Game theory
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Game theory is a branch of applied mathematics that is used in the social sciences (most notably economics), biology, engineering, political science, international relations, computer science, and philosophy. 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 games), 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 nonhuman players (computers, animals, plants)" (Aumann 1987).
Traditional applications of game theory attempt to find equilibria in these games. In an equilibrium, each player of the game has adopted a strategy that they are unlikely to change. Many equilibrium concepts have been developed (most famously the Nash 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 by John von Neumann and Oskar 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 won Nobel prizes in economics, and John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology.
Contents 
[edit] Representation of games
The games studied in game theory are welldefined 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.
[edit] 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 simultaneousmove 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.
[edit] Normal form
Player 2 chooses Left 
Player 2 chooses Right 

Player 1 chooses Up 
4, 3  –1, –1 
Player 1 chooses Down 
0, 0  3, 4 
Normal form or payoff matrix of a 2player, 2strategy game 
The normal (or strategic form) game is usually represented by a matrix which shows the players, strategies, and payoffs (see the example to the right). More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions. In the accompanying example there are two players; one chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in our example). Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff of 4, and Player 2 gets 3.
When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices of other players, the game is usually presented in extensive form.
[edit] Characteristic function form
In cooperative games with transferable 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 games, assumed that when a coalition C forms, it plays against the complementary coalition () as if they were playing a 2player 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 TUgame) is given as a pair (N,v), where N denotes a set of players and is a characteristic function.
The characteristic function form has been generalised to games without the assumption of transferable utility.
[edit] 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 (Thrall & Lucas 1963).
[edit] 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 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 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 as Plato.^{[1]}
[edit] Political science
The application of game theory to political science is focused in the overlapping areas of fair division, political economy, public choice, 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 (Downs 1957), he applies a hotelling firm location model to the political process. In the Downsian model, political candidates commit to ideologies on a onedimensional 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 by Steven Brams, George Tsebelis, Gene M. Grossman and Elhanan Helpman, or David AustenSmith and Jeffrey S. Banks.
A gametheoretic 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 (Levy & Razin 2003).
[edit] Economics and business
Economists have long used game theory to analyze a wide array of economic phenomena, including auctions, bargaining, duopolies, fair division, oligopolies, social network formation, and voting systems. 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 noncooperative 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.
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.
[edit] 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, 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 ideal akin to the models used by physicists. 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 the centipede game, guess 2/3 of the average game, and the dictator game, people regularly do not play Nash equilibria. There is an ongoing debate regarding the importance of these experiments.^{[2]}
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 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).
[edit] Prescriptive or normative analysis
Cooperate  Defect  
Cooperate  1, 1  10, 0 
Defect  0, 10  5, 5 
The Prisoner's Dilemma 
On 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's best 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 nonequilibrium strategy if one expects others to play nonequilibrium 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 his own selfinterest leads both players to be worse off than had they not pursued their own selfinterests.
[edit] Biology
Hawk  Dove  
Hawk  v−c, v−c  2v, 0 
Dove  0, 2v  v, v 
The hawkdove game 
Unlike 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 by evolutionary forces. The best known equilibrium in biology is known as the evolutionarily stable strategy (or ESS), and was first introduced in (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 ratios. (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 (Harper & Maynard Smith 2003). The analysis of signaling games and other communication games has provided some 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.
Biologists have used the hawkdove game (also known as chicken) to analyze fighting behavior and territoriality.
Maynard Smith, in the preface to Evolution and 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.^{[3]}
One such phenomenon is known as biological altruism. This is a situation in which an organism appears to act in a way that benefits other organisms and is detrimental to itself. This is distinct from traditional notions of altruism because such actions are not conscious, but appear to be evolutionary adaptations to increase overall fitness. Examples can be found in species ranging from vampire bats that regurgitate blood they have obtained from a night’s hunting and give it to group members who have failed to feed, to worker bees that care for the queen bee for their entire lives and never mate, to Vervet monkeys that warn group members of a predator's approach, even when it endangers that individual's chance of survival.^{[4]} 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 incidence 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, has a coefficient of ½, because 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.^{[4]}
[edit] 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 computations. Also, game theory provides a theoretical basis to the field of multiagent systems.
Separately, game theory has played a role in online algorithms. In particular, the kserver problem, which has in the past been referred to as games with moving costs and requestanswer games (Ben David, Borodin & Karp et al. 1994). Yao's principle is a gametheoretic technique for proving lower bounds on the computational complexity of randomized algorithms, and especially of online algorithms.
The field of algorithmic game theory combines computer science concepts of complexity and algorithm design with game theory and economic theory. The emergence of the internet has motivated the development of algorithms for finding equilibria in games, markets, computational auctions, peertopeer systems, and security and information markets.^{[5]}
[edit] Philosophy
Stag  Hare  
Stag  3, 3  0, 2 
Hare  2, 0  2, 2 
Stag hunt 
Game theory has been put to several uses in philosophy. Responding to two papers by W.V.O. Quine (1960, 1967), 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 in coordination games. In addition, he first suggested that one can understand meaning in terms of signaling games. This later suggestion has been pursued by several philosophers since Lewis (Skyrms (1996), Grim, Kokalis, and AlaiTafti et al. (2004)). Following Lewis (1969) gametheoretic account of conventions, Ullmann Margalit (1977) and Bicchieri (2006) have developed theories of social norms that define them as Nash equilibria that result from transforming a mixedmotive game into a coordination game.^{[6]}
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),^{[7]} Skyrms (1990),^{[8]} and Stalnaker (1999).^{[9]}
In ethics, some authors have attempted to pursue the project, begun by Thomas Hobbes, of deriving morality from selfinterest. Since games like the Prisoner's dilemma present an apparent conflict between morality and selfinterest, explaining why cooperation is required by selfinterest is an important component of this project. This general strategy is a component of the general social contract view in political philosophy (for examples, see Gauthier (1986) and Kavka (1986).^{[10]}
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., Skyrms (1996, 2004) and Sober and Wilson (1999)).
Some assumptions used in some parts of game theory have been challenged in philosophy; psychological egoism states that rationality reduces to selfinterest—a claim debated among philosophers. (see Psychological egoism#Criticism)
[edit] Types of games
[edit] Cooperative or noncooperative
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 (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 socalled Nashprogramme has already established many of the cooperative solutions as noncooperative equilibria.
Hybrid games contain cooperative and noncooperative elements. For instance, coalitions of players are formed in a cooperative game, but these play in a noncooperative fashion.
[edit] Symmetric and asymmetric
E  F  
E  1, 2  0, 0 
F  0, 0  1, 2 
An asymmetric 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 chicken, the prisoner's dilemma, and the stag 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 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.
[edit] Zerosum and nonzerosum
A  B  
A  –1, 1  3, –3 
B  0, 0  –2, 2 
A zerosum game 
Zerosum games are a special case of constantsum games, in which choices by players can neither increase nor decrease the available resources. In zerosum 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 zerosum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose. Other zerosum games include matching pennies and most classical board games including Go and chess.
Many games studied by game theorists (including the famous prisoner's dilemma) are nonzerosum games, because some outcomes have net results greater or less than zero. Informally, in nonzerosum games, a gain by one player does not necessarily correspond with a loss by another.
Constantsum 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) zerosum game by adding an additional dummy player (often called "the board"), whose losses compensate the players' net winnings.
[edit] Simultaneous 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.
[edit] 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 imperfectinformation games, although there are some interesting examples of perfectinformation games, including the ultimatum game and centipede game. Perfectinformation games include also chess, go, mancala, and arimaa.
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.
[edit] Infinitely long games
Games, as studied by economists and realworld 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 in descriptive set theory.
[edit] 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 games allow players to choose a strategy from a continuous strategy set. For instance, Cournot competition is typically modeled with players' strategies being any nonnegative quantities, including fractional quantities.
Differential games such as the continuous pursuit and evasion game are continuous games.
[edit] Oneplayer and manyplayer games
Individual decision problems are sometimes considered "oneplayer games". While these situations are not game theoretical, they are modeled using many of the same tools within the discipline of decision theory. It is only with two or more players that a problem becomes game theoretical. A randomly acting player who makes "chance moves", also known as "moves by nature", is often added (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. Games with an infinite number of players are often called nperson games (Luce & Raiffa 1957).
[edit] Metagames
These are games the play of which is the development of the rules for another game, the target or subject game. Metagames seek to maximize the utility value of the rule set developed. The theory of metagames is related to mechanism design theory.
[edit] 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 twoperson version of the card game le Her.
James Madison made what we now recognize as a gametheoretic analysis of the ways states can be expected to behave under different systems of taxation.^{[11]}^{[12]}
It was not until the publication of Antoine 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 a duopoly and presents a solution that is a restricted version of the Nash 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 mathematician Émile 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 farreaching 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 and Oskar Morgenstern. This profound work contains the method for finding mutually consistent solutions for twoperson zerosum games. During this 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.
In 1950, the first discussion of the prisoner's dilemma appeared, and an experiment was undertaken on this game at the RAND corporation. Around this same time, 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, allowing for the analysis of noncooperative games 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 games, 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 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 games. 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 of John Maynard Smith and his evolutionarily stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge^{[13]} were introduced and analysed.
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, Roger Myerson, together with Leonid Hurwicz and Eric Maskin, was awarded of the Nobel Prize in Economics "for having laid the foundations of mechanism design theory." Among his contributions, is also the notion of proper equilibrium, and an important graduate text: Game Theory, Analysis of Conflict (Myerson 1997).
[edit] See also
[edit] Notes
 ^ Ross, Don. "Game Theory". The Stanford Encyclopedia of Philosophy (Spring 2008 Edition). Edward N. Zalta (ed.). http://plato.stanford.edu/archives/spr2008/entries/gametheory/. Retrieved on 20080821.
 ^ Experimental work in game theory goes by many names, experimental economics, behavioral economics, and behavioural game theory are several. For a recent discussion on this field see Camerer (2003).
 ^ Evolutionary Game Theory (Stanford Encyclopedia of Philosophy)
 ^ ^{a} ^{b} Biological Altruism (Stanford Encyclopedia of Philosophy)
 ^ Algorithmic Game Theory. http://www.cambridge.org/journals/nisan/downloads/Nisan_Nonprintable.pdf.
 ^ E. Ullmann Margalit, The Emergence of Norms, Oxford University Press, 1977. C. Bicchieri, The Grammar of Society: the Nature and Dynamics of Social Norms, Cambridge University Press, 2006.
 ^ "SelfRefuting Theories of Strategic Interaction: A Paradox of Common Knowledge ", Erkenntnis 30, 1989: 6985. See also Rationality and Coordination, Cambridge University Press, 1993.
 ^ The Dynamics of Rational Deliberation, Harvard University Press, 1990.
 ^ "Knowledge, Belief, and Counterfactual Reasoning in Games." In Cristina Bicchieri, Richard Jeffrey, and Brian Skyrms, eds., The Logic of Strategy. New York: Oxford University Press, 1999.
 ^ For a more detailed discussion of the use of Game Theory in ethics see the Stanford Encyclopedia of Philosophy's entry game theory and ethics.
 ^ James Madison, Vices of the Political System of the United States, April, 1787. Link
 ^ Jack Rakove, "James Madison and the Constitution", History Now, Issue 13 September 2007. Link
 ^ 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 Robert Aumann's work in the 1970s.
[edit] References
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[edit] Textbooks and general references
 Aumann, Robert J. (1987), "game theory,", The New Palgrave: A Dictionary of Economics, 2, pp. 460–82.
 _____ (2008). "game theory," The New Palgrave Dictionary of Economics. 2nd Edition. Abstract.
 Dutta, Prajit K. (1999), Strategies and games: theory and practice, MIT Press, ISBN 9780262041690. Suitable for undergraduate and business students.
 Fernandez, L F.; Bierman, H S. (1998), Game theory with economic applications, AddisonWesley, ISBN 9780201847581. Suitable for upperlevel undergraduates.
 Fudenberg, Drew; Tirole, Jean (1991), Game theory, MIT Press, ISBN 9780262061414. Acclaimed reference text, public description.
 Gibbons, Robert D. (1992), Game theory for applied economists, Princeton University Press, ISBN 9780691003955. Suitable for advanced undergraduates.

 Published in Europe as Robert Gibbons (2001), A Primer in Game Theory, London: Harvester Wheatsheaf, ISBN 9780745011592.
 Gintis, Herbert (2000), Game theory evolving: a problemcentered introduction to modeling strategic behavior, Princeton University Press, ISBN 9780691009438
 Green, Jerry R.; MasColell, Andreu; Whinston, Michael D. (1995), Microeconomic theory, Oxford University Press, ISBN 9780195073409. Presents game theory in formal way suitable for graduate level.
 Gul, Faruk (2008). "behavioural economics and game theory," The New Palgrave Dictionary of Economics. 2nd Edition. Abstract.
 edited by Vincent F. Hendricks, Pelle G. Hansen. (2007), Hansen, Pelle G.; Hendricks, Vincent F., eds., Game Theory: 5 Questions, New York, London: Automatic Press / VIP, ISBN 9788799101344. Snippets from interviews.
 Isaacs, Rufus (1999), Differential Games: A Mathematical Theory With Applications to Warfare and Pursuit, Control and Optimization, New York: Dover Publications, ISBN 9780486406824
 LeytonBrown, Kevin; Shoham, Yoav (2008), Essentials of Game Theory: A Concise, Multidisciplinary Introduction, San Rafael, CA: Morgan & Claypool Publishers, ISBN 9781598295931, http://www.gtessentials.org. An 88page mathematical introduction; free online at many universities.
 Miller, James H. (2003), Game theory at work: how to use game theory to outthink and outmaneuver your competition, New York: McGrawHill, ISBN 9780071400206. Suitable for a general audience.
 Myerson, Roger B. (1997), Game theory: analysis of conflict, Harvard University Press, ISBN 9780674341166
 Osborne, Martin J. (2004), An introduction to game theory, Oxford University Press, ISBN 9780195128956. Undergraduate textbook.
 Osborne, Martin J.; Rubinstein, Ariel (1994), A course in game theory, MIT Press, ISBN 9780262650403. A modern introduction at the graduate level.
 Poundstone, William (1992), Prisoner's Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb, Anchor, ISBN 9780385415804. A general history of game theory and game theoreticians.
 Rasmusen, Eric (2006), Games and Information: An Introduction to Game Theory (4th ed.), WileyBlackwell, ISBN 9781405136662, http://www.rasmusen.org/GI/index.html
 Shoham, Yoav; LeytonBrown, Kevin (2009), Multiagent Systems: Algorithmic, GameTheoretic, and Logical Foundations, New York: Cambridge University Press, ISBN 9780521899437, http://www.masfoundations.org. A comprehensive reference from a computational perspective; downloadable free online.
 Williams, John Davis (1954) (PDF), The Compleat Strategyst: Being a Primer on the Theory of Games of Strategy, Santa Monica: RAND Corp., ISBN 9780833042224, http://www.rand.org/pubs/commercial_books/2007/RAND_CB1131.pdf Praised primer and popular introduction for everybody, never out of print.
[edit] Historically important texts
 Aumann, R.J. and Shapley, L.S. (1974), Values of NonAtomic Games, Princeton University Press
 Cournot, A. Augustin (1838), "Recherches sur les principles mathematiques de la théorie des richesses", Libraire des sciences politiques et sociales (Paris: M. Rivière & C.ie)
 Edgeworth, Francis Y. (1881), Mathematical Psychics, London: Kegan Paul
 Fisher, Ronald (1930), The Genetical Theory of Natural Selection, Oxford: Clarendon Press

 reprinted edition: R.A. Fisher ; edited with a foreword and notes by J.H. Bennett. (1999), The Genetical Theory of Natural Selection: A Complete Variorum Edition, Oxford University Press, ISBN 9780198504405
 Luce, R. Duncan; Raiffa, Howard (1957), Games and decisions: introduction and critical survey, New York: Wiley

 reprinted edition: R. Duncan Luce ; Howard Raiffa (1989), Games and decisions: introduction and critical survey, New York: Dover Publications, ISBN 9780486659435
 Maynard Smith, John (1982), Evolution and the theory of games, Cambridge University Press, ISBN 9780521288842
 Smith, John Maynard; Price, George R. (1973), "The logic of animal conflict", Nature 246: 15–18, doi:
 Nash, John (1950), "Equilibrium points in nperson games", Proceedings of the National Academy of Sciences of the United States of America 36 (1): 48–49, doi:, http://www.pnas.org/cgi/search?sendit=Search&pubdate_year=&volume=&firstpage=&DOI=&author1=nash&author2=&title=equilibrium&andorexacttitle=and&titleabstract=&andorexacttitleabs=and&fulltext=&andorexactfulltext=and&fmonth=Jan&fyear=1915&tmonth=Feb&tyear=2008&fdatedef=15+January+1915&tdatedef=6+February+2008&tocsectionid=all&RESULTFORMAT=1&hits=10&hitsbrief=25&sortspec=relevance&sortspecbrief=relevance
 Shapley, L.S. (1953), A Value for nperson 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. 10951100.
 von Neumann, John (1928), "Zur Theorie der Gesellschaftspiele" (^{[dead link]} – ^{Scholar search}), Mathematische Annalen 100 (1): 295–320, doi:, http://www.digizeitschriften.de/home/services/pdfterms/?ID=363311
 von Neumann, John; Morgenstern, Oskar (1944), Theory of games and economic behavior, Princeton University Press
 Zermelo, Ernst (1913), "Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels", Proceedings of the Fifth International Congress of Mathematicians 2: 501–4
[edit] Other print references
 Ben David, S.; Borodin, Allan; Karp, Richard; Tardos, G.; Wigderson, A. (1994), "On the Power of Randomization in Online Algorithms" (PDF), Algorithmica 11 (1): 2–14, doi:, http://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/BORODIN/paper.pdf
 Bicchieri, Cristina (1993, 2nd. edition, 1997), Rationality and Coordination, Cambridge University Press, ISBN 0521574447
 Camerer, Colin (2003), Behavioral game theory: experiments in strategic interaction, Russesll Sage Foundation, ISBN 9780691090399
 Downs, Anthony (1957), An Economic theory of Democracy, New York: Harper
 Gauthier, David (1986), Morals by agreement, Oxford University Press, ISBN 9780198249924
 Grim, Patrick; Kokalis, Trina; AlaiTafti, Ali; Kilb, Nicholas; St Denis, Paul (2004), "Making meaning happen", Journal of Experimental & Theoretical Artificial Intelligence 16 (4): 209–243, doi:
 Harper, David; Maynard Smith, John (2003), Animal signals, Oxford University Press, ISBN 9780198526858
 Harsanyi, John C. (1974), "An equilibrium point interpretation of stable sets", Management Science 20: 1472–1495, doi:
 Kavka, Gregory S. (1986), Hobbesian moral and political theory, Princeton University Press, ISBN 9780691027654
 Levy, Gilat; Razin, Ronny (2003), "It Takes Two: An Explanation of the Democratic Peace", Working Paper, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=433844
 Lewis, David (1969), Convention: A Philosophical Study, ISBN 9780631232575 (2002 edition)
 McDonald, John (1950  1996), Strategy in Poker, Business & War, W. W. Norton, ISBN 039331457X. A layman's introduction.
 Quine, W.v.O (1967), "Truth by Convention", Philosophica Essays for A.N. Whitehead, Russel and Russel Publishers, ISBN 9780846209706
 Quine, W.v.O (1960), "Carnap and Logical Truth", Synthese 12 (4): 350–374, doi:
 Siegfried, Tom (2006), A Beautiful Math, Joseph Henry Press, ISBN 0309101921
 Skyrms, Brian (1990), The Dynamics of Rational Deliberation, Harvard University Press, ISBN 067421885X
 Skyrms, Brian (1996), Evolution of the social contract, Cambridge University Press, ISBN 9780521555838
 Skyrms, Brian (2004), The stag hunt and the evolution of social structure, Cambridge University Press, ISBN 9780521533928
 Sober, Elliott; Wilson, David Alec (1999), Unto others: the evolution and psychology of unselfish behavior, Harvard University Press, ISBN 9780674930476
 Thrall, Robert M.; Lucas, William F. (1963), "nperson games in partition function form", Naval Research Logistics Quarterly 10 (4): 281–298, doi:
[edit] Websites
 Paul Walker: History of Game Theory Page.
 David Levine: Game Theory. Papers, Lecture Notes and much more stuff.
 Alvin Roth: Game Theory and Experimental Economics page  Comprehensive list of links to game theory information on the Web
 Adam Kalai: Game Theory and Computer Science  Lecture notes on Game Theory and Computer Science
 Mike Shor: Game Theory .net  Lecture notes, interactive illustrations and other information.
 Jim Ratliff's Graduate Course in Game Theory (lecture notes).
 Valentin Robu's software tool for simulation of bilateral negotiation (bargaining)
 Don Ross: Review Of Game Theory in the Stanford Encyclopedia of Philosophy.
 Bruno Verbeek and Christopher Morris: Game Theory and Ethics
 Chris Yiu's Game Theory Lounge
 Elmer G. Wiens: Game Theory  Introduction, worked examples, play online twoperson zerosum games.
 Marek M. Kaminski: Game Theory and Politics  syllabuses and lecture notes for game theory and political science.
 Web sites on game theory and social interactions
 Kesten Green's 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) Gambit: Software Tools for Game Theory.
 Ben Polak: Open Course on Game Theory at Yale
