Game theory

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Game theory is the study of strategic decision making. Specifically, it is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers".[1] An alternative term suggested "as a more descriptive name for the discipline" is interactive decision theory.[2] Game theory is mainly used in economics, political science, and psychology, as well as logic, computer science, and biology. The subject first addressed zero-sum games, such that one person's gains exactly equal net losses of the other participant or participants. Today, however, game theory applies to a wide range of behavioral relations, and has developed into an umbrella term for the logical side of decision science, including both humans and non-humans (e.g. computers, insects/animals).

Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used Brouwer 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 the 1944 book Theory of Games and Economic Behavior, co-written 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. With the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole in 2014, eleven game-theorists have now won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology.

Representation of games

The games studied in game theory are well-defined mathematical objects. To be fully defined, a game must specify the following elements: the players of the game, the information and actions available to each player at each decision point, and the payoffs for each outcome. (Rasmusen refers to these four "essential elements" by the acronym "PAPI".)[3] A game theorist typically uses these elements, along with a solution concept of their choosing, to deduce a set of equilibrium strategies for each player such that, when these strategies are employed, no player can profit by unilaterally deviating from their strategy. These equilibrium strategies determine an equilibrium to the game—a stable state in which either one outcome occurs or a set of outcomes occur with known probability.

Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.

Extensive form

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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. Template:Harv

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

Template:Payoff matrix {{#invoke:main|main}} 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.Template:Harv

Characteristic function form

{{#invoke:main|main}} In games that possess removable utility separate rewards are not given; rather, the characteristic function decides the payoff of each unity. The idea is that the unity that is 'empty', so to speak, does not receive a reward at all.

The origin of this form is to be found in John von Neumann and Oskar Morgenstern's book; when looking at these instances, they guessed that when a union ${\displaystyle \mathbf {C} }$ appears, it works against the fraction ${\displaystyle \left({\frac {\mathbf {N} }{\mathbf {C} }}\right)}$ as if two individuals were playing a normal game. The balanced payoff of C is a basic function. Although there are differing examples that help determine coalitional amounts from normal games, not all appear that in their function form can be derived from such.

Formally, a characteristic function is seen as: (N,v), where N represents the group of people and ${\displaystyle v:2^{N}\to \mathbf {R} }$ is a normal utility.

Such characteristic functions have expanded to describe games where there is no removable utility.

General and applied uses

As a method of applied mathematics, game theory has been used to study a wide variety of human and animal behaviors. It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers. The first use of game-theoretic analysis was by Antoine Augustin Cournot in 1838 with his solution of the Cournot duopoly. The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well.

Although pre-twentieth century naturalists such as Charles Darwin made game-theoretic kinds of statements,Template:Example needed the use of game-theoretic analysis in biology began with Ronald Fisher's studies of animal behavior during the 1930s. This work predates the name "game theory", but it shares many important features with this field. The developments in economics were later applied to biology largely by John Maynard Smith in his book Evolution and the Theory of Games.

In addition to being used to describe, predict, and explain behavior, game theory has also been used to develop theories of ethical or normative behavior and to prescribe such behavior.[4] 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 four-stage centipede game.

The first known use is to describe and model how human populations behave. SomeTemplate:Who 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, in the centipede game, guess 2/3 of the average game, and the dictator game, people regularly do not play Nash equilibria. These experiments have demonstrated that individuals do not play equilibrium strategies. There is an ongoing debate regarding the importance of these experiments.[6]

Alternatively, someTemplate:Who 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, following the work of John Maynard Smith and George R. Price, have turned to evolutionary game theory in order to resolve these issues. These models presume either no rationality or bounded rationality on the part of players. Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning (for example, fictitious play dynamics).

Prescriptive or normative analysis

Template:Payoff matrix On the other hand, someTemplate:Who scholars see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave. Since a strategy, corresponding to a Nash equilibrium of a game constitutes one's best response to the actions of the other players – provided they are in (the same) Nash equilibrium – playing a strategy that is part of a Nash equilibrium seems appropriate. However, the rationality of such a decision has been proved only for special cases. This normative use of game theory has also come under criticism. First, in some cases it is appropriate to play a non-equilibrium strategy if one expects others to play non-equilibrium strategies as well. For an example, see guess 2/3 of the average.

Second, the prisoner's dilemma presents another potential counterexample. In the prisoner's dilemma, each player pursuing their own self-interest leads both players to be worse off than had they not pursued their own self-interests.

Template:Incomplete Game theory is a major method used in mathematical economics and business for modeling competing behaviors of interacting agents.[7] Applications include a wide array of economic phenomena and approaches, such as auctions, bargaining, mergers & acquisitions pricing,[8] fair division, duopolies, oligopolies, social network formation, agent-based computational economics,[9] general equilibrium, mechanism design,[10] and voting systems;[11] and across such broad areas as experimental economics,[12] behavioral economics,[13] information economics,[3] industrial organization,[14] and political economy.[15][16]

This research usually focuses on particular sets of strategies known as "solution concepts" or "equilibria" based on what is required by norms of (ideal) 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.[17][18]

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.[19]

A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of a particular economic situation. One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type. Naturally one might wonder to what use this information should be put. Economists and business professors suggest two primary uses (noted above): descriptive and prescriptive.[4]

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.

Early examples of game theory applied to political science are provided by Anthony Downs. In his book An Economic Theory of Democracy,Template:Harvard citations 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. Downs first shows how the political candidates will converge to the ideology preferred by the median voter if voters are fully informed, but then argues that voters choose to remain rationally ignorant which allows for candidate divergence.

It has also been proposed that game theory explains the stability of any form of political government. Taking the simplest case of a monarchy, for example, the king, being only one person, does not and cannot maintain his authority by personally exercising physical control over all or even any significant number of his subjects. Sovereign control is instead explained by the recognition by each citizen that all other citizens expect each other to view the king (or other established government) as the person whose orders will be followed. Coordinating communication among citizens to replace the sovereign is effectively barred, since conspiracy to replace the sovereign is generally punishable as a crime. Thus, in a process that can be modeled by variants of the prisoner's dilemma, during periods of stability no citizen will find it rational to move to replace the sovereign, even if all the citizens know they would be better off if they were all to act collectively.[20]

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 Template:Harvard citations.

Game theory could also help predict nation's responses when there is a new rule or law to be applied to that nation. One example would be Peter John Wood's (2013) research when he looked into what nations could do to help reduce climate change. Wood thought this could be accomplished by making treaties with other nations to reduce green house gas emissions. However, he concluded that this idea could not work because it would create a prisoner's dilemma to the nations.[21]

Biology

Template:Payoff matrix Unlike those in 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 and more on ones that would be maintained by evolutionary forces. The best known equilibrium in biology is known as the evolutionarily stable strategy (ESS), first introduced in Template:Harv. 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 as a model to understand many different phenomena. It was first used to explain the evolution (and stability) of the approximate 1:1 sex ratios. Template:Harv 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 Template:Harv. The analysis of signaling games and other communication games has provided insight into the evolution of communication among animals. For example, the mobbing behavior of many species, in which a large number of prey animals attack a larger predator, seems to be an example of spontaneous emergent organization. Ants have also been shown to exhibit feed-forward behavior akin to fashion (see Paul Ormerod's Butterfly Economics).

Biologists have used the game of chicken to analyze fighting behavior and territoriality.[22]

According to Maynard Smith, in the preface to Evolution and the Theory of Games, "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.[23]

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.[24] 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 rationale behind this selection with the equation c<b*r where the cost (c) to the altruist must be less than the benefit (b) to the recipient multiplied by the coefficient of relatedness (r). The more closely related two organisms are causes the incidences of altruism to increase because they share many of the same alleles. This means that the altruistic individual, by ensuring that the alleles of its close relative are passed on, (through survival of its offspring) can forgo the option of having offspring itself because the same number of alleles are passed on. Helping a sibling for example (in diploid animals), has a coefficient of ½, because (on average) an individual shares ½ of the alleles in its sibling's offspring. Ensuring that enough of a sibling’s offspring survive to adulthood precludes the necessity of the altruistic individual producing offspring.[24] The coefficient values depend heavily on the scope of the playing field; for example if the choice of whom to favor includes all genetic living things, not just all relatives, we assume the discrepancy between all humans only accounts for approximately 1% of the diversity in the playing field, a co-efficient that was ½ in the smaller field becomes 0.995. Similarly if it is considered that information other than that of a genetic nature (e.g. epigenetics, religion, science, etc.) persisted through time the playing field becomes larger still, and the discrepancies smaller.

Computer science and logic

Game theory has come to play an increasingly important role in logic and in computer science. Several logical theories have a basis in game semantics. In addition, computer scientists have used games to model interactive 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 Template:Harvard citations. Yao's principle is a game-theoretic technique for proving lower bounds on the computational complexity of randomized algorithms, especially online algorithms.

The emergence of the internet has motivated the development of algorithms for finding equilibria in games, markets, computational auctions, peer-to-peer systems, and security and information markets. Algorithmic game theory[25] and within it algorithmic mechanism design[26] combine computational algorithm design and analysis of complex systems with economic theory.[27]

Philosophy

Template:Payoff matrix Game theory has been put to several uses in philosophy. Responding to two papers by Template:Harvard citations, Template:Harvtxt 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 (Template:Harvtxt, Template:Harvard citations). Following Template:Harvtxt game-theoretic account of conventions, Edna 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.[28][29]

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),[30] Skyrms (1990),[31] and Stalnaker (1999).[32]

In ethics, someTemplate:Who authors have attempted to pursue Thomas Hobbes' project 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 Template:Harvtxt and Template:Harvtxt).[33]

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., Template:Harvard citations and Template:Harvard citations).

Some assumptions used in some parts of game theory have been challenged in philosophy; for example, psychological egoism states that rationality reduces to self-interest—a claim debated among philosophers. (see Psychological egoism#Criticisms)

Types of games

Cooperative / Non-cooperative

{{#invoke:main|main}} 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 non-cooperative ones. However, this classification on two binary criteria has been questioned, and sometimes rejected Template:Harv.

Of the two types of games, noncooperative games are able to model situations to the finest details, producing accurate results. Cooperative games focus on the game at large. Considerable efforts have been made to link the two approaches. The so-called Nash-programme (Nash program is the research agenda for investigating on the one hand axiomatic bargaining solutions and on the other hand the equilibrium outcomes of strategic bargaining procedures)[34] 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 / Asymmetric

Template:Payoff matrix {{#invoke:main|main}} 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. SomeTemplate:Who 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 / Non-zero-sum

Template:Payoff matrix {{#invoke:main|main}} 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 infamous prisoner's dilemma) are non-zero-sum games, because the outcome has net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.

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

Simultaneous / Sequential

{{#invoke:main|main}} Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players' actions (making them effectively simultaneous). Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions. This need not be perfect information about every action of earlier players; it might be very little knowledge. For instance, a player may know that an earlier player did not perform one particular action, while he does not know which of the other available actions the first player actually performed.

The difference between simultaneous and sequential games is captured in the different representations discussed above. Often, normal form is used to represent simultaneous games, while extensive form is used to represent sequential ones. The transformation of extensive to normal form is one way, meaning that multiple extensive form games correspond to the same normal form. Consequently, notions of equilibrium for simultaneous games are insufficient for reasoning about sequential games; see subgame perfection.

In short, the differences between sequential and simultaneous games are as follows:

Sequential Simultaneous
Normally denoted by Decision trees Payoff matrices
Template:Longitem Yes No
Time axis? Yes No
Also known as Template:Longitem Template:Longitem

Perfect information and imperfect information

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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 because players in simultaneous games do not know the actions of the other players. Most games studied in game theory are imperfect-information games. Interesting examples of perfect-information games include the ultimatum game and centipede game. Recreational games of perfect information games include chess, go and mancala. Many card games are games of imperfect information, such as poker or contract bridge.

Perfect information is often confused with complete information, which is a similar concept. Complete information requires that every player know the strategies and payoffs available to the other players but not necessarily the actions taken. Games of incomplete information can be reduced, however, to games of imperfect information by introducing "moves by nature" Template:Harv.

Combinatorial games

Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games. Examples include chess and go. Games that involve imperfect or incomplete information may also have a strong combinatorial character, for instance backgammon. There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve particular problems and answer general questions.[35]

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 "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.[36][37] 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 estimating the computational difficulty of finding optimal strategies.[38]

Research in artificial intelligence has addressed both perfect and imperfect (or incomplete) information games that have very complex combinatorial structures (like chess, go, or backgammon) for which no provable optimal strategies have been found. The practical solutions involve computational heuristics, like alpha-beta pruning or use of artificial neural networks trained by reinforcement learning, which make games more tractable in computing practice.[35][39]

Infinitely long games

{{#invoke:main|main}} 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 the best way to play such a game, but whether one player has a winning strategy. (It can be proven, using the axiom of choice, that there are gamesTemplate:Spaced ndasheven with perfect information and where the only outcomes are "win" or "lose"Template:Spaced ndashfor 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

Differential games such as the continuous pursuit and evasion game are continuous games where the evolution of the players' state variables is governed by differential equations. The problem of finding an optimal strategy in a differential game is closely related to the optimal control theory. In particular, there are two types of strategies: the open-loop strategies are found using the Pontryagin maximum principle while the closed-loop strategies are found using Bellman's Dynamic Programming method.

A particular case of differential games are the games with a random time horizon.[40] In such games, the terminal time is a random variable with a given probability distribution function. Therefore, the players maximize the mathematical expectation of the cost function. It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval.

Many-player and population games

Games with an arbitrary, but finite, number of players are often called n-person games Template:Harv. 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 Template:Harv.

Individual decision problems with stochastic outcomes are sometimes considered "one-player games". These situations are not considered game theoretical by some authors.Template: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).{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" ("moves by nature") Template:Harv. 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.[41] (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.[41] 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 Template:Harvard citation whereby a situation is framed as a strategic game in which stakeholders try to realise their objectives by means of the options available to them. Subsequent developments have led to the formulation of confrontation analysis. History 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.[42] 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.[43][44] In his 1838 Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth), Antoine Augustin Cournot considered a duopoly and presents a solution that is a restricted version of the Nash equilibrium. The Danish mathematician Zeuthen proved that the mathematical model had a winning strategy by using Brouwer's fixed point theorem. In his 1938 book Applications aux Jeux de Hasard and earlier notes, Émile Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix was symmetric. Borel conjectured that non-existence of mixed-strategy equilibria in two-person zero-sum games would occur, a conjecture that was proved false. Game theory did not really exist as a unique field until John von Neumann published a paper in 1928.[45] Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book Theory of Games and Economic Behavior. The second edition of this book provided an axiomatic theory of utility, which reincarnated Daniel Bernoulli's old theory of utility (of the money) as an independent discipline. Von Neumann's work in game theory culminated in this 1944 book. This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. During the following time period, work on game theory was primarily focused on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.[46] In 1950, the first mathematical discussion of the prisoner's dilemma appeared, and an experiment was undertaken by notable mathematicians Merrill M. Flood and Melvin Dresher, as part of the RAND corporation's investigations into game theory. Rand pursued the studies because of possible applications to global nuclear strategy.[42] 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 knowledge[47] were introduced and analyzed. In 2005, game theorists Thomas Schelling and Robert Aumann followed Nash, Selten and Harsanyi as Nobel Laureates. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, introducing an equilibrium coarsening, correlated equilibrium, and developing an extensive formal analysis of the assumption of common knowledge and of its consequences. In 2007, Leonid Hurwicz, together with Eric Maskin and Roger Myerson, was awarded the Nobel Prize in Economics "for having laid the foundations of mechanism design theory." Myerson's contributions include the notion of proper equilibrium, and an important graduate text: Game Theory, Analysis of Conflict Template:Harv. Hurwicz introduced and formalized the concept of incentive compatibility. In 2012, Alvin E. Roth and Lloyd S. Shapley were awarded the Nobel Prize in Economics "for the theory of stable allocations and the practice of market design." In popular culture Based on the book by Sylvia Nasar,[48] the life story of game theorist and mathematician John Nash was turned into the biopic A Beautiful Mind starring Russell Crowe.[49] "Games theory" and "theory of games" are mentioned in the military science fiction novel Starship Troopers by Robert A. Heinlein.[50] In the 1997 film of the same name, the character Carl Jenkins refers to his assignment to military intelligence as to "games and theory." One of the main gameplay decision-making mechanics of the video game Zero Escape: Virtue's Last Reward is based on game theory. Some of the characters even reference the prisoner's dilemma. The film Dr. Strangelove satirizes game theoretic ideas about deterrence theory. For example, nuclear deterrence depends on the threat to retaliate catastrophically if a nuclear attack is detected. A game theorist might argue that such threats can fail to be credible, in the sense that they can lead to subgame imperfect equilibria. The movie takes this idea one step further, with the Russians irrevocably committing to a catastrophic nuclear response without making the threat public. See also Lists Notes 1. Roger B. Myerson (1991). Game Theory: Analysis of Conflict, Harvard University Press, p. 1. Chapter-preview links, pp. vii–xi. 2. R. J. Aumann ([1987] 2008). "game theory," Introduction, The New Palgrave Dictionary of Economics, 2nd Edition. Abstract. 3. • Eric Rasmusen (2007). Games and Information, 4th ed. Description and chapter-preview. • David M. Kreps (1990). Game Theory and Economic Modelling. Description. • R. Aumann and S. Hart, ed. (1992, 2002). Handbook of Game Theory with Economic Applications v. 1, ch. 3–6 and v. 3, ch. 43. 4. Colin F. Camerer (2003). Behavioral Game Theory: Experiments in Strategic Interaction, pp. 5–7 (scroll to at 1.1 What Is Game Theory Good For?). 5. Template:Cite web 6. Experimental work in game theory goes by many names, experimental economics, behavioral economics, and behavioural game theory are several. For a recent discussion, see Colin F. Camerer (2003). Behavioral Game Theory: Experiments in Strategic Interaction (description and Introduction, pp. 1–25). 7. • At JEL:C7 of the Journal of Economic Literature classification codes. • R.J. Aumann (2008). "game theory," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract. • Martin Shubik (1981). "Game Theory Models and Methods in Political Economy," in Kenneth Arrow and Michael Intriligator, ed., Handbook of Mathematical Economics, v. 1, pp. 285–330 Template:Hide in printTemplate:Only in print. • Carl Shapiro (1989). "The Theory of Business Strategy," RAND Journal of Economics, 20(1), pp. 125–137 Template:Hide in printTemplate:Only in print. 8. N. Agarwal and P. Zeephongsekul. Psychological Pricing in Mergers & Acquisitions using Game Theory, School of Mathematics and Geospatial Sciences, RMIT University, Melbourne 9. Leigh Tesfatsion (2006). "Agent-Based Computational Economics: A Constructive Approach to Economic Theory," ch. 16, Handbook of Computational Economics, v. 2, pp. 831-880 Template:Hide in printTemplate:Only in print. • Joseph Y. Halpern (2008). "computer science and game theory," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract. 10. • From The New Palgrave Dictionary of Economics (2008), 2nd Edition: Roger B. Myerson. "mechanism design." Abstract. _____. "revelation principle." Abstract. • Tuomas Sandholm. "computing in mechanism design." Abstract. • Noam Nisan and Amir Ronen (2001). "Algorithmic Mechanism Design," Games and Economic Behavior, 35(1–2), pp. 166–196. • Noam Nisan et al., ed. (2007). Algorithmic Game Theory, Cambridge University Press. Description. 11. R. Aumann and S. Hart, ed., 1994. Handbook of Game Theory with Economic Applications, v. 2, outline links, ch. 30: "Voting Procedures" & ch. 31: "Social Choice." 12. Vernon L. Smith, 1992. "Game Theory and Experimental Economics: Beginnings and Early Influences," in E. R. Weintraub, ed., Towards a History of Game Theory, pp. 241–282. • _____, 2001. "Experimental Economics," International Encyclopedia of the Social & Behavioral Sciences, pp. 5100–5108. Abstract per sect. 1.1 & 2.1. • Charles R. Plott and Vernon L. Smith, ed., 2008. Handbook of Experimental Economics Results, v. 1, Elsevier, Part 4, Games, ch. 45–66. • Vincent P. Crawford (1997). "Theory and Experiment in the Analysis of Strategic Interaction," in Advances in Economics and Econometrics: Theory and Applications, pp. 206–242. Cambridge. Reprinted in Colin F. Camerer et al., ed. (2003). Advances in Behavioral Economics, Princeton. 1986–2003 papers. Description, preview, Princeton, ch. 12. • Martin Shubik, 2002. "Game Theory and Experimental Gaming," in R. Aumann and S. Hart, ed., Handbook of Game Theory with Economic Applications, Elsevier, v. 3, pp. 2327–2351. Template:Hide in printTemplate:Only in print. 13. From The New Palgrave Dictionary of Economics (2008), 2nd Edition: • Faruk Gul. "behavioural economics and game theory." Abstract. • Colin F. Camerer. "behavioral game theory." Abstract. • _____ (1997). "Progress in Behavioral Game Theory," Journal of Economic Perspectives, 11(4), p. 172, pp. 167–188. • _____ (2003). Behavioral Game Theory, Princeton. Description, preview ([ctrl]+), and ch. 1 link. • _____, George Loewenstein, and Matthew Rabin, ed. (2003). Advances in Behavioral Economics, Princeton. 1986–2003 papers. Description, contents, and preview. • Drew Fudenberg (2006). "Advancing Beyond Advances in Behavioral Economics," Journal of Economic Literature, 44(3), pp. 694–711 Template:Hide in printTemplate:Only in print. 14. Jean Tirole (1988). The Theory of Industrial Organization, MIT Press. Description and chapter-preview links, pp. vii–ix, "General Organization," pp. 5–6, and "Non-Cooperative Game Theory: A User's Guide Manual,' " ch. 11, pp. 423–59. • Kyle Bagwell and Asher Wolinsky (2002). "Game theory and Industrial Organization," ch. 49, Handbook of Game Theory with Economic Applications, v. 3, pp. 1851–1895. • Martin Shubik (1959). Strategy and Market Structure: Competition, Oligopoly, and the Theory of Games, Wiley. Description and review extract. • _____ with Richard Levitan (1980). Market Structure and Behavior, Harvard University Press. Review extract. 15. • Martin Shubik (1981). "Game Theory Models and Methods in Political Economy," in Handbook of Mathematical Economics, v. 1, pp. 285–330 Template:Hide in printTemplate:Only in print. •_____ (1987). A Game-Theoretic Approach to Political Economy. MIT Press. Description. 16. • Martin Shubik (1978). "Game Theory: Economic Applications," in W. Kruskal and J.M. Tanur, ed., International Encyclopedia of Statistics, v. 2, pp. 372–78. • Robert Aumann and Sergiu Hart, ed. Handbook of Game Theory with Economic Applications (scrollable to chapter-outline or abstract links): 1992. v. 1; 1994. v. 2; 2002. v. 3. 17. Options Games: Balancing the trade-off between flexibility and commitment. Europeanfinancialreview.com (2012-02-15). Retrieved on 2013-01-03. 18. Expected Utility in the Context of a Game 19. Morrison, Andrew Stumpff, "Eminent Legal Philosophers, or Yes, Law is the Command of the Sovereign", at 8-12 (2013). Available at SSRN 20. Wood, Peter John. (2011, February). "Climate change and game theory," Ecological Economics Review 1219: 153-70. 21. Template:Cite doi 22. Evolutionary Game Theory (Stanford Encyclopedia of Philosophy). Plato.stanford.edu. Retrieved on 2013-01-03. 23. Biological Altruism (Stanford Encyclopedia of Philosophy). Seop.leeds.ac.uk. Retrieved on 2013-01-03. 24. Noam Nisan et al., ed. (2007). 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A Game-theoretic Approach. Chapter 4. The Nash Program. ISBN 978-1-4020-7183-6. 34. {{#invoke:citation/CS1|citation |CitationClass=citation }} 35. {{#invoke:citation/CS1|citation |CitationClass=citation }} 36. {{#invoke:citation/CS1|citation |CitationClass=citation }} 37. {{#invoke:citation/CS1|citation |CitationClass=citation }} 38. {{#invoke:citation/CS1|citation |CitationClass=citation }} 1. REDIRECT Template:Link language Petrosjan, L.A. and Murzov, N.V. (1966). Game-theoretic problems of mechanics. Litovsk. Mat. Sb. 6, 423–433. 39. Hugh Brendan McMahan (2006), Robust Planning in Domains with Stochastic Outcomes, Adversaries, and Partial Observability, CMU-CS-06-166, pp. 3–4 40. {{#invoke:citation/CS1|citation |CitationClass=citation }} Cite error: Invalid <ref> tag; name "GT-PD-01" defined multiple times with different content 41. James Madison, Vices of the Political System of the United States, April 1787. 42. Jack Rakove, "James Madison and the Constitution", History Now, Issue 13, September 2007. 43. {{#invoke:citation/CS1|citation |CitationClass=citation }} English translation: {{#invoke:citation/CS1|citation |CitationClass=citation }} 44. {{#invoke:citation/CS1|citation |CitationClass=citation }} 45. 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. 46. Sylvia Nasar, A Beautiful Mind, Simon & Schuster, 1998. ISBN 0-684-81906-6. 47. Simon Singh "Between Genius and Madness", New York Times, June 14, 1998. 48. {{#invoke:citation/CS1|citation |CitationClass=citation }} References and further reading Textbooks and general references • {{#invoke:citation/CS1|citation |CitationClass=citation }}. "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. • {{#invoke:citation/CS1|citation |CitationClass=citation }} Description and Introduction, pp. 1–25. • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }}. Suitable for undergraduate and business students. • {{#invoke:citation/CS1|citation |CitationClass=citation }}. Suitable for upper-level undergraduates. • {{#invoke:citation/CS1|citation |CitationClass=citation }}. Acclaimed reference text. Description. • {{#invoke:citation/CS1|citation |CitationClass=citation }}. Suitable for advanced undergraduates. • Published in Europe as {{#invoke:citation/CS1|citation |CitationClass=citation }}. • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }}. Presents game theory in formal way suitable for graduate level. • {{#invoke:citation/CS1|citation |CitationClass=citation }}. Snippets from interviews. • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }}. An 88-page mathematical introduction; free online at many universities. • {{#invoke:citation/CS1|citation |CitationClass=citation }}. Suitable for a general audience. • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }}. Undergraduate textbook. • {{#invoke:citation/CS1|citation |CitationClass=citation }}. Primer for business men and women. • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }}. A modern introduction at the graduate level. • {{#invoke:citation/CS1|citation |CitationClass=citation }}. A general history of game theory and game theoreticians. • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }}. A comprehensive reference from a computational perspective; downloadable free online. • {{#invoke:citation/CS1|citation |CitationClass=citation }} Praised primer and popular introduction for everybody, never out of print. |CitationClass=citation }} 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 0-7167-6630-2. Textbook suitable for undergraduates in applied fields; numerous examples, fewer formalisms in concept presentation. • {{#invoke:citation/CS1|citation |CitationClass=book }} • {{#invoke:citation/CS1|citation |CitationClass=book }} • {{#invoke:citation/CS1|citation |CitationClass=book }} Historically important texts • Aumann, R.J. and Shapley, L.S. (1974), Values of Non-Atomic Games, Princeton University Press • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • reprinted edition: {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • 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. 1095–1100. • {{#invoke:citation/CS1|citation |CitationClass=citation }} 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. 42. Princeton University Press. • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} Other print references • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • Allan Gibbard, "Manipulation of voting schemes: a general result", Econometrica, Vol. 41, No. 4 (1973), pp. 587–601. • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }}, ISBN 978-0-631-23257-5 (2002 edition) • {{#invoke:citation/CS1|citation |CitationClass=citation }}. A layman's introduction. • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • Mark A. Satterthwaite, "Strategy-proofness and Arrow's Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions", Journal of Economic Theory 10 (April 1975), 187–217. • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} • {{#invoke:citation/CS1|citation |CitationClass=citation }} Websites • {{#invoke:citation/CS1|citation |CitationClass=citation }} {{ safesubst:#invoke:Unsubst||$N=Use dmy dates |date=__DATE__ |\$B= }}

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