For other uses, see Game theory (disambiguation) or Game (disambiguation).

Game theory is a branch of applied mathematics that is often used in the context of economics. It studies strategic interactions between agents. In strategic games, agents choose strategies which will maximize their return, given the strategies the other agents choose. The essential feature is that it provides a formal modelling approach to social situations in which decision makers interact with other agents. Game theory extends the simpler optimisation approach developed in neoclassical economics.

The field of game theory came into being with the 1944 classic Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. A major center for the development of game theory was RAND Corporation where it helped to define nuclear strategies.

Game theory has played, and continues to play a large role in the social sciences, and is now also used in many diverse academic fields. Beginning in the 1970s, game theory has been applied to animal behaviour, including evolutionary theory. Many games, especially the prisoner's dilemma, are used to illustrate ideas in political science and ethics. Game theory has recently drawn attention from computer scientists because of its use in artificial intelligence and cybernetics.

In addition to its academic interest, game theory has received attention in popular culture. A Nobel Prize–winning game theorist, John Nash, was the subject of the 1998 biography by Sylvia Nasar and the 2001 film A Beautiful Mind. Game theory was also a theme in the 1983 film WarGames. Several game shows have adopted game theoretic situations, including Friend or Foe? and to some extent Survivor. The character Jack Bristow on the television show Alias is one of the few fictional game theorists in popular culture.^[1]

Although some game theoretic analyses appear similar to decision theory, game theory studies decisions made in an environment in which players interact. In other words, game theory studies choice of optimal behavior when costs and benefits of each option depend upon the choices of other individuals.

Representation of games

The games studied by game theory are well-defined mathematical objects. A game consists of a set of players, a set of moves (or strategies) available to those players, and a specification of payoffs for each combination of strategies. Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.

Extensive form

Main article: Extensive form game

The extensive form can be used to formalize games with some important order. Games here are often presented as trees (as pictured to the left). Here each vertex (or node) represents a point of choice for a player. The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree.

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

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

Normal form

	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 2-player, 2-strategy game

Main article: Normal form 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.

Characteristic function form

Main article: Cooperative game

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 the in the seminal book of von Neumann and Morgenstern who, when studying coalitional normal form games, assumed that when a coalition forms, it plays against the complementary coalition () as if they were playing a 2-player game. The equilibrium payoff of is characteristic. Now there are different models to derive coalitional values from normal form games, but not all games in characteristic function form can be derived from normal form games.

Formally, a characteristic function form game (also known as a TU-game) is given as a pair , where 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.

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 .

Types of games

Cooperative or noncooperative

Main articles: Cooperative game and 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. This classification on two binary criteria has been rejected .

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

Hybrid games contain cooperative and non-cooperative elements. For instance, coalitions of players are formed in a cooperative game, but these play in a non-cooperative fashion.

Symmetric and asymmetric

	E	F
E	1, 2	0, 0
F	0, 0	1, 2
An asymmetric game

Main article: 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 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.

Zero sum and non-zero sum

	A	B
A	–1, 1	3, –3
B	0, 0	–2, 2
A zero-sum game

Main article: Zero-sum

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 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 and chess.

Many games studied by game theorists (including the famous prisoner's dilemma) are non-zero-sum games, because some outcomes have net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.

Constant sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential gains from trade. It is possible to transform any game into a (possibly asymmetric) zero-sum game by adding an additional dummy player (often called "the board"), whose losses compensate the players' net winnings.

Simultaneous and sequential

Main article: Sequential game

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. Normal form is used to represent simultaneous games, and extensive form is used to represent sequential ones.

Perfect information and imperfect information

Main article: Perfect information

An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players know the moves previously made by all other players. Thus, only sequential games can be games of perfect information, since in simultaneous games not every player knows the actions of the others. Most games studied in game theory are imperfect information games, although there are some interesting examples of perfect information games, including the ultimatum game and centipede game. Perfect information 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 knows the strategies and payoffs of the other players but not necessarily the actions.

Infinitely long (nonterminating) games

Main article: Determinacy

Games, as studied by economists and real-world game players, are generally finished in a finite number of moves. Pure mathematicians are not so constrained, and set theorists in particular study games that last for 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 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

Most of the objects treated in most branches of game theory are discrete, with a finite number of players, moves, events, outcomes, etc. However, the concepts can be extended into the realm of real numbers. This branch has sometimes been called "differential" games, because they map to a real line, usually time, although the behaviors may be mathematically discontinuous. Much of this is discussed under such subjects as "optimization theory" and extends into many fields of engineering and physics.

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.

Application and challenges of game theory

Games in one form or another are widely used in many different disciplines.

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, 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 (1957), he applies a Hotelling firm location model to the political process. In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. The theorist shows how the political candidates will converge to the ideology preferred by the median voter. For more recent examples, see the books by George Tsebelis, Gene M. Grossman and Elhanan Helpman, or David Austen-Smith and Jeffrey S. Banks.

A game-theoretic explanation for democratic peace is that public and open debate in democracies send clear and reliable information regarding their intentions to other states. In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a nondemocracy.[1]

Game theory provides a theoretical description for a variety of observable consequences of changes in governmental policies. For example, in a static world where producers were not themselves decision makers attempting to optimize their own expenditure of resources while assuming risks, response to an increase in tax rates would imply an increase in revenues and vice versa. Game Theory inclusively weights the decision making of all participants and thus explains the contrary results illustrated by the Laffer Curve.

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. 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.

Descriptive

The first 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 rationally to maximize their wins (the Homo economicus model), but real humans often act either irrationally, or act rationally to maximize the wins of some larger group of people (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).

Prescriptive or Normative analysis

	Cooperate	Defect
Cooperate	2, 2	0, 3
Defect	3, 0	1, 1
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 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 his own self-interest leads both players to be worse off than had they not pursued their own self-interests.

Biology

	Hawk	Dove
Hawk	v−c, v−c	2v, 0
Dove	0, 2v	v, v
The hawk-dove 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 most well-known equilibrium in biology is known as the Evolutionary stable strategy or (ESS), and was first introduced by John Maynard Smith (described in his 1982 book). 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. Ronald 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 (Maynard Smith & Harper, 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.

Finally, biologists have used the hawk-dove game (also known as chicken) to analyze fighting behavior and territoriality.

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.

Separately, game theory has played a role in online algorithms. In particular, the k-server problem, which has in the past been referred to as games with moving costs and request-answer games.^[3]

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), David 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 et al. 2004).

In ethics, some 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 Gauthier 1987 and Kavka 1986).^[4]

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; Sober and Wilson 1999).

History of game theory

The first known discussion of game theory occurred in a letter written by James Waldegrave in 1713. In this letter, Waldegrave provides a minimax mixed strategy solution to a two-person version of the card game le Her. It was not until the publication of Antoine Augustin Cournot's 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 Borel did some earlier work on games, Von Neumann can rightfully be credited as the inventor of game theory. Von Neumann was a brilliant mathematician whose work was far-reaching from set theory to his calculations that were key to development of both the Atom and Hydrogen bombs and finally to his work developing computers. Von Neumann's work in game theory culminated in the 1944 book The Theory of Games and Economic Behavior by von Neumann and Oskar Morgenstern. This profound work contains the method for finding optimal solutions for two-person zero-sum 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 definition of an "optimum" strategy for multiplayer games where no such optimum was previously defined, known as Nash equilibrium. This equilibrium is sufficiently general, allowing for the analysis of non-cooperative 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 evolutionary stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge^[5] 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, published in 1991.

Notes

References

Textbooks and general references

• Aumann, R.J. (1987). "game theory ," , v. 2, pp. 460-82.
• Bierman, H. S. and L. Fernandez (1998). Game Theory with economic applications, Addison-Wesley. (suitable for upper-level undergraduates)
• Dutta, Prajit (2000). Strategies and Games: Theory and Practice, MIT Press, ISBN 0-262-04169-3 (suitable for undergraduate and business students)
• Fudenberg, Drew and Jean Tirole (1991). Game Theory, MIT Press, , ISBN 0-262-06141-4 (acclaimed reference text -- (pub. description)
• Gibbons, Robert (1992). Game Theory for Applied Economists, Princeton University Press ISBN 0-691-00395-5 (suitable for advanced undergraduates. Published in Europe by Harvester Wheatsheaf (London) with the title A primer in game theory)
• Gintis, Herbert (2000). Game Theory Evolving, Princeton University Press ISBN 0-691-00943-0
• Hendricks, Vincent F. & Hansen, Pelle G., eds. (2007) Game Theory: 5 Questions, New York, London: Automatic Press / VIP. Read snippets from interviews http://www.gametheorists.com. ISBN 87-991013-4-3
• Mas-Colell, Whinston and Green (1995): Microeconomic Theory. Oxford University Press, ISBN 0-19-507340-1. (Presents game theory in formal way suitable for graduate level)
• Miller, James (2003). ''Game Theory At Work, McGraw-Hill ISBN 0-07-140020-6. (Suitable for a general audience.)
• Myerson Roger B.. Game Theory: Analysis of Conflict, Harvard University Press, Cambridge, 1991, ISBN 0-674-34116-3
• Osborne, Martin J. (2004). An Introduction to Game Theory, Oxford University Press, New York. ISBN 0-19-512895-8 (undergraduate textbook)
• Osborne, Martin J. and Ariel Rubinstein (1994). A Course in Game Theory, MIT Press. ISBN 0-262-65040-1 (a modern introduction at the graduate level)
• Poundstone, William (1992). Prisoner's Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb, ISBN 0-385-41580-X (a general history of game theory and game theoreticians)
• Rasmusen, Eric (2006). Games and information, 4th edition, Blackwell, Available online http://www.rasmusen.org/GI/index.html.
• Rufus Isaacs. Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. ISBN 0486406822

Historically important texts

• Cournot, A. Augustin (1838). Recherches sur les principles mathematiques de la théorie des richesses, Libraire des sciences politiques et sociales, M. Rivière & C.ie, Paris.
• Edgeworth, Francis Y. (1881). Mathematical Psychics, Kegan Paul, London.
• Zermelo, Enst (1913): Über eine anwendung der mengenlehre auf die theorie des schachspiel, Proceedings Fifth International Congress of Mathematicians, vol. 2, 501-504.
• von Neumann, John (1928). Zur Theorie der Gesellschaftspiele, Mathematische Annalen 100:295-320.
• Fisher, Ronald (1930). The Genetical Theory of Natural Selection Clarendon Press, Oxford.
• von Neumann, John, and Oskar Morgenstern (1944). Theory of Games and Economic Behavior Princeton University Press
• Nash, John (1950). "Equilibrium points in n-person games", Proceedings of the National Academy of the USA 36(1):48–49.
• Luce, Duncan and Howard Raiffa (1957). Games and Decisions: Introduction and Critical Survey, Wiley, New York. Reprinted by Dover, New York, 1989, ISBN 0-486-65943-7.
• Maynard Smith, John (1982). Evolution and the Theory of Games, Cambridge University Press

Other print references

• Camerer, Colin (2003). Behavioral Game Theory Princeton University Press ISBN 0-691-09039-4
• Gauthier, David (1987). Morals by Agreement Oxford University Press ISBN 0-19-824992-6
• Green, Kesten C. (2002). Forecasting decisions in conflict situations: A comparison of game theory, role-playing, and unaided judgement. International Journal of Forecasting, 18, 321–344.
• Green, Kesten C. (2005). Game theory, simulated interaction, and unaided judgment for forecasting decisions in conflicts: Further evidence, International Journal of Forecasting, 21, 463–472.
• Grim, Patrick, Trina Kokalis, Ali Alai-Tafti, Nicholas Kilb, and Paul St Denis (2004) "Making meaning happen." Journal of Experimental & Theoretical Artificial Intelligence 16(4): 209–243.
• Harsanyi, John C. (1974.) An equilibrium point interpretation of stable sets, Management Science, 20, 1472-1495.
• Kaminski, Marek M. (2004) Games Prisoners Play Princeton University Press. ISBN 0-691-11721-7 http://webfiles.uci.edu/mkaminsk/www/book.html
• Kavka, Gregory (1986) Hobbesian Moral and Political Theory Princeton University Press. ISBN 0-691-02765-X
• Lewis, David (1969) Convention: A Philosophical Study
• Maynard Smith, J. and Harper, D. (2003) Animal Signals. Oxford University Press. ISBN 0-19-852685-7
• Quine, W.v.O (1967) "Truth by Convention" in Philosophica Essays for A.N. Whitehead Russel and Russel Publishers. ISBN 0-8462-0970-5
• Quine, W.v.O (1960) "Carnap and Logical Truth" Synthese 12(4):350–374.
• Skyrms, Brian (1996) Evolution of the Social Contract Cambridge University Press. ISBN 0-521-55583-3
• Skyrms, Brian (2004) The Stag Hunt and the Evolution of Social Structure Cambridge University Press. ISBN 0-521-53392-9.
• Sober, Elliot and David Sloan Wilson (1999) Unto Others: The Evolution and Psychology of Unselfish Behavior Harvard University Press. ISBN 0-674-93047-9
• id="CITEREFThrallLucas1963">Thrall, Robert M. & William F. Lucas (1963), "-person games in partition function form", Naval Research Logistics Quarterly 10 (4): 281-298
  
  

  Websites

  • Game Theory: 5 Questions, New York, London: Automatic Press / VIP. Read snippets from interviews http://www.gametheorists.com.
  • 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
  • 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 two-person zero-sum 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.
  
  

  	Topics in game theory
  Definitions	Normal form game Extensive form game Cooperative game Information set Preference
  Equilibrium concepts	Nash equilibrium Subgame perfection Bayesian-Nash Perfect Bayesian Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance
  Strategies	Dominant strategies Mixed strategy Tit for tat Grim trigger Collusion
  Classes of games	Symmetric game Perfect information Dynamic game Repeated game Signaling game Cheap talk Zero-sum game Mechanism design Stochastic game Nontransitive game
  Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Minority game Rock, Paper, Scissors Pirate game Dictator game Public goods game Nash bargaining game Blotto games War of attrition
  Theorems	Minimax theorem Purification theorems Folk theorem Revelation principle Arrow's theorem

  
  
  
  
  Game theory can refer to:
  
  Mathematics

  • Game theory, the study of participants' behaviour in strategic situations.
  • Combinatorial game theory, the study of move combinations in games like nim, chess, and go.

  
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  A game is a recreational activity with a set of rules.
  
  Game may also refer to:
  
  In animals:

  • Game (food), any nondomesticated animal hunted for food and sport

  
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  Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains.
  ..... Click the link for more information.
  Economics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Greek for oikos (house) and nomos (custom or law), hence "rules of the house(hold).
  ..... Click the link for more information.
  In economics, an agent is an actor in a model that (generally) solves an optimization problem. In this sense, it is equivalent to the term player, which is also used in economics, but is more common in game theory.
  ..... Click the link for more information.
  A strategy is a long term plan of action designed to achieve a particular goal, most often "winning". Strategy is differentiated from tactics or immediate actions with resources at hand by its nature of being extensively premeditated, and often practically rehearsed.
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  Neoclassical economics refers to a general approach in economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand.
  ..... Click the link for more information.
  Theory of Games and Economic Behavior
  
  60th anniversary edition, 2004
  Author John von Neumann, Oskar Morgenstern
  Country United States
  Language English
  Subject(s)
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  John von Neumann
  
  John von Neumann in the 1940s
  Born November 28 1903(1903--)
  Budapest, Austria-Hungary
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  Oskar Morgenstern
  
  Oskar Morgenstern
  Born January 24 1902(1902--)
  Görlitz, Germany
  Died July 26 1977 (aged 75)
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  Rand may refer to a number of places, people, organizations, and acronyms.
  
  Places named Rand include:

  • Rand, New South Wales, a small town in Australia
  • Rand, Lincolnshire, a small village in Lincolnshire, England

  
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  Nuclear strategy involves the development of doctrines and strategies for the production and use of nuclear weapons.
  
  As a sub-branch of military strategy, nuclear strategy attempts to match nuclear weapons as means to political ends.
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  The social sciences are a group of academic disciplines that study human aspects of the world. They diverge from the arts and humanities in that the social sciences tend to emphasize the use of the scientific method in the study of humanity, including quantitative and qualitative
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  Deconstructing Creole. Amsterdam: Benjamins.
  • Croft, W. (2000). Explaining language change: An Evolutionary Approach. London: Longman.
  • Mufwene, S.S. (1991). Pidgins, creoles, typology, and markedness. In Byrne, F. & T. Huebner (eds.) 1991).
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  prisoner's dilemma (sometimes abbreviated PD) is a type of non-zero-sum game in which two players may each "cooperate" with or "defect" (i.e. betray) the other player.
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  Political science is a branch of social science concerned with theory, description, analysis and prediction of political behavior, political systems and politics broadly-construed.
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  Ethics (via Latin ethica from the Ancient Greek ἠθική [φιλοσοφία]
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  Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems.
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  artificial intelligence (or AI) is "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions which maximizes its chances of success.
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  Cybernetics was defined by Norbert Wiener, in his book of that title, as the study of control and communication in the animal and the machine. Stafford Beer called it the science of effective organization and Gordon Pask extended it to include information flows "in all
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  The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel, commonly called the Nobel Prize in Economics, is a prize awarded each year for outstanding intellectual contributions in the field of economics.
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  John Forbes Nash Jr.
  
  John Nash in 2006.
  Born May 13 1928 (1928--) (age 79)
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  Sylvia Nasar
  Born: 1947
  Rosenheim, Germany
  Occupation: Journalist
  Biogragher
  Professor of Journalism
  
  Sylvia Nasar (born 1947 in Rosenheim, Germany, she is the John S. and James L.
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  All Movie Guide profile
  IMDb profile
  A Beautiful Mind is a 2001 American biographical film about John Forbes Nash, the Nobel Laureate (Economics) mathematician. The film was directed by Ron Howard and written by Akiva Goldsman.
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  All Movie Guide profile
  / IMDb profile
  
  

  ''This article is about the 1983 US movie. For other uses see War Games.

  
  
  WarGames is a 1983 suspense film written by Lawrence Lasker and Walter F. Parkes and directed by John Badham.
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  game show involves members of the public or celebrities, sometimes as part of a team, playing a game, perhaps involving answering quiz questions, for points or prizes. In some shows contestants compete against other players or another team whilst other shows involve contestants
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  Friend or Foe? was an American game show based on knowledge and trust, which aired on Game Show Network. The catch was whether members of a two-person team would think that their partner would be able to fairly split their cash winnings.
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  This article may be too long.
  Please discuss this issue on the talk page and help summarize or split the content into subarticles of an article series.

  This article is about the television show. For other uses, see Survivor.

  
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  Alias character
  
  Jack Bristow
  Gender Male
  Affiliation(s) Alliance of Twelve
  SD-6
  CIA
  APO
  Held ranks SD-6 Director of Operations
  CIA senior officer
  Director of CIA LA Branch (briefly)
  APO Deputy Director
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  Television (often abbreviated to TV, T.V., or more recently, tv; sometimes called telly, the tube, boob tube, or idiot box in British English) is a widely used telecommunication system for broadcasting and receiving moving pictures
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