siegfried | first1=Tom | title=A Beautiful Math | publisher=Joseph Henry Press | isbn=0-309-10192-1| year=2006}}
* {{Citation | last1=Skyrms | first1=Brian|author1-link=Brian Skyrms| title=The Dynamics of Rational Deliberation | publisher=[[Harvard University Press]] | isbn=0-674-21885-X | year=1990}}
* {{Citation | last1=Skyrms | first1=Brian | title=Evolution of the social contract | publisher=Cambridge University Press | isbn=978-0-521-55583-8 | year=1996}}
* {{Citation | last1=Skyrms | first1=Brian | title=The stag hunt and the evolution of social structure | publisher=Cambridge University Press | isbn=978-0-521-53392-8 | year=2004}}
* {{Citation | last1=Sober | first1=Elliott | last2=Wilson | first2=David Sloan | title=Unto others: the evolution and psychology of unselfish behavior | publisher=Harvard University Press | isbn=978-0-674-93047-6 | year=1998}}
* {{Citation
| last =Thrall
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===Websites===
* {{springer|title=Games, theory of|id=p/g043280}}
* 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] &mdash; 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] &mdash; Lecture notes on Game Theory and Computer Science
* Mike Shor: [http://www.gametheory.net Game Theory .net] &mdash; Lecture notes, interactive illustrations and other information.
* Jim Ratliff's [http://virtualperfection.com/gametheory/ Graduate Course in Game Theory] (lecture notes).
* 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]
* Elmer G. Wiens: [http://www.egwald.ca/operationsresearch/gameintroduction.php Game Theory] &mdash; 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] &mdash; 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] &mdash; See [http://www.forecastingprinciples.com/paperpdf/Greenforecastinginconflict.pdf Papers] for [http://www.decisionworkshops.com/#/graphs-of-findings/4553562008 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]''.
* Benjamin Polak: [http://oyc.yale.edu/economics/game-theory Open Course on Game Theory at Yale] [https://www.youtube.com/view_play_list?p=6EF60E1027E1A10B videos of the course]
* Benjamin Moritz, Bernhard Könsgen, Danny Bures, Ronni Wiersch, (2007) ''[http://www.spieltheorie-software.de Spieltheorie-Software.de: An application for Game Theory implemented in JAVA]''.
 
{{Game theory}}
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[[Category:Game theory| ]]
[[Category:Artificial intelligence]]
[[Category:Economics]]
[[Category:Formal sciences]]
[[Category:Mathematical economics]]
[[Category:Mathematical and quantitative methods (economics)]]
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Latest revision as of 04:24, 11 January 2015

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