Game theory is the study of

1

Chapter-preview links, pp

vii–xi

It has applications in all fields of

Psychological Pricing in Mergers & Acquisitions using Game Theory

School of Mathematics and Geospatial Sciences, RMIT University, MelbourneNash equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equ ...

. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. 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

Creative Commons Attribution 4.0 International License

in his review provides several examples where game theory is used to model project management scenarios. For instance, an investor typically has several investment options, and each option will likely result in a different project, and thus one of the investment options has to be chosen before the project charter can be produced. Similarly, any large project involving subcontractors, for instance, a construction project, has a complex interplay between the main contractor (the project manager) and subcontractors, or among the subcontractors themselves, which typically has several decision points. For example, if there is an ambiguity in the contract between the contractor and subcontractor, each must decide how hard to push their case without jeopardizing the whole project, and thus their own stake in it. Similarly, when projects from competing organizations are launched, the marketing personnel have to decide what is the best timing and strategy to market the project, or its resultant product or service, so that it can gain maximum traction in the face of competition. In each of these scenarios, the required decisions depend on the decisions of other players who, in some way, have competing interests to the interests of the decision-maker, and thus can ideally be modeled using game theory. Piraveenan summarises that two-player games are predominantly used to model project management scenarios, and based on the identity of these players, five distinct types of games are used in project management. * Government-sector–private-sector games (games that model

prisoner's dilemma
The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...

, 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.
A game-theoretic explanation for democratic peace is that public and open debate in democracies sends 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.
However, game theory predicts that two countries may still go to war even if their leaders are cognizant of the costs of fighting. War may result from asymmetric information; two countries may have incentives to mis-represent the amount of military resources they have on hand, rendering them unable to settle disputes agreeably without resorting to fighting. Moreover, war may arise because of commitment problems: if two countries wish to settle a dispute via peaceful means, but each wishes to go back on the terms of that settlement, they may have no choice but to resort to warfare. Finally, war may result from issue indivisibilities.
Game theory could also help predict a nation's responses when there is a new rule or law to be applied to that nation. One example is Peter John Wood's (2013) research looking into what nations could do to help reduce climate change. Wood thought this could be accomplished by making treaties with other nations to reduce

Nash equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equ ...

, 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

prisoner's dilemma
The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...

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

prisoner's dilemma
The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...

in a scene where he asks passengers in two different ferries to bomb the other one to save their own.
* In the 2018 film ''

Description

* . Suitable for undergraduate and business students. * . Suitable for upper-level undergraduates. * . Suitable for advanced undergraduates. ** Published in Europe as . * * . Presents game theory in formal way suitable for graduate level. * Joseph E. Harrington (2008) ''Games, strategies, and decision making'', Worth, . Textbook suitable for undergraduates in applied fields; numerous examples, fewer formalisms in concept presentation. * * *Maschler, Michael; Solan, Eilon; Zamir, Shmuel (2013), ''Game Theory'', Cambridge University Press, . Undergraduate textbook. * . Suitable for a general audience. * . Undergraduate textbook. * . A modern introduction at the graduate level. * * . A leading textbook at the advanced undergraduate level. * * Consistent treatment of game types usually claimed by different applied fields, e.g. Markov decision processes.

42.

Princeton University Press. * *

Introductory Game Theory Videos

* * Paul Walker

* David Levine

Game Theory. Papers, Lecture Notes and much more stuff.

* Alvin Roth: — Comprehensive list of links to game theory information on the Web * Adam Kalai

Game Theory and Computer Science

— Lecture notes on Game Theory and Computer Science * Mike Shor

GameTheory.net

— Lecture notes, interactive illustrations and other information. * Jim Ratliff'

Graduate Course in Game Theory

(lecture notes). * Don Ross

Review Of Game Theory

in the ''Stanford Encyclopedia of Philosophy''. * Bruno Verbeek and Christopher Morris

Game Theory and Ethics

* Elmer G. Wiens

Game Theory

— Introduction, worked examples, play online two-person zero-sum games. * Marek M. Kaminski

— Syllabuses and lecture notes for game theory and political science.

* Kesten Green's — Se

Papers

fo

evidence on the accuracy of forecasts from game theory and other methods

. * McKelvey, Richard D., McLennan, Andrew M., and Turocy, Theodore L. (2007)

Gambit: Software Tools for Game Theory

'. * Benjamin Polak

Open Course on Game Theory at Yale

videos of the course

* Benjamin Moritz, Bernhard Könsgen, Danny Bures, Ronni Wiersch, (2007)

Spieltheorie-Software.de: An application for Game Theory implemented in JAVA

'. * Antonin Kucera

Stochastic Two-Player Games

* Yu-Chi Ho

What is Mathematical Game Theory

What is Mathematical Game Theory (#2)

What is Mathematical Game Theory (#3)

What is Mathematical Game Theory (#4)-Many person game theory

What is Mathematical Game Theory ?( #5) – Finale, summing up, and my own view

{{DEFAULTSORT:Game Theory Artificial intelligence Formal sciences Mathematical economics John von Neumann

mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics ...

s of strategic interactions among rational agent
A rational agent or rational being is a person or entity that always aims to perform optimal actions based on given premises and information. A rational agent can be anything that makes decisions, typically a person, firm, machine, or software.
T ...

s. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1

Chapter-preview links, pp

vii–xi

It has applications in all fields of

social science
Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of soc ...

, as well as in logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premi ...

, systems science and computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...

. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term
In linguistics, semantics, general semantics, and ontologies, hyponymy () is a semantic relation between a hyponym denoting a subtype and a hypernym or hyperonym (sometimes called umbrella term or blanket term) denoting a supertype. In other ...

for the science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence ...

of logical decision making in humans, animals, as well as computers.
Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

. Von Neumann's original proof used the Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simple ...

on continuous mappings into compact convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...

s, which became a standard method in game theory and mathematical economics
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference a ...

. His paper was followed by the 1944 book ''Theory of Games and Economic Behavior
''Theory of Games and Economic Behavior'', published in 1944 by Princeton University Press, is a book by mathematician John von Neumann and economist Oskar Morgenstern which is considered the groundbreaking text that created the interdisciplinar ...

'', 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.
Game theory was developed extensively in the 1950s by many scholars. It was explicitly applied to evolution
Evolution is change in the heredity, heritable Phenotypic trait, characteristics of biological populations over successive generations. These characteristics are the Gene expression, expressions of genes, which are passed on from parent to ...

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
The Nobel Memorial Prize in Economic Sciences, officially the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel ( sv, Sveriges riksbanks pris i ekonomisk vetenskap till Alfred Nobels minne), is an economics award administered ...

going to game theorists Paul Milgrom and Robert B. Wilson, fifteen game theorists have won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of evolutionary game theory.
History

Precursors

Discussions on the mathematics of games began long before the rise of modern mathematical game theory. Cardano's work on games of chance in ''Liber de ludo aleae'' (''Book on Games of Chance''), which was written around 1564 but published posthumously in 1663, formulated some of the field's basic ideas. In the 1650s, Pascal and Huygens developed the concept of expectation on reasoning about the structure of games of chance, and Huygens published his gambling calculus in ''De ratiociniis in ludo aleæ'' (''On Reasoning in Games of Chance'') in 1657. In 1713, a letter attributed to Charles Waldegrave analyzed a game called "le Her". He was an active Jacobite and uncle to James Waldegrave, a British diplomat. In this letter, Waldegrave provided aminimax
Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. When d ...

mixed strategy solution to a two-person version of the card game le Her, and the problem is now known as Waldegrave problem. 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
A duopoly (from Greek δύο, ''duo'' "two" and πωλεῖν, ''polein'' "to sell") is a type of oligopoly where two firms have dominant or exclusive control over a market. It is the most commonly studied form of oligopoly due to its simplicit ...

and presented a solution that is the Nash equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equ ...

of the game.
In 1913, Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic se ...

published ''Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels'' (''On an Application of Set Theory to the Theory of the Game of Chess''), which proved that the optimal chess strategy is strictly determined. This paved the way for more general theorems.
In 1938, the Danish mathematical economist Frederik 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
Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability.
Biography
Borel wa ...

proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix is symmetric and provided a solution to a non-trivial infinite game (known in English as Blotto game). Borel conjectured the non-existence of mixed-strategy equilibria in finite two-person zero-sum games, a conjecture that was proved false by von Neumann.
Birth and early developments

Game theory did not exist as a unique field untilJohn von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

published the paper ''On the Theory of Games of Strategy'' in 1928. Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...

s, which became a standard method in game theory and mathematical economics
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference a ...

. His paper was followed by his 1944 book ''Theory of Games and Economic Behavior
''Theory of Games and Economic Behavior'', published in 1944 by Princeton University Press, is a book by mathematician John von Neumann and economist Oskar Morgenstern which is considered the groundbreaking text that created the interdisciplinar ...

'' co-authored with Oskar Morgenstern. The second edition of this book provided an axiomatic theory of utility, which reincarnated Daniel Bernoulli's old theory of utility (of 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. Subsequent work focused primarily on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.
In 1950, the first mathematical discussion of the prisoner's dilemma
The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...

appeared, and an experiment was undertaken by notable mathematicians Merrill M. Flood
Merrill Meeks Flood (1908 – 1991) was an American mathematician, notable for developing, with Melvin Dresher, the basis of the game theoretical Prisoner's dilemma model of cooperation and conflict while being at RAND in 1950 ( Albert W. Tucker ...

and Melvin Dresher
Melvin Dresher (born Dreszer; March 13, 1911 – June 4, 1992) was a Polish-born American mathematician, notable for developing, with Merrill Flood, the game theoretical model of cooperation and conflict known as the Prisoner's dilemma while ...

, as part of the RAND Corporation's investigations into game theory. RAND pursued the studies because of possible applications to global nuclear strategy
Nuclear strategy involves the development of doctrines and strategies for the production and use of nuclear weapons.
As a sub-branch of military strategy, nuclear strategy attempts to match nuclear weapons as means to political ends. In additi ...

. Around this same time, John Nash developed a criterion for mutual consistency of players' strategies known as the Nash equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equ ...

, applicable to a wider variety of games than the criterion proposed by von Neumann and Morgenstern. Nash proved that every finite n-player, non-zero-sum (not just two-player zero-sum) non-cooperative game has what is now known as a Nash equilibrium in mixed strategies.
Game theory experienced a flurry of activity in the 1950s, during which the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a uni ...

were developed. The 1950s also saw the first applications of game theory to philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. S ...

and political science
Political science is the scientific study of politics. It is a social science dealing with systems of governance and power, and the analysis of political activities, political thought, political behavior, and associated constitutions an ...

.
Prize-winning achievements

In 1965,Reinhard Selten
Reinhard Justus Reginald Selten (; 5 October 1930 – 23 August 2016) was a German economist, who won the 1994 Nobel Memorial Prize in Economic Sciences (shared with John Harsanyi and John Nash). He is also well known for his work in bound ...

introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium. Later he would introduce trembling hand perfection
A tremor is an involuntary, somewhat rhythmic, muscle contraction and relaxation involving oscillations or twitching movements of one or more body parts. It is the most common of all involuntary movements and can affect the hands, arms, eyes, fa ...

as well. In 1994 Nash, Selten and Harsanyi became Economics Nobel Laureates for their contributions to economic game theory.
In the 1970s, game theory was extensively applied in biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...

, largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy
An evolutionarily stable strategy (ESS) is a strategy (or set of strategies) that is ''impermeable'' when adopted by a population in adaptation to a specific environment, that is to say it cannot be displaced by an alternative strategy (or set o ...

. In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge
Common knowledge is knowledge that is publicly known by everyone or nearly everyone, usually with reference to the community in which the knowledge is referenced. Common knowledge can be about a broad range of subjects, such as science, literat ...

were introduced and analyzed.
In 2005, game theorists Thomas Schelling and Robert Aumann
Robert John Aumann (Hebrew name: , Yisrael Aumann; born June 8, 1930) is an Israeli-American mathematician, and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew ...

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 equilibrium coarsening and correlated equilibria, and developing an extensive formal analysis of the assumption of common knowledge and of its consequences.
In 2007, Leonid Hurwicz
Leonid Hurwicz (; August 21, 1917 – June 24, 2008) was a Polish-American economist and mathematician, known for his work in game theory and mechanism design. He originated the concept of incentive compatibility, and showed how desired outcomes ...

, Eric Maskin
Eric Stark Maskin (born December 12, 1950) is an American economist and mathematician. He was jointly awarded the 2007 Nobel Memorial Prize in Economic Sciences with Leonid Hurwicz and Roger Myerson "for having laid the foundations of mechanism ...

, and Roger Myerson
Roger Bruce Myerson (born March 29, 1951) is an American economist and professor at the University of Chicago. He holds the title of the David L. Pearson Distinguished Service Professor of Global Conflict Studies at The Pearson Institute for the ...

were 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''. Hurwicz introduced and formalized the concept of incentive compatibility
A mechanism is called incentive-compatible (IC) if every participant can achieve the best outcome to themselves just by acting according to their true preferences.
There are several different degrees of incentive-compatibility:
* The stronger d ...

.
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 2014, the Nobel went to game theorist Jean Tirole.
Game types

Cooperative / non-cooperative

A game is ''cooperative'' if the players are able to form binding commitments externally enforced (e.g. throughcontract law
A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tran ...

). A game is ''non-cooperative'' if players cannot form alliances or if all agreements need to be self-enforcing (e.g. through credible threats).
Cooperative games are often analyzed through the framework of ''cooperative game theory'', which focuses on predicting which coalitions will form, the joint actions that groups take, and the resulting collective payoffs. It is opposed to the traditional ''non-cooperative game theory'' which focuses on predicting individual players' actions and payoffs and analyzing Nash equilibria
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equi ...

. The focus on individual payoff can result in a phenomenon known as Tragedy of the Commons, where resources are used to a collectively inefficient level. The lack of formal negotiation
Negotiation is a dialogue between two or more people or parties to reach the desired outcome regarding one or more issues of conflict. It is an interaction between entities who aspire to agree on matters of mutual interest. The agreement c ...

leads to the deterioration of public goods through over-use and under provision that stems from private incentives.
Cooperative game theory provides a high-level approach as it describes only the structure, strategies, and payoffs of coalitions, whereas non-cooperative game theory also looks at how bargaining procedures will affect the distribution of payoffs within each coalition. As non-cooperative game theory is more general, cooperative games can be analyzed through the approach of non-cooperative game theory (the converse does not hold) provided that sufficient assumptions are made to encompass all the possible strategies available to players due to the possibility of external enforcement of cooperation. While using a single theory may be desirable, in many instances insufficient information is available to accurately model the formal procedures available during the strategic bargaining process, or the resulting model would be too complex to offer a practical tool in the real world. In such cases, cooperative game theory provides a simplified approach that allows analysis of the game at large without having to make any assumption about bargaining powers.
Symmetric / asymmetric

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. That is, 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 ofchicken
The chicken (''Gallus gallus domesticus'') is a domesticated junglefowl species, with attributes of wild species such as the grey and the Ceylon junglefowl that are originally from Southeastern Asia. Rooster or cock is a term for an adu ...

, the prisoner's dilemma
The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...

, and the stag hunt
In game theory, the stag hunt, sometimes referred to as the assurance game, trust dilemma or common interest game, describes a conflict between safety and social cooperation. The stag hunt problem originated with philosopher Jean-Jacques Rousseau ...

are all symmetric games. Some scholars would consider certain asymmetric games as examples of these games as well. However, the most common payoffs for each of these games are symmetric.
The most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the ultimatum game
The ultimatum game is a game that has become a popular instrument of economic experiments. An early description is by Nobel laureate John Harsanyi in 1961. One player, the proposer, is endowed with a sum of money. The proposer is tasked with sp ...

and similarly the dictator game
The dictator game is a popular experimental instrument in social psychology and economics, a derivative of the ultimatum game. The term "game" is a misnomer because it captures a decision by a single player: to send money to another or not. Thus, ...

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 in this section's graphic is asymmetric despite having identical strategy sets for both players.
Zero-sum / non-zero-sum

Zero-sum games (more generally, constant-sum games) are games in which choices by players can neither increase nor decrease the available resources. In zero-sum games, the total benefit goes to all players in a game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the equal expense of others).Poker
Poker is a family of Card_game#Comparing_games, comparing card games in which Card player, players betting (poker), wager over which poker hand, hand is best according to that specific game's rules. It is played worldwide, however in some plac ...

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
Matching pennies is the name for a simple game used in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously ...

and most classical board games including Go and chess
Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dis ...

.
Many games studied by game theorists (including the famed prisoner's dilemma
The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...

) 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
In economics, gains from trade are the net benefits to economic agents from being allowed an increase in voluntary trading with each other. In technical terms, they are the increase of consumer surplus plus producer surplus from lower tariffs or ...

. It is possible to transform any constant-sum 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

Simultaneous game
In game theory, a simultaneous game or static game is a game where each player chooses their action without knowledge of the actions chosen by other players. Simultaneous games contrast with sequential games, which are played by the players takin ...

s are games where both players move simultaneously, or instead 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 they do 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 An extensive-form game is a specification of a game in game theory, allowing (as the name suggests) for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, th ...

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 game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every s ...

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

The Cournot competition model involves players choosing quantity of a homogenous product to produce independently and simultaneously, where marginal cost can be different for each firm and the firm's payoff is profit. The production costs are public information and the firm aims to find their profit-maximising quantity based on what they believe the other firm will produce and behave like monopolies. In this game firms want to produce at the monopoly quantity but there is a high incentive to deviate and produce more, which decreases the market-clearing price. For example, firms may be tempted to deviate from the monopoly quantity if there is a low monopoly quantity and high price, with the aim of increasing production to maximise profit. However this option does not provide the highest payoff, as a firm's ability to maximise profits depends on its market share and the elasticity of the market demand. The Cournot equilibrium is reached when each firm operates on their reaction function with no incentive to deviate, as they have the best response based on the other firms output. Within the game, firms reach the Nash equilibrium when the Cournot equilibrium is achieved.Bertrand Competition

TheBertrand competition Bertrand competition is a model of competition used in economics, named after Joseph Louis François Bertrand (1822–1900). It describes interactions among firms (sellers) that set prices and their customers (buyers) that choose quantities at the p ...

, assumes homogenous products and a constant marginal cost and players choose the prices. The equilibrium of price competition is where the price is equal to marginal costs, assuming complete information about the competitors' costs. Therefore, the firms have an incentive to deviate from the equilibrium because a homogenous product with a lower price will gain all of the market share, known as a cost advantage.
Perfect information and imperfect information

An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players, at every move in the game, know the moves previously made by all other players. In reality, this can be applied to firms and consumers having information about price and quality of all the available goods in a market. An imperfect information game is played when the players do not know all moves already made by the opponent such as a simultaneous move game. Most games studied in game theory are imperfect-information games. Examples of perfect-information games includetic-tac-toe
Tic-tac-toe (American English), noughts and crosses ( Commonwealth English), or Xs and Os (Canadian or Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid with ''X'' or ''O''. ...

, checkers, chess
Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dis ...

, and Go.
Many card games are games of imperfect information, such as poker
Poker is a family of Card_game#Comparing_games, comparing card games in which Card player, players betting (poker), wager over which poker hand, hand is best according to that specific game's rules. It is played worldwide, however in some plac ...

and bridge. Perfect information is often confused with complete information
In economics and game theory, complete information is an economic situation or game in which knowledge about other market participants or players is available to all participants. The utility functions (including risk aversion), payoffs, strategies ...

, 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, whereas perfect information is knowledge of all aspects of the game and players. Games of incomplete information
In economics and game theory, complete information is an economic situation or game in which knowledge about other market participants or players is available to all participants. The utility functions (including risk aversion), payoffs, strategies ...

can be reduced, however, to games of imperfect information by introducing " moves by nature".
Bayesian game

One of the assumptions of the Nash equilibrium is that every player has correct beliefs about the actions of the other players. However, there are many situations in game theory where participants do not fully understand the characteristics of their opponents. Negotiators may be unaware of their opponent's valuation of the object of negotiation, companies may be unaware of their opponent's cost functions, combatants may be unaware of their opponent's strengths, and jurors may be unaware of their colleague's interpretation of the evidence at trial. In some cases, participants may know the character of their opponent well, but may not know how well their opponent knows his or her own character.Bayesian game
In game theory, a Bayesian game is a game that models the outcome of player interactions using aspects of Bayesian probability. Bayesian games are notable because they allowed, for the first time in game theory, for the specification of the soluti ...

means a strategic game with incomplete information. For a strategic game, decision makers are players, and every player has a group of actions. A core part of the imperfect information specification is the set of states. Every state completely describes a collection of characteristics relevant to the player such as their preferences and details about them. There must be a state for every set of features that some player believes may exist.
For example, where Player 1 is unsure whether Player 2 would rather date her or get away from her, while Player 2 understands Player 1's preferences as before. To be specific, supposing that Player 1 believes that Player 2 wants to date her under a probability of 1/2 and get away from her under a probability of 1/2 (this evaluation comes from Player 1's experience probably: she faces players who want to date her half of the time in such a case and players who want to avoid her half of the time). Due to the probability involved, the analysis of this situation requires to understand the player's preference for the draw, even though people are only interested in pure strategic equilibrium.
Combinatorial games

Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games. Examples includechess
Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dis ...

and Go. Games that involve imperfect information may also have a strong combinatorial character, for instance backgammon
Backgammon is a two-player board game played with counters and dice on tables boards. It is the most widespread Western member of the large family of tables games, whose ancestors date back nearly 5,000 years to the regions of Mesopotamia and P ...

. There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve some particular problems and answer some general questions.
Games of perfect information have been studied in combinatorial game theory
Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the players ...

, which has developed novel representations, e.g. surreal numbers
In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals s ...

, as well as combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...

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. A typical game that has been solved this way is Hex. A related field of study, drawing from computational complexity theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by ...

, is game complexity
Combinatorial game theory has several ways of measuring game complexity. This article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity.
Measures of game comp ...

, which is concerned with estimating the computational difficulty of finding optimal strategies.
Research in artificial intelligence
Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech r ...

has addressed both perfect and imperfect 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
Alpha–beta pruning is a search algorithm that seeks to decrease the number of nodes that are evaluated by the minimax algorithm in its search tree. It is an adversarial search algorithm used commonly for machine playing of two-player games ...

or use of artificial neural networks trained by reinforcement learning
Reinforcement learning (RL) is an area of machine learning concerned with how intelligent agents ought to take actions in an environment in order to maximize the notion of cumulative reward. Reinforcement learning is one of three basic machine ...

, which make games more tractable in computing practice.
Infinitely long games

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 awinning strategy
Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and simil ...

. (It can be proven, using the axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...

, that there are gameseven with perfect information and where the only outcomes are "win" or "lose"for which ''neither'' player has a winning strategy.) The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to ot ...

.
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 game
A continuous game is a mathematical concept, used in game theory, that generalizes the idea of an ordinary game like tic-tac-toe (noughts and crosses) or checkers (draughts). In other words, it extends the notion of a discrete game, where the playe ...

s allow players to choose a strategy from a continuous strategy set. For instance, Cournot competition Cournot competition is an economic model used to describe an industry structure in which companies compete on the amount of output they will produce, which they decide on independently of each other and at the same time. It is named after Antoine Au ...

is typically modeled with players' strategies being any non-negative quantities, including fractional quantities.
Differential games

Differential game 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. More specifically, a state variable or variables evolve over time according to a differential equation ...

s 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
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...

. 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.
Evolutionary game theory

Evolutionary game theory studies players who adjust their strategies over time according to rules that are not necessarily rational or farsighted. In general, the evolution of strategies over time according to such rules is modeled as aMarkov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...

with a state variable such as the current strategy profile or how the game has been played in the recent past. Such rules may feature imitation, optimization, or survival of the fittest.
In biology, such models can represent evolution
Evolution is change in the heredity, heritable Phenotypic trait, characteristics of biological populations over successive generations. These characteristics are the Gene expression, expressions of genes, which are passed on from parent to ...

, in which offspring adopt their parents' strategies and parents who play more successful strategies (i.e. corresponding to higher payoffs) have a greater number of offspring. In the social sciences, such models typically represent strategic adjustment by players who play a game many times within their lifetime and, consciously or unconsciously, occasionally adjust their strategies.
Stochastic outcomes (and relation to other fields)

Individual decision problems with stochastic outcomes are sometimes considered "one-player games". They may be modeled using similar tools within the related disciplines of decision theory,operations research
Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve deci ...

, and areas of artificial intelligence
Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech r ...

, particularly AI planning (with uncertainty) and multi-agent system
A multi-agent system (MAS or "self-organized system") is a computerized system composed of multiple interacting intelligent agents.Hu, J.; Bhowmick, P.; Jang, I.; Arvin, F.; Lanzon, A.,A Decentralized Cluster Formation Containment Framework fo ...

. Although these fields may have different motivators, the mathematics involved are substantially the same, e.g. using Markov decision processes (MDP).
Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" (" moves by nature"). 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. (See Black swan theory
The black swan theory or theory of black swan events is a metaphor that describes an event that comes as a surprise, has a major effect, and is often inappropriately rationalized after the fact with the benefit of hindsight. The term is based on ...

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
A gold standard is a monetary system in which the standard economic unit of account is based on a fixed quantity of gold. The gold standard was the basis for the international monetary system from the 1870s to the early 1920s, and from the la ...

" is considered to be partially observable stochastic game (POSG), but few realistic problems are computationally feasible in POSG representation.
Metagames

These are games the play of which is the development of the rules for another game, the target or subject game.Metagame
Metagame, Hypergame, or game about the game, is an approach to a game that transcends or operates outside of the prescribed rules of the game, uses external factors to affect the game, or goes beyond the supposed limits or environment set by t ...

s 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 Metagame analysis involves framing a problem situation as a strategic game in which participants try to realise their objectives by means of the options available to them. The subsequent meta-analysis of this game gives insight in possible strategie ...

is also used to refer to a practical approach developed by Nigel Howard, whereby a situation is framed as a strategic game in which stakeholders try to realize their objectives by means of the options available to them. Subsequent developments have led to the formulation of confrontation analysis
Confrontation analysis (also known as dilemma analysis) is an operational analysis technique used to structure, understand and think through multi-party interactions such as negotiations. It is the underpinning mathematical basis of drama theory. ...

.
Pooling games

These are games prevailing over all forms of society. Pooling games are repeated plays with changing payoff table in general over an experienced path, and their equilibrium strategies usually take a form of evolutionarysocial convention
A convention is a set of agreed, stipulated, or generally accepted standards, norms, social norms, or criteria, often taking the form of a custom.
In a social context, a convention may retain the character of an "unwritten law" of custom (for ex ...

and economic convention. Pooling game theory emerges to formally recognize the interaction between optimal choice in one play and the emergence of forthcoming payoff table update path, identify the invariance existence and robustness, and predict variance over time. The theory is based upon topological transformation classification of payoff table update over time to predict variance and invariance, and is also within the jurisdiction of the computational law of reachable optimality for ordered system.
Mean field game theory

Mean field game theory Mean-field game theory is the study of strategic decision making by small interacting agents in very large populations. It lies at the intersection of game theory with stochastic analysis and control theory. The use of the term "mean field" is insp ...

is the study of strategic decision making in very large populations of small interacting agents. This class of problems was considered in the economics literature by Boyan Jovanovic
Boyan Jovanovic is a professor of economics at New York University and a long-term consultant at the Federal Reserve Bank of Richmond.
Jovanovic, of Serbian descent, received his Bachelor's and master's degrees from the London School of Economic ...

and Robert W. Rosenthal, in the engineering literature by Peter E. Caines, and by mathematicians Pierre-Louis Lions
Pierre-Louis Lions (; born 11 August 1956) is a French mathematician. He is known for a number of contributions to the fields of partial differential equations and the calculus of variations. He was a recipient of the 1994 Fields Medal and the 19 ...

and Jean-Michel Lasry.
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. (Eric Rasmusen refers to these four "essential elements" by the acronym "PAPI".) 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

The extensive form can be used to formalize games with a time sequencing of moves. Games here are played ontrees
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that ar ...

(as pictured here). Here each vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
*Vertex (computer graphics), a data structure that describes the position ...

(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
A decision tree is a decision support tool that uses a tree-like model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. It is one way to display an algorithm that only contains condi ...

. To solve any extensive form game, backward induction
Backward induction is the process of reasoning backwards in time, from the end of a problem or situation, to determine a sequence of optimal actions. It proceeds by examining the last point at which a decision is to be made and then identifying wha ...

must be used. It involves working backward up the game tree to determine what a rational player would do at the last vertex of the tree, what the player with the previous move would do given that the player with the last move is rational, and so on until the first vertex of the tree is reached.
The game pictured consists of two players. The way this particular game is structured (i.e., with sequential decision making and perfect information), ''Player 1'' "moves" first by choosing either or (fair or unfair). Next in the sequence, ''Player 2'', who has now seen ''Player 1''s move, chooses to play either or . Once ''Player 2'' has made their choice, the game is considered finished and each player gets their respective payoff. Suppose that ''Player 1'' chooses and then ''Player 2'' chooses : ''Player 1'' then gets a payoff of "eight" (which in real-world terms can be interpreted in many ways, the simplest of which is in terms of money but could mean things such as eight days of vacation or eight countries conquered or even eight more opportunities to play the same game against other players) and ''Player 2'' gets a payoff of "two".
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

The normal (or strategic form) game is usually represented by amatrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

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.
Characteristic function form

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 $\backslash mathbf$ appears, it works against the fraction $\backslash left(\backslash frac\backslash right)$ 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 $v:2^N\; \backslash to\; \backslash mathbf$ is a normal utility. Such characteristic functions have expanded to describe games where there is no removable utility.Alternative game representations

Alternative game representation forms are used for some subclasses of games or adjusted to the needs of interdisciplinary research. In addition to classical game representations, some of the alternative representations also encode time related aspects.General and applied uses

As a method ofapplied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical ...

, 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
Charles Robert Darwin ( ; 12 February 1809 – 19 April 1882) was an English naturalist, geologist, and biologist, widely known for his contributions to evolutionary biology. His proposition that all species of life have descended f ...

made game-theoretic kinds of statements, 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 1982 book ''Evolution and the Theory of Games
''Evolution and the Theory of Games'' is a book by the British evolutionary biologist John Maynard Smith on evolutionary game theory. The book was initially published in December 1982 by Cambridge University Press.
Overview
In the book, John Ma ...

''.
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. 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
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...

. An alternative version of game theory, called chemical game theory, represents the player's choices as metaphorical chemical reactant molecules called "knowlecules". Chemical game theory then calculates the outcomes as equilibrium solutions to a system of chemical reactions.
Description and modeling

The primary use of game theory is to describe and model how human populations behave. Some scholars believe that by finding the equilibria of games they can predict how actual human populations will behave when confronted with situations analogous to the game being studied. This particular view of game theory has been criticized. It is argued that the assumptions made by game theorists are often violated when applied to real-world situations. Game theorists usually assume players act rationally, but in practice, human behavior often deviates from this model. Game theorists respond by comparing their assumptions to those used inphysics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which re ...

. Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.
Physicists generally are interested in the root or ultimate caus ...

s. However, empirical work has shown that in some classic games, such as the centipede game, guess 2/3 of the average
In game theory, "guess of the average" is a game that explores how a player’s strategic reasoning process takes into account the mental process of others in the game.
In this game, players simultaneously select a real number between 0 and 100, ...

game, and the dictator game
The dictator game is a popular experimental instrument in social psychology and economics, a derivative of the ultimatum game. The term "game" is a misnomer because it captures a decision by a single player: to send money to another or not. Thus, ...

, people regularly do not play Nash equilibria. There is an ongoing debate regarding the importance of these experiments and whether the analysis of the experiments fully captures all aspects of the relevant situation.
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
Bounded rationality is the idea that rationality is limited when individuals make decisions, and under these limitations, rational individuals will select a decision that is satisfactory rather than optimal.
Limitations include the difficulty of ...

on the part of players. Despite the name, evolutionary game theory does not necessarily presume natural selection
Natural selection is the differential survival and reproduction of individuals due to differences in phenotype. It is a key mechanism of evolution, the change in the heritable traits characteristic of a population over generations. Char ...

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

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 strategy, corresponding to aNash equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equ ...

of a game constitutes one's best response
In game theory, the best response is the strategy (or strategies) which produces the most favorable outcome for a player, taking other players' strategies as given (; ). The concept of a best response is central to John Nash's best-known contribu ...

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. This normative use of game theory has also come under criticism.
Economics and business

Game theory is a major method used inmathematical economics
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference a ...

and business for modeling competing behaviors of interacting agents. Applications include a wide array of economic phenomena and approaches, such as auction
An auction is usually a process of buying and selling goods or services by offering them up for bids, taking bids, and then selling the item to the highest bidder or buying the item from the lowest bidder. Some exceptions to this definition ex ...

s, bargaining
In the social sciences, bargaining or haggling is a type of negotiation in which the buyer and seller of a good or service debate the price or nature of a transaction. If the bargaining produces agreement on terms, the transaction takes pl ...

, mergers and acquisitions pricing,N. Agarwal and P. ZeephongsekulPsychological Pricing in Mergers & Acquisitions using Game Theory

School of Mathematics and Geospatial Sciences, RMIT University, Melbourne

fair division
Fair division is the problem in game theory of dividing a set of resources among several people who have an entitlement to them so that each person receives their due share. That problem arises in various real-world settings such as division of in ...

, duopolies, oligopolies
An oligopoly (from Greek ὀλίγος, ''oligos'' "few" and πωλεῖν, ''polein'' "to sell") is a market structure in which a market or industry is dominated by a small number of large sellers or producers. Oligopolies often result fro ...

, social network
A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for an ...

formation, agent-based computational economics
Agent-based computational economics (ACE) is the area of computational economics that studies economic processes, including whole economies, as dynamic systems of interacting agents. As such, it falls in the paradigm of complex adaptive systems. ...

, general equilibrium
In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an ov ...

, mechanism design, and voting system
An electoral system or voting system is a set of rules that determine how elections and referendums are conducted and how their results are determined. Electoral systems are used in politics to elect governments, while non-political elections m ...

s; and across such broad areas as experimental economics
Experimental economics is the application of experimental methods to study economic questions. Data collected in experiments are used to estimate effect size, test the validity of economic theories, and illuminate market mechanisms. Economic exp ...

, behavioral economics, information economics, industrial organization, and political economy.
This research usually focuses on particular sets of strategies known as "solution concepts" or "equilibria". A common assumption is that players act rationally. In non-cooperative games, the most famous of these is the utility
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophe ...

of individual players.
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. Economists and business professors suggest two primary uses (noted above): ''descriptive'' and ''prescriptive
Linguistic prescription, or prescriptive grammar, is the establishment of rules defining preferred usage of language. These rules may address such linguistic aspects as spelling, pronunciation, vocabulary, syntax, and semantics. Sometimes i ...

''.
The Chartered Institute of Procurement & Supply
The Chartered Institute of Procurement & Supply (CIPS), formerly the Chartered Institute of Purchasing & Supply, is a global professional body working for the procurement and supply profession in many regions of the world. It promotes best prac ...

(CIPS) promotes knowledge and use of game theory within the context of business procurement
Procurement is the method of discovering and agreeing to terms and purchasing goods, services, or other works from an external source, often with the use of a tendering or competitive bidding process. When a government agency buys goods or se ...

. CIPS and TWS Partners have conducted a series of surveys designed to explore the understanding, awareness and application of game theory among procurement
Procurement is the method of discovering and agreeing to terms and purchasing goods, services, or other works from an external source, often with the use of a tendering or competitive bidding process. When a government agency buys goods or se ...

professionals. Some of the main findings in their third annual survey (2019) include:
*application of game theory to procurement activity has increased – at the time it was at 19% across all survey respondents
*65% of participants predict that use of game theory applications will grow
*70% of respondents say that they have "only a basic or a below basic understanding" of game theory
*20% of participants had undertaken on-the-job training in game theory
*50% of respondents said that new or improved software solutions were desirable
*90% of respondents said that they do not have the software they need for their work.
Project management

Sensible decision-making is critical for the success of projects. In project management, game theory is used to model the decision-making process of players, such as investors, project managers, contractors, sub-contractors, governments and customers. Quite often, these players have competing interests, and sometimes their interests are directly detrimental to other players, making project management scenarios well-suited to be modeled by game theory. Piraveenan (2019) Material was copied from this source, which is available underCreative Commons Attribution 4.0 International License

in his review provides several examples where game theory is used to model project management scenarios. For instance, an investor typically has several investment options, and each option will likely result in a different project, and thus one of the investment options has to be chosen before the project charter can be produced. Similarly, any large project involving subcontractors, for instance, a construction project, has a complex interplay between the main contractor (the project manager) and subcontractors, or among the subcontractors themselves, which typically has several decision points. For example, if there is an ambiguity in the contract between the contractor and subcontractor, each must decide how hard to push their case without jeopardizing the whole project, and thus their own stake in it. Similarly, when projects from competing organizations are launched, the marketing personnel have to decide what is the best timing and strategy to market the project, or its resultant product or service, so that it can gain maximum traction in the face of competition. In each of these scenarios, the required decisions depend on the decisions of other players who, in some way, have competing interests to the interests of the decision-maker, and thus can ideally be modeled using game theory. Piraveenan summarises that two-player games are predominantly used to model project management scenarios, and based on the identity of these players, five distinct types of games are used in project management. * Government-sector–private-sector games (games that model

public–private partnership
A public–private partnership (PPP, 3P, or P3) is a long-term arrangement between a government and private sector institutions.Hodge, G. A and Greve, C. (2007), Public–Private Partnerships: An International Performance Review, Public Administ ...

s)
* Contractor–contractor games
* Contractor–subcontractor games
* Subcontractor–subcontractor games
* Games involving other players
In terms of types of games, both cooperative as well as non-cooperative, normal-form as well as extensive-form, and zero-sum as well as non-zero-sum are used to model various project management scenarios.
Political science

The application of game theory topolitical science
Political science is the scientific study of politics. It is a social science dealing with systems of governance and power, and the analysis of political activities, political thought, political behavior, and associated constitutions an ...

is focused in the overlapping areas of fair division
Fair division is the problem in game theory of dividing a set of resources among several people who have an entitlement to them so that each person receives their due share. That problem arises in various real-world settings such as division of in ...

, political economy, public choice
Public choice, or public choice theory, is "the use of economic tools to deal with traditional problems of political science".Gordon Tullock, 9872008, "public choice," ''The New Palgrave Dictionary of Economics''. . Its content includes the s ...

, war bargaining, positive political theory, and social choice theory
Social choice theory or social choice is a theoretical framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a ''collective decision'' or ''social welfare'' in some sense.Amartya Sen (2008). "Soc ...

. 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
Anthony Downs (November 21, 1930October 2, 2021) was an American economist specializing in public policy and public administration. His research focuses included political choice theory, rent control, affordable housing, and transportation ec ...

. In his 1957 book ''An Economic Theory of Democracy
''An Economic Theory of Democracy'' is a treatise of economics written by Anthony Downs, published in 1957. The book set forth a model with precise conditions under which economic theory could be applied to non- market political decision-making. ...

'', 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. Game theory was applied in 1962 to the Cuban Missile Crisis
The Cuban Missile Crisis, also known as the October Crisis (of 1962) ( es, Crisis de Octubre) in Cuba, the Caribbean Crisis () in Russia, or the Missile Scare, was a 35-day (16 October – 20 November 1962) confrontation between the United ...

during the presidency of John F. Kennedy.
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 greenhouse gas emissions
Greenhouse gas emissions from human activities strengthen the greenhouse effect, contributing to climate change. Most is carbon dioxide from burning fossil fuels: coal, oil, and natural gas. The largest emitters include coal in China and lar ...

. However, he concluded that this idea could not work because it would create a prisoner's dilemma for the nations.
Biology

Unlike those in economics, the payoffs for games inbiology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...

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 evolution
Evolution is change in the heredity, heritable Phenotypic trait, characteristics of biological populations over successive generations. These characteristics are the Gene expression, expressions of genes, which are passed on from parent to ...

ary forces. The best-known equilibrium in biology is known as the ''evolutionarily stable strategy
An evolutionarily stable strategy (ESS) is a strategy (or set of strategies) that is ''impermeable'' when adopted by a population in adaptation to a specific environment, that is to say it cannot be displaced by an alternative strategy (or set o ...

'' (ESS), first introduced in . Although its initial motivation did not involve any of the mental requirements of the sex ratio
The sex ratio (or gender ratio) is usually defined as the ratio of males to females in a population. As explained by Fisher's principle, for evolutionary reasons this is typically about 1:1 in species which reproduce sexually. Many species de ...

s. 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. 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
Paul Andrew Ormerod (born 20 March 1950) is a British economist who is a partner at Volterra Partners consultancy. Additionally, he is a visiting professor at UCL Centre for Decision Making Uncertainty.
Research
Ormerod has researched complexity ...

's '' Butterfly Economics'').
Biologists have used the game of chicken to analyze fighting behavior and territoriality.
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.
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 monkey
The vervet monkey (''Chlorocebus pygerythrus''), or simply vervet, is an Old World monkey of the family Cercopithecidae native to Africa. The term "vervet" is also used to refer to all the members of the genus ''Chlorocebus''. The five distinct ...

s that warn group members of a predator's approach, even when it endangers that individual's chance of survival. 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
Kin selection is the evolutionary strategy that favours the reproductive success of an organism's relatives, even when at a cost to the organism's own survival and reproduction. Kin altruism can look like altruistic behaviour whose evolution i ...

. Altruists discriminate between the individuals they help and favor relatives. Hamilton's rule
Kin selection is the evolutionary strategy that favours the reproductive success of an organism's relatives, even when at a cost to the organism's own survival and reproduction. Kin altruism can look like altruistic behaviour whose evolution i ...

explains the evolutionary rationale behind this selection with the equation , where the cost to the altruist must be less than the benefit to the recipient multiplied by the coefficient of relatedness . 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. For example, helping a sibling (in diploid animals) has a coefficient of , because (on average) an individual shares half 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. 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 coefficient that was in the smaller field becomes 0.995. Similarly if it is considered that information other than that of a genetic nature (e.g. epigenetics, religion, science, etc.) persisted through time the playing field becomes larger still, and the discrepancies smaller.
Computer science and logic

Game theory has come to play an increasingly important role inlogic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premi ...

and in computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...

. Several logical theories have a basis in game semantics
Game semantics (german: dialogische Logik, translated as '' dialogical logic'') is an approach to formal semantics that grounds the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a player ...

. In addition, computer scientists have used games to model interactive computation
In computer science, interactive computation is a mathematical model for computation that involves input/output communication with the external world ''during'' computation.
Uses
Among the currently studied mathematical models of computation th ...

s. Also, game theory provides a theoretical basis to the field of multi-agent system
A multi-agent system (MAS or "self-organized system") is a computerized system composed of multiple interacting intelligent agents.Hu, J.; Bhowmick, P.; Jang, I.; Arvin, F.; Lanzon, A.,A Decentralized Cluster Formation Containment Framework fo ...

s.
Separately, game theory has played a role in online algorithm
In computer science, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start.
In contrast, an o ...

s; in particular, the -server problem, which has in the past been referred to as ''games with moving costs'' and ''request-answer games''. Yao's principle is a game-theoretic technique for proving lower bounds on the computational complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations ...

of randomized algorithm
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performa ...

s, 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
Algorithmic game theory (AGT) is an area in the intersection of game theory and computer science, with the objective of understanding and design of algorithms in strategic environments.
Typically, in Algorithmic Game Theory problems, the input to ...

and within it algorithmic mechanism design Algorithmic mechanism design (AMD) lies at the intersection of economic game theory, optimization, and computer science. The prototypical problem in mechanism design is to design a system for multiple self-interested participants, such that the part ...

combine computational algorithm design
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

and analysis of complex system
A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication s ...

s with economic theory.
Philosophy

Game theory has been put to several uses inphilosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. S ...

. Responding to two papers by , used game theory to develop a philosophical account of convention. In so doing, he provided the first analysis of common knowledge
Common knowledge is knowledge that is publicly known by everyone or nearly everyone, usually with reference to the community in which the knowledge is referenced. Common knowledge can be about a broad range of subjects, such as science, literat ...

and employed it in analyzing play in coordination game A coordination game is a type of simultaneous game found in game theory. It describes the situation where a player will earn a higher payoff when they select the same course of action as another player. The game is not one of pure conflict, which r ...

s. 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. Following game-theoretic account of conventions, Edna Ullmann-Margalit (1977) and Bicchieri (2006) have developed theories of social norms
Social norms are shared standards of acceptance, acceptable behavior by groups. Social norms can both be informal understandings that govern the behavior of members of a society, as well as be codified into wikt:rule, rules and laws. Social normat ...

that define them as Nash equilibria that result from transforming a mixed-motive game into a coordination game.
Game theory has also challenged philosophers to think in terms of interactive epistemology
Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics.
Episte ...

: 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 the interactions of agents. Philosophers who have worked in this area include Bicchieri (1989, 1993), Skyrms (1990), and Stalnaker (1999).
In ethics
Ethics or moral philosophy is a branch of philosophy that "involves systematizing, defending, and recommending concepts of right and wrong behavior".''Internet Encyclopedia of Philosophy'' The field of ethics, along with aesthetics, concerns m ...

, some (most notably David Gauthier, Gregory Kavka, and Jean Hampton) authors have attempted to pursue Thomas Hobbes' project of deriving morality from self-interest. Since games like the social contract
In moral and political philosophy, the social contract is a theory or model that originated during the Age of Enlightenment and usually, although not always, concerns the legitimacy of the authority of the state over the individual.
Social ...

view in political philosophy
Political philosophy or political theory is the philosophical study of government, addressing questions about the nature, scope, and legitimacy of public agents and institutions and the relationships between them. Its topics include politics, ...

(for examples, see and ).
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
In game theory, the stag hunt, sometimes referred to as the assurance game, trust dilemma or common interest game, describes a conflict between safety and social cooperation. The stag hunt problem originated with philosopher Jean-Jacques Rousseau ...

, and the Nash bargaining game Cooperative bargaining is a process in which two people decide how to share a surplus that they can jointly generate. In many cases, the surplus created by the two players can be shared in many ways, forcing the players to negotiate which division o ...

as providing an explanation for the emergence of attitudes about morality (see, e.g., and ).
Retail and consumer product pricing

Game theory applications are often used in the pricing strategies of retail and consumer markets, particularly for the sale of inelastic goods. With retailers constantly competing against one another for consumer market share, it has become a fairly common practice for retailers to discount certain goods, intermittently, in the hopes of increasing foot-traffic in brick and mortar locations (websites visits fore-commerce
E-commerce (electronic commerce) is the activity of electronically buying or selling of products on online services or over the Internet. E-commerce draws on technologies such as mobile commerce, electronic funds transfer, supply chain manage ...

retailers) or increasing sales of ancillary or complimentary products.
Black Friday, a popular shopping holiday in the US, is when many retailers focus on optimal pricing strategies to capture the holiday shopping market. In the Black Friday scenario, retailers using game theory applications typically ask "what is the dominant competitor's reaction to me?" In such a scenario, the game has two players: the retailer, and the consumer. The retailer is focused on an optimal pricing strategy, while the consumer is focused on the best deal. In this closed system, there often is no dominant strategy as both players have alternative options. That is, retailers can find a different customer, and consumers can shop at a different retailer. Given the market competition that day, however, the dominant strategy
In game theory, strategic dominance (commonly called simply dominance) occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The o ...

for retailers lies in outperforming competitors. The open system assumes multiple retailers selling similar goods, and a finite number of consumers demanding the goods at an optimal price. A blog by a Cornell University
Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to teac ...

professor provided an example of such a strategy, when Amazon
Amazon most often refers to:
* Amazons, a tribe of female warriors in Greek mythology
* Amazon rainforest, a rainforest covering most of the Amazon basin
* Amazon River, in South America
* Amazon (company), an American multinational technology c ...

priced a Samsung TV $100 below retail value, effectively undercutting competitors. Amazon made up part of the difference by increasing the price of HDMI cables, as it has been found that consumers are less price discriminatory when it comes to the sale of secondary items.
Retail markets continue to evolve strategies and applications of game theory when it comes to pricing consumer goods. The key insights found between simulations in a controlled environment and real-world retail experiences show that the applications of such strategies are more complex, as each retailer has to find an optimal balance between pricing
Pricing is the process whereby a business sets the price at which it will sell its products and services, and may be part of the business's marketing plan. In setting prices, the business will take into account the price at which it could acqui ...

, supplier relations, brand image, and the potential to cannibalize the sale of more profitable items.
Epidemiology

Since the decision to take a vaccine for a particular disease is often made by individuals, who may consider a range of factors and parameters in making this decision (such as the incidence and prevalence of the disease, perceived and real risks associated with contracting the disease, mortality rate, perceived and real risks associated with vaccination, and financial cost of vaccination), game theory has been used to model and predict vaccination uptake in a society.In popular culture

* Based on the 1998 book bySylvia Nasar
Sylvia Nasar (born 17 August 1947) is an Uzbek German-born American journalist. She is best known for her biography of John Forbes Nash Jr., '' A Beautiful Mind'', for which she won the National Book Critics Circle Award for Biography. Nasar cu ...

, the life story of game theorist and mathematician John Nash was turned into the 2001 biopic
A biographical film or biopic () is a film that dramatizes the life of a non-fictional or historically-based person or people. Such films show the life of a historical person and the central character's real name is used. They differ from docu ...

'' A Beautiful Mind'', starring Russell Crowe as Nash.
* The 1959 military science fiction novel ''Starship Troopers
''Starship Troopers'' is a military science fiction novel by American writer Robert A. Heinlein. Written in a few weeks in reaction to the US suspending nuclear tests, the story was first published as a two-part serial in ''The Magazine of ...

'' by Robert A. Heinlein
Robert Anson Heinlein (; July 7, 1907 – May 8, 1988) was an American science fiction author, aeronautical engineer, and naval officer. Sometimes called the "dean of science fiction writers", he was among the first to emphasize scientific accu ...

mentioned "games theory" and "theory of games". In the 1997 film of the same name, the character Carl Jenkins referred to his military intelligence assignment as being assigned to "games and theory".
* The 1964 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 Soviet Union irrevocably committing to a catastrophic nuclear response without making the threat public.
* The 1980s power pop band Game Theory was founded by singer/songwriter Scott Miller, who described the band's name as alluding to "the study of calculating the most appropriate action given an adversary... to give yourself the minimum amount of failure"..
* ''Liar Game
''Liar Game'' (stylized as ''LIAR GAME'') is a Japanese manga series written and illustrated by Shinobu Kaitani. It was serialized in Shueisha's ''seinen'' manga magazine ''Weekly Young Jump'' from February 2005 to January 2015. It was ada ...

'', a 2005 Japanese manga
Manga (Japanese: 漫画 ) are comics or graphic novels originating from Japan. Most manga conform to a style developed in Japan in the late 19th century, and the form has a long prehistory in earlier Japanese art. The term ''manga'' is used ...

and 2007 television series, presents the main characters in each episode with a game or problem that is typically drawn from game theory, as demonstrated by the strategies applied by the characters.
* The 1974 novel '' Spy Story'' by Len Deighton
Leonard Cyril Deighton (; born 18 February 1929) is a British author. His publications have included cookery books, history and military history, but he is best known for his spy novels.
After completing his national service in the Royal Air Fo ...

explores elements of Game Theory in regard to cold war army exercises.
* The 2008 novel '' The Dark Forest'' by Liu Cixin
Liu Cixin (, pronounced ; born 23 June 1963) is a Chinese science fiction writer. He is a nine-time winner of China's Galaxy Award and has also received the 2015 Hugo Award for his novel '' The Three-Body Problem'' as well as the 2017 L ...

explores the relationship between extraterrestrial life, humanity, and game theory.
* The prime antagonist Joker in the movie ''The Dark Knight
''The Dark Knight'' is a 2008 superhero film directed by Christopher Nolan from a screenplay he co-wrote with his brother Jonathan. Based on the DC Comics superhero, Batman, it is the sequel to ''Batman Begins'' (2005) and the second inst ...

'' presents game theory concepts—notably the Crazy Rich Asians
''Crazy Rich Asians'' is a satirical 2013 romantic comedy novel by Kevin Kwan. Kwan stated that his intention in writing the novel was to "introduce a contemporary Asia to a North American audience". He claimed the novel was loosely based on hi ...

'', the female lead Rachel Chu is a professor of economics and game theory at New York University
New York University (NYU) is a private research university in New York City. Chartered in 1831 by the New York State Legislature, NYU was founded by a group of New Yorkers led by then-Secretary of the Treasury Albert Gallatin.
In 1832, t ...

. At the beginning of the film she is seen in her NYU classroom playing a game of poker
Poker is a family of Card_game#Comparing_games, comparing card games in which Card player, players betting (poker), wager over which poker hand, hand is best according to that specific game's rules. It is played worldwide, however in some plac ...

with her teaching assistant and wins the game by bluffing; then in the climax of the film, she plays a game of mahjong with her boy friend's disapproving mother Eleanor, losing the game to Eleanor on purpose but winning her approval as a result.
See also

*Applied ethics
Applied ethics refers to the practical aspect of moral considerations. It is ethics with respect to real-world actions and their moral considerations in the areas of private and public life, the professions, health, technology, law, and leadersh ...

* Bandwidth-sharing game
A bandwidth-sharing game is a type of resource allocation game designed to model the real-world allocation of bandwidth to many users in a network. The game is popular in game theory
Game theory is the study of mathematical models of strateg ...

* Chainstore paradox
* Collective intentionality
* Glossary of game theory
* Intra-household bargaining Intra-household bargaining refers to negotiations that occur between members of a household in order to arrive at decisions regarding the household unit, like whether to spend or save, whether to study or work.
Bargaining is traditionally defined i ...

* Kingmaker scenario
In game theory, a kingmaker scenario in a game of three or more players, is an endgame situation where a player who is unable to win has the capacity to determine which player among others will win. This player is referred to as the ''kingmaker'' ...

* Law and economics
Law and economics, or economic analysis of law, is the application of microeconomic theory to the analysis of law, which emerged primarily from scholars of the Chicago school of economics. Economic concepts are used to explain the effects of laws ...

* Outline of artificial intelligence
The following outline is provided as an overview of and topical guide to artificial intelligence:
Artificial intelligence (AI) – intelligence exhibited by machines or software. It is also the name of the scientific field which studies how to ...

* Parrondo's paradox
Parrondo's paradox, a paradox in game theory, has been described as: ''A combination of losing strategies becomes a winning strategy''. It is named after its creator, Juan Parrondo, who discovered the paradox in 1996. A more explanatory descript ...

* Precautionary principle
The precautionary principle (or precautionary approach) is a broad epistemological, philosophical and legal approach to innovations with potential for causing harm when extensive scientific knowledge on the matter is lacking. It emphasizes cauti ...

* Quantum refereed game
* Risk management
* Self-confirming equilibrium
* Tragedy of the commons
* Wilson doctrine (economics)
Lists
* List of cognitive biases
* List of emerging technologies
This is a list of emerging technologies, in-development technical innovations with significant potential in their applications. The criteria for this list is that the technology must:
# Exist in some way; purely hypothetical technologies can ...

* List of games in game theory
Game theory studies strategic interaction between individuals in situations called games. Classes of these games have been given names. This is a list of the most commonly studied games
Explanation of features
Games can have several features, ...

Notes

References

Further reading

Textbooks and general literature

* . *Description

* . Suitable for undergraduate and business students. * . Suitable for upper-level undergraduates. * . Suitable for advanced undergraduates. ** Published in Europe as . * * . Presents game theory in formal way suitable for graduate level. * Joseph E. Harrington (2008) ''Games, strategies, and decision making'', Worth, . Textbook suitable for undergraduates in applied fields; numerous examples, fewer formalisms in concept presentation. * * *Maschler, Michael; Solan, Eilon; Zamir, Shmuel (2013), ''Game Theory'', Cambridge University Press, . Undergraduate textbook. * . Suitable for a general audience. * . Undergraduate textbook. * . A modern introduction at the graduate level. * * . A leading textbook at the advanced undergraduate level. * * Consistent treatment of game types usually claimed by different applied fields, e.g. Markov decision processes.

Historically important texts

* * * * * :*reprinted edition: * * * * 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. * 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, p42.

Princeton University Press. * *

Other material

* * * *Allan Gibbard
Allan may refer to:
People
* Allan (name), a given name and surname, including list of people and characters with this name
* Allan (footballer, born 1984) (Allan Barreto da Silva), Brazilian football striker
* Allan (footballer, born 1989) (A ...

, "Manipulation of voting schemes: a general result", ''Econometrica'', Vol. 41, No. 4 (1973), pp. 587–601.
*
*
* , (2002 edition)
* . A layman's introduction.
* .
*
*
*
*
*
*
*
*
*
*
*
External links

* James Miller (2015)Introductory Game Theory Videos

* * Paul Walker

* David Levine

Game Theory. Papers, Lecture Notes and much more stuff.

* Alvin Roth: — Comprehensive list of links to game theory information on the Web * Adam Kalai

Game Theory and Computer Science

— Lecture notes on Game Theory and Computer Science * Mike Shor

GameTheory.net

— Lecture notes, interactive illustrations and other information. * Jim Ratliff'

Graduate Course in Game Theory

(lecture notes). * Don Ross

Review Of Game Theory

in the ''Stanford Encyclopedia of Philosophy''. * Bruno Verbeek and Christopher Morris

Game Theory and Ethics

* Elmer G. Wiens

Game Theory

— Introduction, worked examples, play online two-person zero-sum games. * Marek M. Kaminski

— Syllabuses and lecture notes for game theory and political science.

* Kesten Green's — Se

Papers

fo

evidence on the accuracy of forecasts from game theory and other methods

. * McKelvey, Richard D., McLennan, Andrew M., and Turocy, Theodore L. (2007)

Gambit: Software Tools for Game Theory

'. * Benjamin Polak

Open Course on Game Theory at Yale

videos of the course

* Benjamin Moritz, Bernhard Könsgen, Danny Bures, Ronni Wiersch, (2007)

Spieltheorie-Software.de: An application for Game Theory implemented in JAVA

'. * Antonin Kucera

Stochastic Two-Player Games

* Yu-Chi Ho

What is Mathematical Game Theory

What is Mathematical Game Theory (#2)

What is Mathematical Game Theory (#3)

What is Mathematical Game Theory (#4)-Many person game theory

What is Mathematical Game Theory ?( #5) – Finale, summing up, and my own view

{{DEFAULTSORT:Game Theory Artificial intelligence Formal sciences Mathematical economics John von Neumann