Game theory is a branch of applied mathematics that is used in the social sciences (most notably economics), biology, political science, computer science, and philosophy. 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 The social sciences comprise academic disciplines concerned with the study of the social life of human groups and individuals including Anthropology, Communication studies Economics is the social science that studies the production distribution, and consumption of goods and services. Foundations of modern biology There are five unifying principles Political science is a branch of Social sciences that deals with the theory and practice of Politics and the description and analysis of Political systems Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Game theory attempts to mathematically capture behavior in strategic situations, in which an individual's success in making choices depends on the choices of others. While initially developed to analyze competitions in which one individual does better at another's expense (zero sum games), it has been expanded to treat a wide class of interactions, which are classified according to several criteria. In Game theory and Economic theory, zero-sum describes a situation in which a participant's gain or loss is exactly balanced by the losses or gains of the other Game theory is a branch of Applied mathematics that is used in the Social sciences (most notably Economics) Biology, Engineering, 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 attempt to find equilibria in these games—sets of strategies in which individuals are unlikely to change their behavior. Many equilibrium concepts have been developed (most famously the Nash equilibrium) in an attempt to capture this idea. In Game theory, a solution concept is a formal rule for predicting how the game will be played In Game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it is a Solution concept of a game involving two or more players in which These equilibrium concepts are motivated differently depending on the field of application, although they often overlap or coincide. This methodology is not without criticism, and debates continue over the appropriateness of particular equilibrium concepts, the appropriateness of equilibria altogether, and the usefulness of mathematical models more generally.

Although some developments occurred before it, the field of game theory came into being with the 1944 book Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. Theory of Games and Economic Behavior, published in 1944 by Princeton University Press, is a book by Mathematician John von Neumann Oskar Morgenstern ( January 24, 1902 – July 26, 1977) was a German born Austrian economist. This theory was developed extensively in the 1950s by many scholars. Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. Eight game theorists have won Nobel prizes in economics, and John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. The Nobel Prize (Nobelpriset (Nobelprisen is a Swedish prize established in the 1895 will of Swedish chemist Alfred Nobel; it was first awarded in Peace, Literature Maynard Smith redirects here -- for other uses see Maynard Smith (disambiguation Professor John Maynard Smith, F The annual Crafoord Prize is a science prize established in 1980 by Holger Crafoord, a Swedish industrialist and his wife Anna-Greta Crafoord

## Representation of games

The games studied by game theory are well-defined mathematical objects. Game theory studies strategic interaction between individuals in situations called games 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. A player of a Game is a participant therein The term 'player' is used with this same meaning both in Game theory and in ordinary recreational Games In Game theory, a player's strategy in a game is a complete plan of action for whatever situation might arise this fully determines the player's behaviour 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 some important order. Games here are often presented as trees (as pictured to the left). In Graph theory, a tree is a graph in which any two vertices are connected by exactly one path. Here each vertex (or node) represents a point of choice for a player. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. 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. In Game theory, an information set is a set that for a particular player establishes all the possible moves that could have taken place in the game so far given what that e. , the players do not know at which point they are), or a closed line is drawn around them.

### Normal form

 Player 2chooses Left Player 2chooses Right Player 1chooses Up 4, 3 –1, –1 Player 1chooses 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). In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally 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

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 ($N\setminus C$) as if they were playing a 2-player game. Oskar Morgenstern ( January 24, 1902 – July 26, 1977) was a German born Austrian economist. The equilibrium payoff of C is characteristic. Now there are different models to derive coalitional values from normal form games, but not all games in characteristic function form can be derived from normal form games.

Formally, a characteristic function form game (also known as a TU-game) is given as a pair (N,v), where N denotes a set of players and $v:2^N\longrightarrow\mathbb{R}$ is a characteristic function.

The characteristic function form has been generalised to games without the assumption of transferable utility. Transferable utility is a term used in cooperative Game theory and in Economics.

### Partition function form

The characteristic function form ignores the possible externalities of coalition formation. In Economics, an externality is an impact on any party not directly involved in an economic decision 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).

## Application and challenges

Game theory has been used to study a wide variety of human and animal behaviors. It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers. Economics is the social science that studies the production distribution, and consumption of goods and services. The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociology, 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). Sir Ronald Aylmer Fisher, FRS ( 17 February 1890 – 29 July 1962) was an English Statistician, Evolutionary Charles Robert Darwin (February 12 1809 &ndash April 19 1882 was an English naturalist, who realised and demonstrated that all Species of life 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. Maynard Smith redirects here -- for other uses see Maynard Smith (disambiguation Professor John Maynard Smith, F Evolution and the Theory of Games is a 1982 Book by the British evolutionary biologist John Maynard Smith on Evolutionary

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. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Game theoretic arguments of this type can be found as far back as Plato. Biography Early life Birth and family Plato was born in Athens Greece

### 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. Political science is a branch of Social sciences that deals with the theory and practice of Politics and the description and analysis of Political systems "Cake cutting" redirects here For the wedding tradition see Wedding reception#Wedding_cake. Political economy originally was the term for studying production buying and selling and their relations with law custom and government Public choice in economic theory is the use of modern Economic tools to study problems that are traditionally in the province of Political science. Positive political theory or explanatory political theory is the study of Politics using Formal methods such as Set theory, Statistical analysis Social choice theory studies voting rules that govern and describe how individual preferences are aggregated to form a collective preference 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. Anthony Downs is a noted scholar in Public policy and Public administration, and since 1977 is a Senior Fellow at the Brookings Institution in Washington In his book An Economic Theory of Democracy (Downs 1957), he applies a hotelling firm location model to the political process. An Economic Theory of Democracy is a Political science treatise written by Anthony Downs, published in 1957 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 Steven Brams, George Tsebelis, Gene M. Steven J Brams (born November 28, 1940) is a game theorist and Political scientist at the New York University department of politics George Tsebelis is a Professor of Political Science at the University of Michigan. Grossman and Elhanan Helpman, or David Austen-Smith and Jeffrey S. Elhanan Helpman (born March 30, 1946 in Jalal-Abad in the Fergana Valley, former Soviet Union) is an Israeli American 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. The democratic peace theory (or liberal peace theory or simply the democratic peace) holds that democracies &mdash usually liberal democracies In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a nondemocracy (Levy & Razin 2003).

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. In Economics, the Laffer curve is used to illustrate the idea that increases in the rate of Taxation may sometimes decrease Tax revenue.

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. "Auctioneer" redirects here For the DC Comics supervillain see Auctioneer (comics. Bargaining or Haggling is a type of Negotiation in which the buyer and seller of a good or service dispute the price which will be paid and the exact nature A true duopoly is a specific type of Oligopoly where only two producers exist in one Market. "Cake cutting" redirects here For the wedding tradition see Wedding reception#Wedding_cake. An oligopoly is a Market form in which a Market or Industry is dominated by a small number of sellers (oligopolists A social network is a Social structure made of nodes (which are generally individuals or organizations that are tied by one or more specific types of interdependency such as A voting system allows voters to choose between options often in an Election where candidates are selected for public office. This research usually focuses on particular sets of strategies known as equilibria in games. In Game theory, a solution concept is a formal rule for predicting how the game will be played These "solution concepts" are usually based on what is required by norms of rationality. In Economics and Game theory, the participants are sometimes considered to have perfect rationality: that is they always act in a way that maximizes their Utility In non-cooperative games, the most famous of these is the Nash equilibrium. In Game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it is a Solution concept of a game involving two or more players in which 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. In Economics, utility is a measure of the relative satisfaction from or desirability of Consumption of various Goods and services. 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

A three stage Centipede Game

The first known use is to inform us about how actual human populations behave. In Game theory, the centipede game, first introduced by Rosenthal (1981 is an Extensive form game in which two players take turns choosing either to take Some scholars believe that by finding the equilibria of games they can predict how actual human populations will behave when confronted with situations analogous to the game being studied. This particular view of game theory has come under recent criticism. First, it is criticized because the assumptions made by game theorists are often violated. Game theorists may assume players always act in a way to directly maximize their wins (the Homo economicus model), but in practice, humans behaviour is often contrary to this model. Homo economicus, or Economic man, is the concept in some Economic theories of man (that is a Human) as a rational, perfectly informed and Explanations of this phenomenon are many; irrationality, new models of deliberation, or even different motives (like that of altruism). Irrationality is talking or acting without regard of Rationality. This article refers to legal deliberation for other meanings of the word refer to its Wiktionary entry. Altruism is selfless concern for the welfare of others It is a traditional Virtue in many cultures and central to many religious traditions 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. Idealization ( British English: idealisation) is the process by which scientific models assume facts about the phenomenon being modeled that are certainly A physicist is a Scientist who studies or practices Physics. Physicists study a wide range of physical phenomena in many branches of physics spanning 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. In Game theory, the centipede game, first introduced by Rosenthal (1981 is an Extensive form game in which two players take turns choosing either to take In Game theory, Guess 2/3 of the average is a game where several people guess what 2/3 of the average of their guesses will be and where the numbers are restricted to the The dictator game is a very simple game in Experimental economics, similar to the Ultimatum game. There is an ongoing debate regarding the importance of these experiments. [1]

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. Evolutionary game theory (EGT is the application of interaction dependent strategy drift in populations to Game theory. These models presume either no rationality or bounded rationality on the part of players. Some models of Human behavior in the Social sciences assume that Humans can be reasonably approximated or described as " rational " entities (see Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Natural selection is the process by which favorable Heritable traits become more common in successive Generations of a Population of Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning (for example, fictitious play dynamics). In Game theory, fictitious play is a learning rule first introduced by G

#### 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. In Game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it is a Solution concept of a game involving two or more players in which In Game theory, the best response is the strategy (or strategies which produces the most favorable outcome for a player taking other players' strategies 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. In Game theory, Guess 2/3 of the average is a game where several people guess what 2/3 of the average of their guesses will be and where the numbers are restricted to the

Second, the Prisoner's dilemma presents another potential counterexample. The Prisoner's Dilemma constitutes a problem in Game theory. It was originally framed by Merrill Flood and Melvin Dresher 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. Foundations of modern biology There are five unifying principles Fitness (often denoted w in Population genetics models is a central concept in evolutionary theory. 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. In Game theory, a solution concept is a formal rule for predicting how the game will be played eVolution is the third Album by eLDee, it was due to be released in 2008 The best known equilibrium in biology is known as the Evolutionarily stable strategy or (ESS), and was first introduced by John Maynard Smith (described in his 1982 book). In Game theory and Behavioural ecology, an evolutionarily stable strategy (ESS is a strategy which if adopted by a population of players Maynard Smith redirects here -- for other uses see Maynard Smith (disambiguation Professor John Maynard Smith, F Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium. In Game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it is a Solution concept of a game involving two or more players in which

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. Sex ratio is the Ratio of Males to Females in a Population. The primary sex ratio is the ratio at the time of conception secondary sex ratio is 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. Sir Ronald Aylmer Fisher, FRS ( 17 February 1890 – 29 July 1962) was an English Statistician, Evolutionary

Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication (Maynard Smith & Harper, 2003). Evolutionary game theory (EGT is the application of interaction dependent strategy drift in populations to Game theory. Animal communication is any Behaviour on the part of one Animal that has an effect on the current or future behaviour of another animal Maynard Smith redirects here -- for other uses see Maynard Smith (disambiguation Professor John Maynard Smith, F The analysis of signaling games and other communication games has provided some insight into the evolution of communication among animals. Signalling games are Dynamic games with two players the sender (S and the receiver (R In Game theory, cheap talk is communication between players which does not directly affect the payoffs of the game 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. Mobbing behavior is an antipredator behavior which occurs when individuals of a certain species mob a Predator by cooperatively attacking or

Biologists have used the hawk-dove game (also known as chicken) to analyze fighting behavior and territoriality. The game of Chicken, also known as the Hawk-Dove or Snowdrift game is an influential model of conflict for two players in Game theory.

Maynard Smith, in the preface to Evolution of the Theory of Games writes, "[p]aradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behaviour for which it was originally designed. " Evolutionary game theory has been used to explain many seemingly incongruous phenomena in nature. [2]

One such phenomena is known as biological altruism. This is a situation in which an organism appears to act in a way that benefits other organisms and is detrimental to itself. This is distinct from traditional notions of altruism because such actions are not conscious, but appear to be evolutionary adaptations to increase overall fitness. Examples can be found in species ranging from vampire bats that regurgitate blood they have obtained from a night’s hunting and give it to group members who have failed to feed, to worker bees that care for the queen bee for their entire lives and never mate, to Vervet monkeys that warn group members of a predator’s approach, even when it endangers that individual’s chance of survival. [3] All of these actions increase the overall fitness of a group, but occur at a cost to the individual.

Evolutionary game theory explains this altruism with the idea of kin selection. Altruists discriminate between the individuals they help and favor relatives. Hamilton’s rule explains the evolutionary reasoning behind this selection with the equation c<b*r where the cost ( c ) to the altruist must be less than the benefit ( b ) to the recipient multiplied by the coefficient of relatedness ( r ). The more closely related two organisms are causes the incidence of altruism to increase because they share many of the same alleles. This means that the altruistic individual, by ensuring that the alleles of its close relative are passed on, (through survival of its offspring) can forgo the option of having offspring itself because the same number of alleles are passed on. Helping a sibling for example, has a coefficient of ½, because an individual shares ½ of the alleles in its sibling’s offspring. Ensuring that enough of a sibling’s offspring survive to adulthood precludes the necessity of the altruistic individual producing offspring. [3]

Recent applications of biological game theory to humans has garnered some criticism because evolutionary analysis cannot provide a value-neutral evaluation of a given cultural situation. The valuations of whether an action is good or bad constitute a normative judgment of whether an action is altruistic. Altruism also has a different socially constructed meaning in the context of human society because altruistic actions within culture are not all instinctually driven and do not always result in increased fitness for a group. [2]

### Computer science and logic

Game theory has come to play an increasingly important role in logic and in computer science. Logic is the study of the principles of valid demonstration and Inference. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Several logical theories have a basis in game semantics. Game semantics ( German: dialogische Logik) is an approach to Formal semantics that grounds the concepts of Truth or Validity on In addition, computer scientists have used games to model interactive computations. Interactive computation involves communication with the external world during the computation Also, game theory provides a theoretical basis to the field of multi-agent systems. A multi-agent system ( MAS) is a system composed of multiple interacting Intelligent agents Multi-agent systems can be used to solve problems which are difficult or

Separately, game theory has played a role in online algorithms. In Computer science, an online algorithm is one that can process its input piece-by-piece without having the entire input available from the start 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. The k-server problem is a problem of theoretical computer science in the category of online algorithms, one of two abstract problems on Metric spaces that are central  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. Yao's principle states that the expected cost of any Randomized algorithm for solving a given problem on the Worst case input for that algorithm can be no better Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources A randomized algorithm or probabilistic algorithm is an Algorithm which employs a degree of randomness as part of its logic

### 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. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Responding to two papers by W.V.O. Quine (1960, 1967), Lewis (1969) used game theory to develop a philosophical account of convention. Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van" A convention is a set of agreed, stipulated or generally accepted Standards norms social norms or criteria, often taking the form of In so doing, he provided the first analysis of common knowledge and employed it in analyzing play in coordination games. Common knowledge is a special kind of Knowledge for a group of agents There is common knowledge of p in a group of agents G In Game theory, coordination games are a class of games with multiple Pure strategy Nash equilibria in which players choose the same or corresponding In addition, he first suggested that one can understand meaning in terms of signaling games. In Semiotics, the meaning of a sign is its place in a Sign relation, in other words the set of roles that it occupies within a given sign relation Signalling games are Dynamic games with two players the sender (S and the receiver (R This later suggestion has been pursued by several philosophers since Lewis (Skyrms (1996), Grim, Kokalis, and Alai-Tafti et al.  (2004)).

In ethics, some authors have attempted to pursue the project, begun by Thomas Hobbes, of deriving morality from self-interest. Ethics is a major branch of Philosophy, encompassing right conduct and good life Thomas Hobbes (born 5 April 1588died 4 December 1679 was an English philosopher, whose famous 1651 book Leviathan established the foundation 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. The Prisoner's Dilemma constitutes a problem in Game theory. It was originally framed by Merrill Flood and Melvin Dresher This general strategy is a component of the general social contract view in political philosophy (for examples, see Gauthier (1986) and Kavka (1986). Social contract describes a broad class of republican theories whose subjects are implied agreements by which people form Nations and maintain a Social order Political philosophy is the study of questions about the City, Government, Politics, Liberty, Justice, Property, Rights [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. Evolutionary game theory (EGT is the application of interaction dependent strategy drift in populations to Game theory. 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. The Prisoner's Dilemma constitutes a problem in Game theory. It was originally framed by Merrill Flood and Melvin Dresher In Game theory, the stag hunt is a game which describes a conflict between safety and social cooperation The Nash bargaining game is a simple two-player game used to model bargaining interactions 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-interest—a claim debated among philosophers. Psychological egoism is the view that humans are always motivated by Self-interest, even in what seem to be acts of Altruism. (see Psychological egoism#Criticism)

## Types of games

### Cooperative or noncooperative

A game is cooperative if the players are able to form binding commitments. Psychological egoism is the view that humans are always motivated by Self-interest, even in what seem to be acts of Altruism. This article is about a part of a game theory For video gaming see Cooperative gameplay. In Game theory, a non-cooperative game is a one in which players can cooperate but any cooperation must be self-enforcing 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 article is about a part of a game theory For video gaming see Cooperative gameplay. 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-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. This article is about a part of a game theory For video gaming see Cooperative gameplay.

### 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. In Game theory, 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. The game of Chicken, also known as the Hawk-Dove or Snowdrift game is an influential model of conflict for two players in Game theory. The Prisoner's Dilemma constitutes a problem in Game theory. It was originally framed by Merrill Flood and Melvin Dresher In Game theory, the stag hunt is a game which describes a conflict between safety and social cooperation 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. The ultimatum game is an Experimental economics game in which two players interact to decide how to divide a sum of money that is given to them The dictator game is a very simple game in Experimental economics, similar to the Ultimatum game. 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 Game theory and Economic theory, zero-sum describes a situation in which a participant's gain or loss is exactly balanced by the losses or gains of the other 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). In Game theory and Economic theory, zero-sum describes a situation in which a participant's gain or loss is exactly balanced by the losses or gains of the other Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose. Poker is a type of Card game in which players bet on the value of the card combination (" hand " in their possession by placing a bet into Other zero sum games include matching pennies and most classical board games including Go and chess. Matching pennies is the name for a simple example game used in Game theory. Chess is a recreational and competitive Game played between two players.

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. The Prisoner's Dilemma constitutes a problem in Game theory. It was originally framed by Merrill Flood and Melvin Dresher In Game theory, an outcome is a set of moves or strategies taken by the players or their Payoffs resulting from the actions or 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). In Game theory, a sequential game is a game where one player chooses his action before the others choose theirs 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. Perfect information is a term used in Economics and Game theory to describe a state of complete knowledge about the actions of other players that is instantaneously For instance, a player may know that an earlier player did not perform one particular action, while he does not know which of the other available actions the first player actually performed.

The difference between simultaneous and sequential games is captured in the different representations discussed above. Often, normal form is used to represent simultaneous games, and extensive form is used to represent sequential ones; although this isn't a strict rule in a technical sense.

### Perfect information and imperfect information

A game of imperfect information (the dotted line represents ignorance on the part of player 2)
Main article: Perfect information

An important subset of sequential games consists of games of perfect information. Perfect information is a term used in Economics and Game theory to describe a state of complete knowledge about the actions of other players that is instantaneously 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. The ultimatum game is an Experimental economics game in which two players interact to decide how to divide a sum of money that is given to them In Game theory, the centipede game, first introduced by Rosenthal (1981 is an Extensive form game in which two players take turns choosing either to take Perfect information games include also chess, go, mancala, and arimaa. Chess is a recreational and competitive Game played between two players. Mancala is a family of board games played around the world sometimes called " Sowing " games or "count-and-capture" games which describes the Arimaa is a two-player Abstract strategy Board game that can be played using the same equipment as Chess.

Perfect information is often confused with complete information, which is a similar concept. Complete information is a term used in Economics and Game theory to describe an economic situation or game in which knowledge about other market participants or players Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions.

### Infinitely long 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 an infinite number of moves, with the winner (or other payoff) not known until after all those moves are completed.

The focus of attention is usually not so much on what is the best way to play such a game, but simply on whether one or the other player has a winning strategy. (It can be proven, using the axiom of choice, that there are games—even with perfect information, and where the only outcomes are "win" or "lose"—for which neither player has a winning strategy. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. ) The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory. In Mathematical logic, descriptive set theory is the study of certain classes of " Well-behaved " subsets of the Real line and other

### 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. In Game theory, differential games are a group of problems related to the modeling and analysis of conflict in the context of a dynamical system A typical example of a differential game is the continuous pursuit and evasion game. Pursuit-evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in Mathematics and Computer science Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics. In Mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function

### 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. Metagaming is a broad term usually used to define any strategy action or method used in a game which transcends a prescribed ruleset uses external factors to affect the game or The theory of metagames is related to mechanism design theory. In Economics and Game theory, mechanism design is the study of designing rules of a game or system to achieve a specific outcome even though each

## History

The first known discussion of game theory occurred in a letter written by James Waldegrave in 1713. James Waldegrave 1st Earl Waldegrave KG, PC (1684&ndash 11 April 1741) was a British Ambassador. In this letter, Waldegrave provides a minimax mixed strategy solution to a two-person version of the card game le Her. Minimax (sometimes minmax) is a decision rule used in Decision theory, Game theory, Statistics and Philosophy for mini mizing In Game theory, a player's strategy in a game is a complete plan of action for whatever situation might arise this fully determines the player's behaviour It was not until the publication of Antoine Augustin Cournot's Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth) in 1838 that a general game theoretic analysis was pursued. Antoine Augustin Cournot ( 28 August 1801 ‑ 31 March 1877) was a French economist Philosopher and Mathematician In this work Cournot considers a duopoly and presents a solution that is a restricted version of the Nash equilibrium. A true duopoly is a specific type of Oligopoly where only two producers exist in one Market. In Game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it is a Solution concept of a game involving two or more players in which

Although Cournot's analysis is more general than Waldegrave's, game theory did not really exist as a unique field until John von Neumann published a series of papers in 1928. While the French mathematician Émile Borel did some earlier work on games, Von Neumann can rightfully be credited as the inventor of game theory. Félix Édouard Justin Émile Borel ( January 7, 1871 in Saint-Affrique, France &ndash February 3, 1956 in Paris Von Neumann was a brilliant mathematician whose work was far-reaching from set theory to his calculations that were key to development of both the Atom and Hydrogen bombs and finally to his work developing computers. Von Neumann's work in game theory culminated in the 1944 book Theory of Games and Economic Behavior by von Neumann and Oskar Morgenstern. Theory of Games and Economic Behavior, published in 1944 by Princeton University Press, is a book by Mathematician John von Neumann Oskar Morgenstern ( January 24, 1902 – July 26, 1977) was a German born Austrian economist. This profound work contains the method for finding mutually consistent solutions for two-person zero-sum games. During this time period, work on game theory was primarily focused on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies. This article is about a part of a game theory For video gaming see Cooperative gameplay.

In 1950, the first discussion of the prisoner's dilemma appeared, and an experiment was undertaken on this game at the RAND corporation. The Prisoner's Dilemma constitutes a problem in Game theory. It was originally framed by Merrill Flood and Melvin Dresher The RAND Corporation ( R esearch AN d D evelopment is a Nonprofit global policy Think tank first formed to offer research and analysis 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. In Game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it is a Solution concept of a game involving two or more players in which This equilibrium is sufficiently general, allowing for the analysis of non-cooperative games in addition to cooperative ones. In Game theory, a non-cooperative game is a one in which players can cooperate but any cooperation must be self-enforcing

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. The core is the set of feasible allocations that cannot be improved upon by a subset (a coalition) of the economy's consumers In Game theory, fictitious play is a learning rule first introduced by G In Game theory, a repeated game (or iterated game) is an Extensive form game which consists in some number of repetitions of some base game (called a In Game theory, a Shapley value, named in honour of Lloyd Shapley, who introduced it in 1953, describes one approach to the fair allocation of gains obtained In addition, the first applications of Game theory to philosophy and political science occurred during this time. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Political science is a branch of Social sciences that deals with the theory and practice of Politics and the description and analysis of Political systems

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). Reinhard Selten ( October 5, 1930) is a German economist. Selten was born in Breslau (Wrocław in Lower Silesia In Game theory, a solution concept is a formal rule for predicting how the game will be played In Game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in Dynamic In Game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it is a Solution concept of a game involving two or more players in which This article is about games For the cocktail see Perfect Equilibrium (Cocktail Trembling hand perfect equilibrium is a refinement of In 1967, John Harsanyi developed the concepts of complete information and Bayesian games. John Charles Harsanyi ( Harsányi János Károly) (born May 29, 1920 in Budapest, Complete information is a term used in Economics and Game theory to describe an economic situation or game in which knowledge about other market participants or players In Game theory, a Bayesian game is one in which information about characteristics of the other players (i Nash, Selten and Harsanyi became Economics Nobel Laureates in 1994 for their contributions to economic game theory. The Nobel Memorial Prize in Economic Sciences, officially named The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel (Sveriges riksbanks pris i ekonomisk

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. Foundations of modern biology There are five unifying principles Maynard Smith redirects here -- for other uses see Maynard Smith (disambiguation Professor John Maynard Smith, F In Game theory and Behavioural ecology, an evolutionarily stable strategy (ESS is a strategy which if adopted by a population of players In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge[5] were introduced and analysed. In Game theory, a correlated equilibrium is a Solution concept that is more general than the well known Nash equilibrium. Common knowledge is a special kind of Knowledge for a group of agents There is common knowledge of p in a group of agents G

In 2005, game theorists Thomas Schelling and Robert Aumann followed Nash, Selten and Harsanyi as Nobel Laureates. Thomas Crombie Schelling (born 14 April 1921) is an American Economist and professor of Foreign affairs, National security Robert John Aumann ( Hebrew name: Yisrael Aumann he ישראל אומן (born June 8, 1930) is an Israeli Mathematician and a Schelling worked on dynamic models, early examples of evolutionary game theory. Evolutionary game theory (EGT is the application of interaction dependent strategy drift in populations to 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 Game theory, a solution concept is a formal rule for predicting how the game will be played Common knowledge is a special kind of Knowledge for a group of agents There is common knowledge of p in a group of agents G

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. Roger Bruce Myerson (born March 29 1951) is an American Economist and Nobel laureate recognised with Leonid Hurwicz Leonid “Leo” Hurwicz ( August 21, 1917 June 24, 2008) was an American Economist and Mathematician. Eric Stark Maskin (born December 12, 1950) is a American Economist and Nobel laureate recognized with Leonid Hurwicz In Economics and Game theory, mechanism design is the study of designing rules of a game or system to achieve a specific outcome even though each " Among his contributions, is also the notion of proper equilibrium, and an important graduate text: Game Theory, Analysis of Conflict, published in 1991. Proper equilibrium is a refinement of Nash Equilibrium due to Roger B

The Utilitarianism series,
part of the Politics series
Portal:Politics

## Notes

1. ^ Experimental work in game theory goes by many names, experimental economics, behavioral economics, and behavioural game theory are several. Utilitarianism is the idea that the moral worth of an action is solely determined by its contribution to overall Utility, that is its contribution to happiness Politics Politics is the process by which groups of people make decisions This is an incomplete list of advocates of Utilitarianism. John Austin Jeremy Bentham Richard Brandt Jeremy Bentham ( IPA: or) (15 February 1748&ndash6 June 1832 was an English Jurist, Philosopher, and legal and Social reformer John Stuart Mill (20 May 1806 &ndash 8 May 1873 British Philosopher, political economist, civil servant and Member of Parliament, was an influential Henry Sidgwick ( May 31, 1838 – August 28, 1900) was an English Utilitarian Philosopher. Peter Albert David Singer (born July 6, 1946 in Melbourne, Victoria, Australia) is an Australian philosopher. Preference utilitarianism is quite probably the most popular form of Utilitarianism in contemporary philosophy Rule utilitarianism is a form of Utilitarianism which states that moral actions are those which conform to the rules which lead to the greatest good or that "the rightness Act utilitarianism is a Utilitarian theory of Ethics which states that the morally right action is the one which produces the greatest amount of happiness for the Two-level utilitarianism is a utilitarian theory of Ethics developed by R All proponents of Utilitarianism believe that the quality of conscious experience is important indeed it is the basis of their consequentialist approach to Ethics All proponents of Utilitarianism believe that the quality of conscious experience is important indeed it is the basis of their consequentialist approach to Ethics Utilitarianism is the idea that the moral worth of an action is solely determined by its contribution to overall Utility, that is its contribution to happiness Animal welfare refers to the viewpoint that it is morally acceptable for humans to use nonhuman animals for food in animal research, as clothing and in entertainment Abolitionism is a Bioethical school and movement which proposes the use of Biotechnology to maximize Happiness and minimize Suffering while Hedonism is the Philosophy that Pleasure is of ultimate importance, the most important pursuit Enlightened self-interest is a philosophy in Ethics which states that persons who act to further the interests of others (or the interests of the group or groups to David Hume (26 April 1711 25 August 1776 Scottish Philosopher, Economist, and Historian is an important figure in Western philosophy William Godwin ( 3 March 1756 &ndash 7 April 1836) was an English journalist political philosopher and Novelist Pain, in the sense of physical pain, is a typical sensory experience that may be described as the unpleasant awareness of a noxious stimulus or bodily harm Suffering, or pain, is an individual's basic Affective experience of unpleasantness and aversion associated with harm or threat of harm Pleasure is commonly conceptualized as a positive experience Happiness, Entertainment, Enjoyment, ecstasy, and euphoria, but is hard In Economics, utility is a measure of the relative satisfaction from or desirability of Consumption of various Goods and services. Happiness is an Emotion associated with feelings ranging from contentment and satisfaction to Bliss and intense Joy. Eudaimonia ( Greek:) is a classical Greek word commonly translated as ' Happiness ' Consequentialism refers to those moral theories which hold that the consequences of a particular action form the basis for any valid moral judgment about that action The felicific calculus is an Algorithm formulated by utilitarian philosopher Jeremy Bentham for calculating the degree or amount of Pleasure The mere addition paradox is a problem in Ethics, identified by Derek Parfit, and appearing in his book Reasons and Persons. The paradox of Hedonism, also called the pleasure paradox, is the idea in the study of Ethics which points out that Pleasure and Happiness The utility monster is a Thought experiment in the study of Ethics. Rational choice theory, also known as rational action theory, is a framework for understanding and often formally modeling social and economic behavior Social choice theory studies voting rules that govern and describe how individual preferences are aggregated to form a collective preference Economics is the social science that studies the production distribution, and consumption of goods and services. An analytic narrative is a social science research method seeking to combine historical narratives with the rigor of Rational choice theory, particularly through the use of The Trap What Happened to Our Dream of Freedom is a BBC Documentary series by English filmmaker Adam Curtis, well known for Adam Curtis (born 1955 is a British Television documentary maker who has during the course of his television career worked as a writer producer Experimental economics is a the application of experimental methods to study economic questions Behavioral economics and behavioral finance are closely related fields which apply scientific research on human and social cognitive and emotional factors to better For a recent discussion on this field see Camerer (2003).
2. ^ a b Evolutionary Game Theory (Stanford Encyclopedia of Philosophy)
3. ^ a b Biological Altruism (Stanford Encyclopedia of Philosophy)
4. ^ 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.
5. ^ 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. David Kellogg Lewis ( September 28, 1941  &ndash October 14, 2001) is considered to have been one of the leading philosophers of the latter Robert John Aumann ( Hebrew name: Yisrael Aumann he ישראל אומן (born June 8, 1930) is an Israeli Mathematician and a

## References

#### Textbooks and general references

• Aumann, Robert J. (1987), Game Theory, vol. Robert John Aumann ( Hebrew name: Yisrael Aumann he ישראל אומן (born June 8, 1930) is an Israeli Mathematician and a 2, The New Palgrave: A Dictionary of Economics, pp. The New Palgrave A Dictionary of Economics (1987 is a 4-volume reference edited by John Eatwell, Murray Milgate and Peter Newman 460–82 .
• Dutta, Prajit K. (1999), Strategies and games: theory and practice, MIT Press, ISBN 978-0-262-04169-0 . The MIT Press is a University press affiliated with the Massachusetts Institute of Technology (MIT in Cambridge Massachusetts ( USA) 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 . Addison-Wesley is a Book publishing imprint of Pearson PLC, best known for computer books Suitable for upper-level undergraduates.
• Fudenberg, Drew & Tirole, Jean (1991), Game theory, MIT Press, ISBN 978-0-262-06141-4 . Jean Marcel Tirole (Aug 9 1953 -) is a French professor of Economics. The MIT Press is a University press affiliated with the Massachusetts Institute of Technology (MIT in Cambridge Massachusetts ( USA) Acclaimed reference text, public description.
• Gibbons, Robert D. (1992), Game theory for applied economists, Princeton University Press, ISBN 978-0-691-00395-5 . The Princeton University Press is an independent publisher with close connections to Princeton University. Suitable for advanced undergraduates.
• Published in Europe as 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. The Princeton University Press is an independent publisher with close connections to Princeton University. ; 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.
• Hansen, Pelle G. & Hendricks, Vincent F. , eds. , Game Theory: 5 Questions, New York, London: Automatic Press / VIP, ISBN 9788799101344 . Snippets from interviews.
• Isaacs, Rufus, Differential Games: A Mathematical Theory With Applications to Warfare and Pursuit, Control and Optimization, New York: Dover Publications, ISBN 978-0-486-40682-4
• Miller, James H. Dover Publications is an American book Publisher founded in 1941 by Hayward Cirker and his wife Blanche (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 . The McGraw-Hill Companies Inc, ( is a Publicly traded corporation headquartered in Rockefeller Center in New York City. Suitable for a general audience.
• Osborne, Martin J. Roger Bruce Myerson (born March 29 1951) is an American Economist and Nobel laureate recognised with Leonid Hurwicz Harvard University Press ( HUP) is a Publishing house, a division of Harvard University, that is highly respected in Academic publishing. (2004), An introduction to game theory, Oxford University Press, ISBN 978-0-19-512895-6 . Undergraduate textbook.
• 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.
• Rubinstein, Ariel & Osborne, Martin J. Ariel Rubinstein (born April 13, 1951) is an Israeli Economist who works in Game theory. (1994), A course in game theory, MIT Press, ISBN 978-0-262-65040-3 . The MIT Press is a University press affiliated with the Massachusetts Institute of Technology (MIT in Cambridge Massachusetts ( USA) A modern introduction at the graduate level.
• Williams, John Davis (1954), The Compleat Strategyst: Being a Primer on the Theory of Games of Strategy, Santa Monica: RAND Corp.   Praised primer and popular introduction for everybody, never out of print.

#### Historically important texts

• Cournot, A. Augustin (1838), “Recherches sur les principles mathematiques de la théorie des richesses”, Libraire des sciences politiques et sociales (Paris: M. Robert John Aumann ( Hebrew name: Yisrael Aumann he ישראל אומן (born June 8, 1930) is an Israeli Mathematician and a Lloyd Stowell Shapley (born June 2 1923 is a distinguished American mathematician and economist Antoine Augustin Cournot ( 28 August 1801 ‑ 31 March 1877) was a French economist Philosopher and Mathematician Rivière & C. ie)
• reprinted edition: The Genetical Theory of Natural Selection: A Complete Variorum Edition, Oxford University Press, 1999, ISBN 978-0-19-850440-5
• reprinted edition: Games and decisions: introduction and critical survey, New York: Dover Publications, 1989, ISBN 978-0-486-65943-5
• Shapley, L.S. (1953), A Value for n-person Games, In: Contributions to the Theory of Games volume II, H. Francis Ysidro Edgeworth (8 February 1845 &ndash 13 February 1926 made significant contributions to the methods of statistics during the 1880s Sir Ronald Aylmer Fisher, FRS ( 17 February 1890 – 29 July 1962) was an English Statistician, Evolutionary The Genetical Theory of Natural Selection is a book by RA Fisher. Robert Duncan Luce (born May 16 1925 is the Distinguished Research Professor of Cognitive Science at the University of California Irvine. Howard Raiffa (1924&ndash ˈreɪfə is the Frank P Ramsey Professor (Emeritus of Managerial Economics, a joint chair held by the Business School Dover Publications is an American book Publisher founded in 1941 by Hayward Cirker and his wife Blanche Maynard Smith redirects here -- for other uses see Maynard Smith (disambiguation Professor John Maynard Smith, F Evolution and the Theory of Games is a 1982 Book by the British evolutionary biologist John Maynard Smith on Evolutionary Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534 The Proceedings of the National Academy of Sciences of the United States of America, usually referred to as PNAS, is the official journal of the United An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. Lloyd Stowell Shapley (born June 2 1923 is a distinguished American mathematician and economist W. Kuhn and A. W. Tucker (eds. )
• Shapley, L.S. (1953), Stochastic Games, Proceedings of National Academy of Science Vol. Lloyd Stowell Shapley (born June 2 1923 is a distinguished American mathematician and economist 39, pp. 1095-1100.
• Zermelo, Ernst (1913), “Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels”, Proceedings of the Fifth International Congress of Mathematicians 2: 501–4

#### Other print references

• Ben David, S. The Mathematische Annalen (abbreviated as Math Ann or Math Annal An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. Oskar Morgenstern ( January 24, 1902 – July 26, 1977) was a German born Austrian economist. Theory of Games and Economic Behavior, published in 1944 by Princeton University Press, is a book by Mathematician John von Neumann The Princeton University Press is an independent publisher with close connections to Princeton University. Ernst Friedrich Ferdinand Zermelo ( July 27 1871, Berlin, German Empire – May 21 1953, Freiburg im Breisgau ; Borodin, Allan; Karp, Richard; Tardos, G. Allan Bertram Borodin is a Mathematician and Computational theorist who has done important work in Computational complexity theory and Algorithms Richard Manning Karp (born 1935 is a Computer scientist and Computational theorist, notable for research in & Wigderson, A. (1994), “On the Power of Randomization in On-line Algorithms”, Algorithmica 11 (1): 2–14
• Camerer, Colin (2003), Behavioral game theory: experiments in strategic interaction, Russesll Sage Foundation, ISBN 978-0-691-09039-9
• 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
• Grim, Patrick; Kokalis, Trina; Alai-Tafti, Ali; Kilb, Nicholas & St Denis, Paul (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
• Kavka, Gregory S. Anthony Downs is a noted scholar in Public policy and Public administration, and since 1977 is a Senior Fellow at the Brookings Institution in Washington David Gauthier (born 1932 is a Canadian-American Philosopher best known for his neo- Hobbesian social contract (contractarian theory of Morality John Charles Harsanyi ( Harsányi János Károly) (born May 29, 1920 in Budapest, (1986), Hobbesian moral and political theory, Princeton University Press, ISBN 978-0-691-02765-4
• Lewis, David (1969), Convention: A Philosophical Study , ISBN 978-0-631-23257-5 (2002 edition)
• Harper, David & Maynard Smith, John (2003), Animal signals, Oxford University Press, ISBN 978-0-19-852685-8
• Levy, Gilat & Razin, Ronny (2003), “It Takes Two: An Explanation of the Democratic Peace”, Working Paper
• Quine, W.v.O (1967), “Truth by Convention”, Philosophica Essays for A. The Princeton University Press is an independent publisher with close connections to Princeton University. David Kellogg Lewis ( September 28, 1941  &ndash October 14, 2001) is considered to have been one of the leading philosophers of the latter Maynard Smith redirects here -- for other uses see Maynard Smith (disambiguation Professor John Maynard Smith, F Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van" N. Whitehead, Russel and Russel Publishers, ISBN 978-0-8462-0970-6
• 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 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 Alec (1999), Unto others: the evolution and psychology of unselfish behavior, Harvard University Press, ISBN 978-0-674-93047-6
• Thrall, Robert M. Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van" Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534 Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534 Harvard University Press ( HUP) is a Publishing house, a division of Harvard University, that is highly respected in Academic publishing. & Lucas, William F. (1963), “n-person games in partition function form”, Naval Research Logistics Quarterly 10 (4): 281–298

## game theory

### -noun

1. (mathematics, economics) A branch of applied mathematics that studies strategic situations in which individuals or organisations choose various actions in an attempt to maximize their returns.
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