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In mathematics game theory models strategic situations or games in which an individual's success in making choices depends on the choices of others (Myerson 1991). It is used in the social sciences (most notably in economics management operations research political science and social psychology) as well as in other formal sciences (logic computer science and statistics) and biology (particularly evolutionary biology and ecology). 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 non-human players (computers animals plants)." (Aumann 1987).

Traditional applications of game theory define and study equilibria in these games. In an equilibrium each player of the game has adopted a strategy that cannot improve his outcome given the others' strategy. Many equilibrium concepts have been developed (most famously the Nash equilibrium) to describe aspects of strategic equilibria. These equilibrium concepts are motivated differently depending on the area of application although they often overlap or coincide. This methodology has received criticism and debates continue over the appropriateness of particular equilibrium concepts the appropriateness of equilibria altogether and the usefulness of mathematical models in the social sciences.

Mathematical game theory had beginnings with some publications by mile Borel which led to his 1938 book Applications aux Jeux de Hasard. However Borel's results were limited and his conjecture about the non-existence of a mixed-strategy equilibria in two-person zero-sum games was wrong. The modern epoch of game theory began with the statement of the theorem on 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. Contents 1 History 2 Representation of games 2.1 Extensive form 2.2 Normal form 2.3 Characteristic function form 2.4 Partition function form 3 General and applied uses 3.1 Description and modeling 3.2 Prescriptive or normative analysis 3.3 Economics and business 3.4 Political science 3.5 Biology 3.6 Computer science and logic 3.7 Philosophy 4 Types of games 4.1 Cooperative or non-cooperative 4.2 Symmetric and asymmetric 4.3 Zero-sum and non-zero-sum 4.4 Simultaneous and sequential 4.5 Perfect information and imperfect information 4.6 Combinatorial games 4.7 Infinitely long games 4.8 Discrete and continuous games 4.9 Many-player and population games 4.10 Stochastic outcomes (and relation to other fields) 4.11 Metagames 5 See also 6 Notes 7 References and further reading 7.1 Textbooks and general references 7.2 Historically important texts 7.3 Other print references 7.4 Websites History 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 in 1713. 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.12 In his 1838 ''Recherches sur les principes mathmatiques de la thorie 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 a mathematical model has a winning strategy by using a Brouwer's fixed point theorem. In his 1938 book Applications aux Jeux de Hasard and earlier notes mile Borel 1938 book 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 a 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 in 1928.3 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. 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 foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. During this time period work on game theory was primarily focused 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 to allow 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 evolutionarily stable strategy. In addition the concepts of correlated equilibrium trembling hand perfection and common knowledge4 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 Roger Myerson together with Leonid Hurwicz and Eric Maskin 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 (Myerson 1997). Representation of games See also: List of games in game theory The games studied in game theory are well-defined mathematical objects. A game consists of a set of players a set of moves (or strategies) available to those players and a specification of payoffs for each combination of strategies. Most cooperative games are presented in the characteristic function form while the extensive and the normal forms are used to define noncooperative games. Extensive form Main article: Extensive form game An extensive form game The extensive form can be used to formalize games with a time sequencing of moves. Games here are played on 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. The extensive form can be viewed as a multi-player generalization of a decision tree. (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 imperfect information section.) 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. 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. (Leyton-Brown & Shoham 2008 p. 35) 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 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 2-player game. The equilibrium payoff of C is characteristic. Now there are different models to derive coalitional values from normal form games but not all games in characteristic function form can be derived from normal form games. Formally a characteristic function form game (also known as a TU-game) is given as a pair (Nv) 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. 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). 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 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.5 Description and modeling A three stage Centipede Game The first known use is to describe and model how 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.6 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 -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 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. Economics and business This article is incomplete and may require expansion or cleanup. Please help to improve the article or discuss the issue on the talk page. (November 2010) 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 and to model across such broad classifications as mathematical economics7 behavioral economics8 political economy9 and industrial organization.10 This research usually focuses on particular sets of strategies known as equilibria in games. These "solution concepts" are usually based on what is required by norms of rationality. In non-cooperative games the most famous of these is 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 and prescriptive. Political science The application of game theory to political science is focused in the overlapping areas of fair division political economy public choice war bargaining positive political theory and social choice theory. In each of these areas researchers have developed game-theoretic models in which the players are often voters states special interest groups and politicians. 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 the 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. 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 non-democracy (Levy & Razin 2003). Biology Hawk Dove Hawk 20 20 80 40 Dove 40 80 60 60 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 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. Ants have also been shown to exhibit feed-forward behavior akin to fashion see Butterfly Economics. Biologists have used the game of chicken to analyze fighting behavior and territoriality.citation needed 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.11 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.12 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 because (on average) an individual shares of the alleles in its sibling's offspring. Ensuring that enough of a siblings offspring survive to adulthood precludes the necessity of the altruistic individual producing offspring.12 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 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 computations. Also game theory provides a theoretical basis to the field of multi-agent systems. Separately game theory has played a role in online algorithms. In particular the k-server problem which has in the past been referred to as games with moving costs and request-answer games (Ben David Borodin & Karp et al. 1994). Yao's principle is a game-theoretic 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 peer-to-peer systems and security and information markets.13 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 Alai-Tafti et al. (2004)). Following Lewis (1969) game-theoretic 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 mixed-motive game into a coordination game.14 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)15 Skyrms (1990)16 and Stalnaker (1999).17 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 (1986) and Kavka (1986).18 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 self-interesta claim debated among philosophers. (see Psychological egoism#Criticisms) Types of games Cooperative or non-cooperative 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 (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-programmeclarification needed 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 22 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 (game theory) 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 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 game (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. 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. Perfect information and imperfect information A game of imperfect information (the dotted line represents ignorance on the part of player 2 formally called an 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 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. Recreational games of perfect information games include chess 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 of the other players but not necessarily the actions. Games of incomplete information can be reduced however to games of imperfect information by introducing "moves by nature". (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 some general questions.19 Games of perfect information have been studied in combinatorial game theory which has developed novel representations e.g. surreal numbers as well as combinatorial and algebraic (and sometimes non-constructive) proof methods to solve games of certain types including some "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.2021 A typical game that has been solved this way is hex. A related field of study drawing from computational complexity theory is game complexity which is concerned with the estimating the computational difficulty of finding optimal strategies.22 Research in artificial intelligence has addressed both perfect and imperfect (or incomplete) information games that have very complex combinatorial structure (like chess go or backgammon) for which no provable optimal strategies have been found. The practical solutions involve computational heuristics like alpha-beta pruning or use of artificial neural networks trained by reinforcement learning which make games more tractable in computing practice.2319 Infinitely long games Main article: 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 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 gameseven 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 games allow players to choose a strategy from a continuous strategy set. For instance Cournot competition is typically modeled with players' strategies being any non-negative quantities including fractional quantities. Differential games such as the continuous pursuit and evasion game are continuous games. Many-player and population games Games with an arbitrary but finite number of players are often called n-person games (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. (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 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 processes (MDP).citation needed Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" also known as "moves by nature" (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 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.24 (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.24 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. The term metagame analysis is also used to refer to a practical approach developed by Nigel Howard (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 drama theory. See also Chainstore paradox Combinatorial game theory Glossary of game theory Intra-household bargaining List of games in game theory Quantum game theory Rationality Reverse Game Theory Self-confirming equilibrium Parrondo's paradox Notes 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 J. v. Neumann (1928). "Zur Theorie der Gesellschaftsspiele" Mathematische Annalen 100(1) pp. 295-320. 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 p. 13-42. 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. Ross Don. "Game Theory". The Stanford Encyclopedia of Philosophy (Spring 2008 Edition). Edward N. Zalta (ed.). http://plato.stanford.edu/archives/spr2008/entries/game-theory/. Retrieved 2008-08-21.  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). Kenneth J. Arrow and Michael D. Intriligator ed. (1981) Handbook of Mathematical Economics (1981) v. 1. Faruk Gul 2008. "behavioural economics and game theory" The New Palgrave Dictionary of Economics 2nd Edition. Abstract. Martin Shubik (1981). "Game Theory Models and Methods in Political Economy" in Handbook of Mathematical Economics v. 1 pp. 285-330. Jean Tirole 1988. The Theory of Industrial Organization MIT Press. Description and chapter-preview links via scroll down. Evolutionary Game Theory (Stanford Encyclopedia of Philosophy) a b Biological Altruism (Stanford Encyclopedia of Philosophy) Algorithmic Game Theory Cambridge University Press http://www.cambridge.org/journals/nisan/downloads/NisanNon-printable.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. "Self-Refuting Theories of Strategic Interaction: A Paradox of Common Knowledge " Erkenntnis 30 1989: 69-85. 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. a b Jrg Bewersdorff (2005). Luck logic and white lies: the mathematics of games. A K Peters Ltd.. pp. ix-xii and chapter 31. ISBN 9781568812106.  Albert Michael H.; Nowakowski Richard J.; Wolfe David (2007). Lessons in Play: In Introduction to Combinatorial Game Theory. A K Peters Ltd. pp. 3-4. ISBN 978-1-56881-277-9.  Beck Jzsef (2008). Combinatorial games: tic-tac-toe theory. Cambridge University Press. pp. 1-3. ISBN 9780521461009.  Robert A. Hearn; Erik D. Demaine (2009). Games Puzzles and Computation. A K Peters Ltd.. ISBN 9781568813226.  M. Tim Jones (2008). Artificial Intelligence: A Systems Approach. Jones & Bartlett Learning. pp. 106118. ISBN 9780763773373.  a b Hugh Brendan McMahan (2006) Robust Planning in Domains with Stochastic Outcomes Adversaries and Partial Observability CMU-CS-06-166 pp. 3-4 References and further reading Wikibooks has a book on the topic of Introduction to Game Theory Wikiversity has learning materials about Game Theory Look up game theory in Wiktionary the free dictionary. Wikimedia Commons has media related to: Game theory Textbooks and general references Aumann Robert J. (1987) game theory The New Palgrave: A Dictionary of Economics 2 pp. 46082 . Aumann Robert J. and Sergiu Hart ed. (1992 1994 2002). Handbook of Game Theory with Economic Applications 3 v. Elsevier. Table of Contents and "Review Article" (Abstract) links. The New Palgrave Dictionary of Economics (2008). 2nd Edition: "game theory" by Robert J. Aumann. Abstract. "game theory in economics origins of" by Robert Leonard. Abstract. "behavioural economics and game theory" by Faruk Gul. Abstract. Camerer Colin (2003) Behavioral Game Theory: Experiments in Strategic Interaction Russell Sage Foundation ISBN 978-0-691-09039-9  Description amd Introduction pp. 125. Dutta Prajit K. (1999) Strategies and games: theory and practice MIT Press ISBN 978-0-262-04169-0 . Suitable for undergraduate and business students. Fernandez L F.; Bierman H S. (1998) Game theory with economic applications Addison-Wesley ISBN 978-0-201-84758-1 . Suitable for upper-level undergraduates. Fudenberg Drew; Tirole Jean (1991) Game theory MIT Press ISBN 978-0-262-06141-4 . Acclaimed reference text. Description. Gibbons Robert D. (1992) Game theory for applied economists Princeton University Press ISBN 978-0-691-00395-5 . Suitable for advanced undergraduates. Published in Europe as Robert Gibbons (2001) A Primer in Game Theory London: Harvester Wheatsheaf ISBN 978-0-7450-1159-2 . Gintis Herbert (2000) Game theory evolving: a problem-centered introduction to modeling strategic behavior Princeton University Press ISBN 978-0-691-00943-8  Green Jerry R.; Mas-Colell Andreu; Whinston Michael D. (1995) Microeconomic theory Oxford University Press ISBN 978-0-19-507340-9 . Presents game theory in formal way suitable for graduate level. 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. Howard Nigel (1971) Paradoxes of Rationality: Games Metagames and Political Behavior Cambridge Massachusetts: The MIT Press ISBN 978-0262582377  Isaacs Rufus (1999) Differential Games: A Mathematical Theory With Applications to Warfare and Pursuit Control and Optimization New York: Dover Publications ISBN 978-0-486-40682-4  Leyton-Brown Kevin; Shoham Yoav (2008) Essentials of Game Theory: A Concise Multidisciplinary Introduction San Rafael CA: Morgan & Claypool Publishers ISBN 978-1-598-29593-1 http://www.gtessentials.org . An 88-page 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: McGraw-Hill ISBN 978-0-07-140020-6 . Suitable for a general audience. Myerson Roger B. (1991) Game theory: analysis of conflict Harvard University Press ISBN 978-0-674-34116-6  Osborne Martin J. (2004) An introduction to game theory Oxford University Press ISBN 978-0-19-512895-6 . Undergraduate textbook. Papayoanou Paul (2010) Game Theory for Business Probabilistic Publishing ISBN 978-09647938-7-3 . Primer for business men and women. Osborne Martin J.; Rubinstein Ariel (1994) A course in game theory MIT Press ISBN 978-0-262-65040-3 . 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 978-0-385-41580-4 . A general history of game theory and game theoreticians. Rasmusen Eric (2006) Games and Information: An Introduction to Game Theory (4th ed.) Wiley-Blackwell ISBN 978-1-4051-3666-2 http://www.rasmusen.org/GI/index.html  Shoham Yoav; Leyton-Brown Kevin (2009) Multiagent Systems: Algorithmic Game-Theoretic and Logical Foundations New York: Cambridge University Press ISBN 978-0-521-89943-7 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/commercialbooks/2007/RANDCB113-1.pdf  Praised primer and popular introduction for everybody never out of print. Roger McCain's Game Theory: A Nontechnical Introduction to the Analysis of Strategy (Revised Edition) Christopher Griffin (2010) Game Theory: Penn State Math 486 Lecture Notes pp. 169 CC-BY-NC-SA license suitable introduction for undergraduates Webb James N. (2007) Game theory: decisions interaction and evolution Springer undergraduate mathematics series Springer ISBN 1846284236  Consistent treatment of game types usually claimed by different applied fields e.g. Markov decision processes. Joseph E. Harrington (2008) Games strategies and decision making Worth ISBN 0716766302. Textbook suitable for undergraduates in applied fields; numerous examples fewer formalisms in concept presentation. Historically important texts Aumann R.J. and Shapley L.S. (1974) Values of Non-Atomic Games Princeton University Press Cournot A. Augustin (1838) "Recherches sur les principles mathematiques de la thorie des richesses" Libraire des sciences politiques et sociales (Paris: M. Rivire & C.ie)  Edgeworth Francis Y. (1881) Mathematical Psychics London: Kegan Paul  Farquharson Robin (1969) Theory of Voting Blackwell (Yale U.P. in the U.S.) ISBN 0631124608  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 978-0-486-65943-5  Maynard Smith John (1982) Evolution and the theory of games Cambridge University Press ISBN 978-0-521-28884-2  Maynard Smith John; Price George R. (1973) "The logic of animal conflict" Nature 246 (5427): 1518 doi:10.1038/246015a0  Nash John (1950) "Equilibrium points in n-person games" Proceedings of the National Academy of Sciences of the United States of America 36 (1): 4849 doi:10.1073/pnas.36.1.48 PMC 1063129 PMID 16588946 http://www.pnas.org/cgi/searchsenditSearch&pubdateyear&volume&firstpage&DOI&author1nash&author2&titleequilibrium&andorexacttitleand&titleabstract&andorexacttitleabsand&fulltext&andorexactfulltextand&fmonthJan&fyear1915&tmonthFeb&tyear2008&fdatedef15+January+1915&tdatedef6+February+2008&tocsectionidall&RESULTFORMAT1&hits10&hitsbrief25&sortspecrelevance&sortspecbriefrelevance  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. 10951100. von Neumann John (1928) "Zur Theorie der Gesellschaftsspiele" Mathematische Annalen 100 (1): p. 295320.  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 p. 13-42. Princeton University Press. von Neumann John; Morgenstern Oskar (1944) Theory of games and economic behavior Princeton University Press  Zermelo Ernst (1913) "ber eine Anwendung der Mengenlehre auf die Theorie des Schachspiels" Proceedings of the Fifth International Congress of Mathematicians 2: 5014  Other print references Ben David S.; Borodin Allan; Karp Richard; Tardos G.; Wigderson A. (1994) "On the Power of Randomization in On-line Algorithms" (PDF) Algorithmica 11 (1): 214 doi:10.1007/BF01294260 http://www.math.ias.edu/avi/PUBLICATIONS/MYPAPERS/BORODIN/paper.pdf  Bicchieri Cristina (1993 2nd. edition 1997) Rationality and Coordination Cambridge University Press ISBN 0-521-57444-7  Downs Anthony (1957) An Economic theory of Democracy New York: Harper  Gauthier David (1986) Morals by agreement Oxford University Press ISBN 978-0-19-824992-4  Allan Gibbard "Manipulation of voting schemes: a general result" Econometrica Vol. 41 No. 4 (1973) pp. 587601. Grim Patrick; Kokalis Trina; Alai-Tafti Ali; Kilb Nicholas; St Denis Paul (2004) "Making meaning happen" Journal of Experimental & Theoretical Artificial Intelligence 16 (4): 209243 doi:10.1080/09528130412331294715  Harper David; Maynard Smith John (2003) Animal signals Oxford University Press ISBN 978-0-19-852685-8  Harsanyi John C. (1974) "An equilibrium point interpretation of stable sets" Management Science 20 (11): 14721495 doi:10.1287/mnsc.20.11.1472  Levy Gilat; Razin Ronny (2003) "It Takes Two: An Explanation of the Democratic Peace" Working Paper http://papers.ssrn.com/sol3/papers.cfmabstractid433844  Lewis David (1969) Convention: A Philosophical Study  ISBN 978-0-631-23257-5 (2002 edition) McDonald John (1950 - 1996) Strategy in Poker Business & War W. W. Norton ISBN 0-393-31457-X . A layman's introduction. Quine W.v.O (1967) "Truth by Convention" Philosophica Essays for A.N. Whitehead Russel and Russel Publishers ISBN 978-0-8462-0970-6  Quine W.v.O (1960) "Carnap and Logical Truth" Synthese 12 (4): 350374 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) 187217. Siegfried Tom (2006) A Beautiful Math Joseph Henry Press ISBN 0-309-10192-1  Skyrms Brian (1990) The Dynamics of Rational Deliberation Harvard University Press ISBN 0-674-21885-X  Skyrms Brian (1996) Evolution of the social contract Cambridge University Press ISBN 978-0-521-55583-8  Skyrms Brian (2004) The stag hunt and the evolution of social structure Cambridge University Press ISBN 978-0-521-53392-8  Sober Elliott; Wilson David Sloan (1998) Unto others: the evolution and psychology of unselfish behavior Harvard University Press ISBN 978-0-674-93047-6  Thrall Robert M.; Lucas William F. (1963) "n-person games in partition function form" Naval Research Logistics Quarterly 10 (4): 281298 doi:10.1002/nav.3800100126  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 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. McKelvey Richard D. McLennan Andrew M. and Turocy Theodore L. (2007) Gambit: Software Tools for Game Theory. Benjamin Polak: Open Course on Game Theory at Yale videos of the course Benjamin Moritz Bernhard Knsgen Danny Bures Ronni Wiersch (2007) Spieltheorie-Software.de: An application for Game Theory implemented in JAVA. v d eTopics in game theory Definitions Normal-form game  Extensive-form game  Cooperative game  Succinct 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  Pareto efficiency  Quantal response equilibrium  Self-confirming equilibrium  Strong Nash equilibrium Strategies Dominant strategies  Pure strategy  Mixed strategy  Tit for tat  Grim trigger  Collusion  Backward induction  Markov strategy Classes of games Symmetric game  Perfect information  Simultaneous game  Sequential game  Repeated game  Signaling game  Cheap talk  Zerosum game  Mechanism design  Bargaining problem  Stochastic game  Large poisson game  Nontransitive game  Global games Games Prisoner's dilemma  Traveler's dilemma  Coordination game  Chicken  Centipede game  Volunteer's dilemma  Dollar auction  Battle of the sexes  Stag hunt  Matching pennies  Ultimatum game  Rock-paper-scissors  Pirate game  Dictator game  Public goods game  Blotto games  War of attrition  El Farol Bar problem  Cake cutting  Cournot game  Deadlock  Diner's dilemma  Guess 2/3 of the average  Kuhn poker  Nash bargaining game  Screening game  Trust game  Princess and monster game  Monty Hall problem Theorems Minimax theorem  Nash's theorem  Purification theorem  Folk theorem  Revelation principle  Arrow's impossibility theorem See also Tragedy of the commons  Tyranny of small decisions  All-pay auction  List of games in game theory v d eAreas of mathematics Areas Arithmetic  Algebra (elementary  linear  multilinear  abstract)  Geometry (Discrete geometry  Algebraic geometry  Differential geometry)  Calculus/Analysis  Set theory  Logic  Category theory  Number theory  Combinatorics  Graph theory  Topology  Lie theory  Differential equations/Dynamical systems  Mathematical physics  Numerical analysis  Computation  Information theory  Probability  Statistics  Optimization  Control theory  Game theory Divisions Pure mathematics  Applied mathematics  Discrete mathematics  Computational mathematics Category   Mathematics portal   Outline  Lists