View source for Game theory

{{Short description|Mathematical models of strategic interactions}}
{{hatnote group|{{about|the mathematical study of strategic behavior|the study of playing games for entertainment|Game studies}}{{redirect|Game theorist|the YouTube series|Game Theorists}}{{other uses}}
}}
{{Use American English|date=July 2018}}
{{Economics sidebar}}
{{Strategy}}

'''Game theory''' is the study of [[mathematical model]]s of strategic interactions.<ref name=Myerson>{{cite book|last=Myerson|first=Roger B.|author-link=Roger B. Myerson|year=1991|title=Game Theory: Analysis of Conflict|publisher=[[Harvard University Press]]|isbn=9780674341166}}</ref> It has applications in many fields of [[social science]], and is used extensively in [[economics]], [[logic]], [[systems science]] and [[computer science]].<ref>{{Cite book|last1=Shapley |first1=Lloyd S. |last2=Shubik |first2=Martin |date=1971-01-01 |title=Game Theory in Economics |chapter = Chapter 1, Introduction, The Use of Models |url=https://www.rand.org/pubs/reports/R0904z1.html |access-date=23 April 2023 |archive-date=23 April 2023 |archive-url=https://web.archive.org/web/20230423130243/https://www.rand.org/pubs/reports/R0904z1.html |url-status=live }}</ref> Initially, game theory addressed two-person [[zero-sum game]]s, in which a participant's gains or losses are exactly balanced by the losses and gains of the other participant. In the 1950s, it was extended to the study of non zero-sum games, and was eventually applied to a wide range of [[Human behavior|behavioral relations]]. It is now an [[umbrella term]] for the [[science]] of rational [[Decision-making|decision making]] in humans, animals, and computers.

Modern game theory began with the idea of [[Mixed Strategy Equilibria|mixed-strategy equilibria]] in two-person zero-sum games and its proof by [[John von Neumann]]. Von Neumann's original proof used the [[Brouwer fixed-point theorem]] on continuous mappings into compact [[convex set]]s, which became a standard method in game theory and [[mathematical economics]]. His paper was followed by ''[[Theory of Games and Economic Behavior]]'' (1944), co-written with [[Oskar Morgenstern]], which considered [[Cooperative game theory|cooperative game]]s of several players.<ref>{{Cite book |url= https://press.princeton.edu/books/paperback/9780691130613/theory-of-games-and-economic-behavior |title=Theory of Games and Economic Behavior |date=2007-04-08 |isbn=978-0-691-13061-3 |last1=Neumann |first1=John von |last2=Morgenstern |first2=Oskar |publisher=Princeton University Press |access-date=23 April 2023 |archive-date=28 March 2023 |archive-url= https://web.archive.org/web/20230328004503/https://press.princeton.edu/books/paperback/9780691130613/theory-of-games-and-economic-behavior |url-status=live }}</ref> The second edition provided an [[Von Neumann–Morgenstern utility theorem|axiomatic theory]] of [[Expected utility hypothesis|expected utility]], which allowed mathematical statisticians and economists to treat decision-making under uncertainty.

Game theory was developed extensively in the 1950s, and was explicitly applied to [[evolution]] 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. [[John Maynard Smith]] was awarded the [[Crafoord Prize]] for his application of [[evolutionary game theory]] in 1999, and fifteen game theorists have won the [[Nobel Memorial Prize in Economic Sciences|Nobel Prize in economics]] as of 2020, including most recently [[Paul Milgrom]] and [[Robert B. Wilson]].

{{Toclimit|3}}

==History==
Discussions on the mathematics of games began long before the rise of modern, mathematical game theory. [[Gerolamo Cardano|Cardano]] wrote on games of chance in ''Liber de ludo aleae'' (''Book on Games of Chance''), written around 1564 but published posthumously in 1663.<ref>{{Cite journal |last=Stigler |first=Stephen M. |date=2007 |title=Chance Is 350 Years Old |url=https://www.tandfonline.com/doi/abs/10.1080/09332480.2007.10722870 |journal=CHANCE |volume=20 |issue=4 |pages=26–30 |doi=10.1080/09332480.2007.10722870 |issn=}}</ref> Influenced by the work of [[Pierre de Fermat|Fermat]] and [[Blaise Pascal|Pascal]] on the [[problem of points]], [[Christiaan Huygens|Huygens]] developed the concept of [[Expected value|expectation]] on reasoning about the structure of games of chance, publishing his gambling calculus in ''De ratiociniis in ludo aleæ'' (''On Reasoning in Games of Chance'') in 1657.<ref>{{Citation |last=Schneider |first=Ivo |title=Christiaan Huygens |date=2001 |work=Statisticians of the Centuries |pages=23–28 |editor-last=Heyde |editor-first=C. C. |url=https://doi.org/10.1007/978-1-4613-0179-0_5 |access-date=2025-10-17 |place=New York, NY |publisher=Springer |language=en |doi=10.1007/978-1-4613-0179-0_5#citeas |isbn=978-1-4613-0179-0 |editor2-last=Seneta |editor2-first=E. |editor3-last=Crépel |editor3-first=P. |editor4-last=Fienberg |editor4-first=S. E.}}</ref>

In 1713, a letter attributed to Charles Waldegrave, an active [[Jacobitism|Jacobite]] and uncle to British diplomat [[James Waldegrave, 1st Earl Waldegrave|James Waldegrave]], analyzed a game called "[[le her]]". Waldegrave provided a [[minimax]] [[mixed strategy]] solution to a two-person version of the card game, and the problem is now known as the [[Waldegrave problem]].<ref name="GT-PD-01">{{citation |url=http://www.jehps.net/Decembre2007/Bellhouse.pdf |archive-url=https://web.archive.org/web/20080820134028/http://www.jehps.net/Decembre2007/Bellhouse.pdf |archive-date=2008-08-20 |url-status=live |title=The Problem of Waldegrave |author=Bellhouse, David R. |journal=Journal Électronique d'Histoire des Probabilités et de la Statistique |trans-work=Electronic Journal of Probability History and Statistics |year=2007 |volume=3 |issue=2}}</ref><ref>{{cite journal |last1=Bellhouse |first1=David R. |title=Le Her and Other Problems in Probability Discussed by Bernoulli, Montmort and Waldegrave |journal=Statistical Science |volume=30 |pages=26–39 |date=2015 |publisher=[[Institute of Mathematical Statistics]] |issue=1 |arxiv=1504.01950 |doi=10.1214/14-STS469 |bibcode=2015arXiv150401950B |s2cid=59066805 }}</ref>

In 1838, [[Antoine Augustin Cournot]] provided a [[Cournot competition|model of competition]] in [[Oligopoly|oligopolies]]. Though he did not refer to it as such, he presented a solution that is the [[Nash equilibrium]] of the game in his {{Lang|fr|Recherches sur les principes mathématiques de la théorie des richesses}} (''Researches into the Mathematical Principles of the Theory of Wealth''). In 1883, [[Joseph Bertrand]] critiqued Cournot's model as unrealistic, providing an alternative model of price competition<ref name=":42">{{Cite journal |last1=Qin |first1=Cheng-Zhong |last2=Stuart |first2=Charles |date=1997 |title=Bertrand versus Cournot Revisited |url=https://www.jstor.org/stable/25055054 |journal=Economic Theory |volume=10 |issue=3 |pages=497–507 |doi=10.1007/s001990050169 |issn=0938-2259 |jstor=25055054 |s2cid=153431949}}</ref> which would later be formalized by [[Francis Ysidro Edgeworth]].<ref>Edgeworth, Francis (1889) "The pure theory of monopoly", reprinted in Collected Papers relating to Political Economy 1925, vol.1, Macmillan.</ref>

In 1913, [[Ernst Zermelo]] published {{Lang|de|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 game|strictly determined]].<ref>{{cite conference |last=Zermelo |first=Ernst |author-link=Ernst Zermelo |date=1913 |editor1-last=Hobson |editor1-first=E. W. |editor2-last=Love |editor2-first=A. E. H. |title=Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels |trans-title=On an Application of Set Theory to the Theory of the Game of Chess |url=https://socio.ethz.ch/content/dam/ethz/special-interest/gess/chair-of-sociology-dam/documents/articles/Zermelo_Uber_eine_Anwendung_der_Mengenlehre_auf_die_Theorie_des_Schachspiels.pdf |conference=Proceedings of the Fifth International Congress of Mathematicians (1912) |language=de |location=Cambridge |publisher=Cambridge University Press |pages=501–504 |archive-url=https://web.archive.org/web/20200731231930/https://socio.ethz.ch/content/dam/ethz/special-interest/gess/chair-of-sociology-dam/documents/articles/Zermelo_Uber_eine_Anwendung_der_Mengenlehre_auf_die_Theorie_des_Schachspiels.pdf |archive-date=31 July 2020 |access-date=29 August 2019 |url-status=dead}}</ref>

===Foundation===
[[File:JohnvonNeumann-LosAlamos.gif|thumb|upright|[[John von Neumann]]]]
The work of [[John von Neumann]] established game theory as its own independent field in the early-to-mid 20th century, with von Neumann publishing his paper ''On the Theory of Games of Strategy'' in 1928.<ref>{{cite journal |first=John |last=von Neumann|s2cid=122961988 |author-link=John von Neumann|year=1928 |title=Zur Theorie der Gesellschaftsspiele |journal=[[Mathematische Annalen]] |trans-work=Mathematical Annals |volume=100 |issue=1 |pages=295–320 |doi=10.1007/BF01448847 |trans-title=On the Theory of Games of Strategy |language=de}}</ref><ref>{{cite book |first=John|last=von Neumann |author-link=John von Neumann |chapter=On the Theory of Games of Strategy |editor1-first=A. W. |editor1-last=Tucker |editor2-first=R. D. |editor2-last=Luce |year=1959 |translator-last=Bargmann|translator-first=Sonya|title=Contributions to the Theory of Games |volume=4 |pages=13–42 |isbn=0-691-07937-4 |chapter-url=https://books.google.com/books?id=9lSVFzsTGWsC&pg=PA13|location=[[Princeton, New Jersey]]|publisher=[[Princeton University Press]]}}</ref> Von Neumann's original proof used [[Brouwer's fixed-point theorem]] on continuous [[Map (mathematics)|mappings]] into compact [[convex set]]s, which became a standard method in game theory and [[mathematical economics]]. Von Neumann's work in game theory culminated in his 1944 book ''[[Theory of Games and Economic Behavior]]'', co-authored with [[Oskar Morgenstern]].<ref>{{cite book |first=Philip |last=Mirowski |author-link=Philip Mirowski |chapter=What Were von Neumann and Morgenstern Trying to Accomplish? |editor-first=E. Roy |editor-last=Weintraub |title=Toward a History of Game Theory |location=Durham |publisher=Duke University Press |year=1992 |isbn=978-0-8223-1253-6 |pages=113–147 |chapter-url=https://books.google.com/books?id=9CHY2Gozh1MC&pg=PA113}}</ref> The second edition of this book provided an [[axiomatic set theory|axiomatic theory of utility]], which reincarnated [[Daniel Bernoulli|Daniel Bernoulli's]] old theory of utility (of money) as an independent discipline. 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|cooperative game]] theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.<ref>{{citation |last=Leonard |first=Robert |title=Von Neumann, Morgenstern, and the Creation of Game Theory |location=New York |publisher=Cambridge University Press |year=2010 |isbn=978-0-521-56266-9 |doi=10.1017/CBO9780511778278}}</ref>

In his 1938 book {{Lang|fr|Applications aux Jeux de Hasard}} and earlier notes, [[Émile Borel]] proved a [[minimax theorem]] for two-person zero-sum matrix games only when the pay-off matrix 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 Game|finite two-person zero-sum games]], a conjecture that was proved false by von Neumann.<ref>{{cite book |url=https://books.google.com/books?id=phOXBQAAQBAJ&pg=PA3 |title=Game theory applications in network design |publisher=IGI Global |year=2014 |isbn=978-1-4666-6051-9 |editor-last=Kim |editor-first=Sungwook |page=3}}</ref>
[[File:John f nash 20061102 3.jpg|thumb|upright|[[John Forbes Nash Jr.|John Nash]]]]
In 1950, [[John Forbes Nash|John Nash]] developed a criterion for mutual consistency of players' strategies known as the [[Nash equilibrium]], 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 (economics)|core]], the [[extensive form game]], [[fictitious play]], [[repeated game]]s, and the [[Shapley value]] were developed. The 1950s also saw the first applications of game theory to [[philosophy]] and [[political science]]. The first mathematical discussion of the [[prisoner's dilemma]] appeared, and an experiment was undertaken by mathematicians [[Merrill M. Flood]] and [[Melvin Dresher]], as part of the [[RAND Corporation]]'s investigations into game theory. RAND pursued the studies because of possible applications to global [[nuclear strategy]].<ref name="stanfordprisoner">{{cite web |last=Kuhn |first=Steven |author-link=Steven Kuhn |date=4 September 1997 |editor-last=Zalta |editor-first=Edward N. |title=Prisoner's Dilemma |url=http://plato.stanford.edu/entries/prisoner-dilemma/ |url-status=live |archive-url=https://web.archive.org/web/20120118191720/http://plato.stanford.edu/entries/prisoner-dilemma/ |archive-date=18 January 2012 |access-date=3 January 2013 |website=Stanford Encyclopedia of Philosophy |publisher=Stanford University}}</ref>

==== Prize-winning achievements ====
In 1965, [[Reinhard Selten]] introduced his [[solution concept]] of [[Subgame perfect equilibrium|subgame perfect equilibria]], which further refined the Nash equilibrium. Later he would introduce [[trembling hand perfection]] as well. In 1994 Nash, Selten and [[John Harsanyi|Harsanyi]] became [[Nobel Memorial Prize in Economic Sciences|Economics Nobel Laureates]] for their contributions to economic game theory.

In the 1970s, game theory was extensively applied in [[biology]], largely as a result of the work of [[John Maynard Smith]] and his [[evolutionarily stable strategy]]. In addition, the concepts of [[correlated equilibrium]], [[Trembling hand perfect equilibrium|trembling hand perfection]] and [[Common knowledge (logic)|common knowledge]]{{efn|Although common knowledge was first discussed by the philosopher [[David Kellogg Lewis|David Lewis]] in his dissertation (and later book) ''Convention'' in the late 1960s, it was not widely considered by economists until [[Robert Aumann]]'s work in the 1970s.}} were introduced and analyzed.

In 1994, John Nash was awarded the Nobel Memorial Prize in the Economic Sciences for his contribution to game theory. Nash's most famous contribution to game theory is the concept of the Nash equilibrium, which is a solution concept for [[Non-cooperative game theory|non-cooperative games]], published in 1951. A Nash equilibrium is a set of strategies, one for each player, such that no player can improve their payoff by unilaterally changing their strategy.

In 2005, game theorists [[Thomas Schelling]] and [[Robert Aumann]] followed Nash, Selten, and Harsanyi as Nobel Laureates. Schelling worked on dynamic models, early examples of [[evolutionary game theory]]. Aumann contributed more to the equilibrium school, introducing 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]], [[Eric Maskin]], and [[Roger Myerson]] 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''.<ref name=Myerson /> Hurwicz introduced and formalized the concept of [[incentive compatibility]].

In 2012, [[Alvin E. Roth]] and [[Lloyd Shapley|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]].

==Different types of games==
{{see also|List of games in game theory}}

===Cooperative / non-cooperative===
{{main|Cooperative game theory|Non-cooperative game}}
A game is ''cooperative'' if the players are able to form binding commitments externally enforced (e.g. through [[contract law]]). A game is ''non-cooperative'' if players cannot form alliances or if all agreements need to be [[self-enforcing agreement|self-enforcing]] (e.g. through [[credible threat]]s).<ref>{{cite web |url=http://www.gametheory.net/dictionary/Non-CooperativeGame.html |title=Non-Cooperative Game |last=Shor |first=Mike |website=GameTheory.net |access-date=2016-09-15 |archive-date=1 April 2014 |archive-url=https://web.archive.org/web/20140401211601/http://www.gametheory.net/Dictionary/Non-CooperativeGame.html |url-status=live }}</ref>

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 different from ''non-cooperative game theory'' which focuses on predicting individual players' actions and payoffs by analyzing [[Nash equilibria]].<ref>{{cite web |url=http://www.utdallas.edu/~chandra/documents/6311/coopgames.pdf |archive-url=https://web.archive.org/web/20160418101712/http://www.utdallas.edu/~chandra/documents/6311/coopgames.pdf |archive-date=2016-04-18 |url-status=live |title=Cooperative Game Theory |last=Chandrasekaran |first=Ramaswamy |publisher=University of Texas at Dallas}}</ref><ref>{{cite web|last=Brandenburger|first=Adam|title=Cooperative Game Theory: Characteristic Functions, Allocations, Marginal Contribution|url=http://www.uib.cat/depart/deeweb/pdi/hdeelbm0/arxius_decisions_and_games/cooperative_game_theory-brandenburger.pdf|url-status=dead|archive-url=https://web.archive.org/web/20170829084624/http://www.uib.cat/depart/deeweb/pdi/hdeelbm0/arxius_decisions_and_games/cooperative_game_theory-brandenburger.pdf|archive-date=29 August 2017|access-date=14 April 2020}}</ref>

Cooperative game theory provides a high-level approach as it describes only the structure and payoffs of coalitions, whereas non-cooperative game theory also looks at how strategic interaction will affect the distribution of payoffs. 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.

===Symmetric / asymmetric===
{{Payoff matrix |Name=An asymmetric game |2L=E |2R=F |1U=E |UL=1, 2 |UR=0, 0 |1D=F |DL=0, 0 |DR=1, 2}}
{{main|Symmetric game}}
A symmetric game is a game where each player earns the same payoff when making the same choice. In other words, the identity of the player does not change the resulting game facing the other player.<ref>{{Cite web |last=Shor |first=Mike |date=2006 |title=Symmetric Game |url=https://www.gametheory.net/dictionary/Games/SymmetricGame.html |website=Game Theory.net}}</ref> Many of the commonly studied 2×2 games are symmetric. The standard representations of [[game of chicken|chicken]], [[Prisoner's dilemma|the prisoner's dilemma]], and the [[stag hunt]] are all symmetric games.

The most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the [[ultimatum game]] and similarly the [[dictator game]] have different strategies for each player. It is possible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the game pictured in this section's graphic is asymmetric despite having identical strategy sets for both players.

===Zero-sum / non-zero-sum===
{{Payoff matrix |Name=A zero-sum game |2L=A |2R=B |1U=A |UL=–1, 1 |UR=3, −3 |1D=B |DL=0, 0 |DR=–2, 2}}
{{main|Zero-sum game}}
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, and always adds to zero (more informally, a player benefits only at the equal expense of others).<ref>{{cite book |title=Game Theory: Third Edition |last=Owen |first=Guillermo |author-link=Guillermo Owen |publisher=Emerald Group Publishing |year=1995 |isbn=978-0-12-531151-9 |location=Bingley |page=11}}</ref> [[Poker]] exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose. Other zero-sum games include [[matching pennies]] and most classical board games including [[Go (board game)|Go]] and [[chess]].

Many games studied by game theorists (including the famed prisoner's dilemma) are non-zero-sum games, because the [[Outcome (game theory)|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.

Furthermore, ''constant-sum games'' correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential [[gains from trade]]. It is possible to transform any 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===
{{main | Simultaneous game | Sequential game}}
[[Simultaneous game]]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 game]]s (a type of dynamic games) are games where players do not make decisions simultaneously, and player's earlier actions affect the outcome and decisions of other players.<ref>{{cite book |doi=10.1016/b978-0-12-398512-5.00002-5 |chapter=Decisions in Engineering Design |title=Design Theory and Methods Using CAD/CAE |date=2015 |last1=Chang |first1=Kuang-Hua |pages=39–101 |isbn=978-0-12-398512-5 }}</ref> 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 game|normal form]] is used to represent simultaneous games, while [[Extensive form game|extensive form]] is used to represent sequential ones. The transformation of extensive to normal form is one way, meaning that multiple extensive form games correspond to the same normal form. Consequently, notions of equilibrium for simultaneous games are insufficient for reasoning about sequential games; see [[subgame perfection]].

In short, the differences between sequential and simultaneous games are as follows:
{| class="wikitable"
|-
! !! Sequential !! Simultaneous 
|-
| Normally denoted by || [[Decision tree]]s || [[Payoff matrix|Payoff matrices]]
|-
| {{longitem|style=line-height:1.3em;padding-right:0.65em|Prior knowledge<br />of opponent's move?}} || {{Yes}} || {{No}}
|-
| Time axis?    || {{Yes}} || {{No}}
|-
| Also known as || {{longitem|style=line-height:1.3em|[[Extensive-form game]]<br />Extensive game}} || {{longitem|style=line-height:1.3em|[[Strategy game]]<br />Strategic game}}
|}

===Perfect information and imperfect information===
{{main|Perfect information}}
[[File:PD with outside option.svg|thumb|upright=1.25|right|A game of imperfect information. The dotted line represents ignorance on the part of player 2, formally called an [[Information set (game theory)|information set]].]]

An important subset of sequential games consists of games of perfect information. A game with perfect information means that all players, at every move in the game, know the previous history of the game and the moves previously made by all other players. 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.<ref name=":2" /> Examples of perfect-information games include [[tic-tac-toe]], [[Draughts|checkers]], [[chess]], and [[Go (board game)|Go]].<ref>{{cite web |url=https://www.math.ucla.edu/~tom/Game_Theory/mat.pdf#page=56 |archive-url=https://web.archive.org/web/20040730140255/http://www.math.ucla.edu/~tom/Game_Theory/mat.pdf |archive-date=2004-07-30 |url-status=live |title=Game Theory |first=Thomas S. |last=Ferguson|author-link= Thomas S. Ferguson |pages=56–57 |publisher=UCLA Department of Mathematics}}</ref><ref>{{cite book |first=Jan |last=Mycielski |author-link=Jan Mycielski |chapter=Games with Perfect Information |title=Handbook of Game Theory with Economic Applications |volume=1 |year=1992 |pages=41–70 |doi=10.1016/S1574-0005(05)80006-2 |isbn=978-0-4448-8098-7}}</ref><ref>{{cite web |url=https://www.youtube.com/watch?v=PN-I6u-AxMg&t=0m25s | archive-url=https://ghostarchive.org/varchive/youtube/20211028/PN-I6u-AxMg| archive-date=2021-10-28|title=Infinite Chess |work=PBS Infinite Series |date=2 March 2017 }}{{cbignore}} Perfect information defined at 0:25, with academic sources {{ArXiv|1302.4377}} and {{ArXiv|1510.08155}}.</ref>

Many card games are games of imperfect information, such as [[poker]] and [[contract bridge|bridge]].<ref>{{Cite book |title=Game Theory: Third Edition |last=Owen |first=Guillermo |publisher=Emerald Group Publishing |year=1995 |isbn=978-0-12-531151-9 |location=Bingley |page=4}}</ref> Perfect information is often confused with [[complete information]], which is a similar concept pertaining to the common knowledge of each player's sequence, strategies, and payoffs throughout gameplay.<ref>{{cite book |doi=10.1007/978-1-349-20181-5_22 |chapter=Perfect Information |title=Game Theory |date=1989 |last1=Mirman |first1=Leonard J. |pages=194–198 |isbn=978-0-333-49537-7 }}</ref> 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.<ref>{{Cite book|last=Mirman|first=Leonard|title=Perfect Information|publisher=Palgrave Macmillan|year=1989|isbn=978-1-349-20181-5|location=London|pages=194–195}}</ref> Games of [[Information asymmetry|incomplete information]] can be reduced, however, to games of imperfect information by introducing "[[Move by nature|moves by nature]]".{{sfnp|Shoham|Leyton-Brown|2008|p=60}}

=== Bayesian game ===
{{main|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.<ref>{{Cite book|last=Osborne|first=Martin J.|title=An Introduction to Game Theory|publisher=Oxford University Press|year=2000|pages=271–272}}</ref>

[[Bayesian game]] 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.<ref>{{Cite book|last=Osborne|first=Martin J|title=An Introduction to Game Theory|publisher=Oxford University Press|year=2020|pages=271–277}}</ref>
[[File:An_example_of_diagram.jpg|thumb|Example of a Bayesian game]]
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 ===
<!-- visible anchor to link to, for example at "Solving chess" -->

Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games. Examples include chess and [[Go (game)|Go]]. Games that involve [[Perfect information|imperfect information]] may also have a strong combinatorial character, for instance [[backgammon]]. There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve some particular problems and answer some general questions.<ref name="Bewersdorff2005" />

Games of perfect information have been studied in [[combinatorial game theory]], which has developed novel representations, e.g. [[surreal numbers]], as well as [[Combinatorics|combinatorial]] and [[Abstract algebra|algebraic]] (and [[Strategy stealing argument|sometimes non-constructive]]) proof methods to [[Solved game|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.<ref>{{citation |last1=Albert |first1=Michael H. |author1-link=Michael H. Albert |last2=Nowakowski |first2=Richard J. |last3=Wolfe |first3=David |isbn=978-1-56881-277-9 |publisher=A K Peters Ltd |title=Lessons in Play: In Introduction to Combinatorial Game Theory |year=2007 |pages=3–4}}</ref><ref>{{cite book |last=Beck |first=József |author-link=József Beck |isbn=978-0-521-46100-9 |publisher=Cambridge University Press |title=Combinatorial Games: Tic-Tac-Toe Theory |title-link=Combinatorial Games: Tic-Tac-Toe Theory |year=2008 |pages=[https://archive.org/details/combinatorialgam00jbec/page/n15 1]–3}}</ref> A typical game that has been solved this way is [[Hex (board game)|Hex]]. A related field of study, drawing from [[computational complexity theory]], is [[game complexity]], which is concerned with estimating the computational difficulty of finding optimal strategies.<ref>{{citation |first1=Robert A. |last1=Hearn|author1-link=Bob Hearn |first2=Erik D. |last2=Demaine |title=Games, Puzzles, and Computation |title-link= Games, Puzzles, and Computation |year=2009 |publisher=A K Peters, Ltd. |isbn=978-1-56881-322-6}}</ref>

Research in [[artificial intelligence]] 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]] or use of [[artificial neural network]]s trained by [[reinforcement learning]], which make games more tractable in computing practice.<ref name="Bewersdorff2005">{{cite book |author=Jörg Bewersdorff |title=Luck, logic, and white lies: the mathematics of games |year=2005 |publisher=A K Peters, Ltd. |isbn=978-1-56881-210-6 |pages=ix–xii |chapter=31 |author-link=Jörg Bewersdorff }}</ref><ref name="Jones2008">{{cite book |first=M. Tim |last=Jones |title=Artificial Intelligence: A Systems Approach |year=2008 |publisher=Jones & Bartlett Learning |isbn=978-0-7637-7337-3 |pages=106–118}}</ref>

===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]]s allow players to choose a strategy from a continuous strategy set. For instance, [[Cournot competition]] is typically modeled with players' strategies being any non-negative quantities, including fractional quantities.

===Differential games===
{{main|Differential game}}
Differential games such as the continuous [[Pursuit-evasion|pursuit and evasion game]] are continuous games where the evolution of the players' state variables is governed by [[differential equation]]s. 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's Minimum Principle|Pontryagin maximum principle]] while the closed-loop strategies are found using [[Hamilton–Jacobi–Bellman equation|Bellman's Dynamic Programming]] method.

A particular case of differential games are the games with a random [[time horizon]].<ref name="PM66">{{cite journal |language=ru |last1=Petrosjan |first1=L. A. |last2=Murzov |first2=N. V. |date=1966 |title=Game-theoretic problems of mechanics |journal=Litovsk. Mat. Sb. |volume=6 |pages=423–433}}</ref> 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===
{{main|Evolutionary game theory}}
Evolutionary game theory studies players who adjust their strategies over time according to rules that are not necessarily rational or farsighted.<ref name=Newton2018renaissance>{{cite journal | last1=Newton | first1=Jonathan | year=2018 | title=Evolutionary Game Theory: A Renaissance | journal=Games | volume=9 | issue=2 | pages=31 | doi=10.3390/g9020031| doi-access=free | hdl=10419/179191 | hdl-access=free }}</ref>  In general, the evolution of strategies over time according to such rules is modeled as a [[Markov chain]] 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]], 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.{{sfnp|Webb|2007}}

===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]], and areas of [[artificial intelligence]], particularly [[AI planning]] (with uncertainty) and [[multi-agent system]]. Although these fields may have different motivators, the mathematics involved are substantially the same, e.g. using [[Markov decision process]]es (MDP).<ref>{{cite book |last1=Lozovanu |first1=D |last2=Pickl |first2=S |title=A Game-Theoretical Approach to Markov Decision Processes, Stochastic Positional Games and Multicriteria Control Models |date=2015 |publisher=Springer, Cham |isbn=978-3-319-11832-1 }}</ref>

Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" ("[[Move by nature|moves by nature]]").{{sfnp|Osborne|Rubinstein|1994}} This player is not typically considered a third player in what is otherwise a two-player game, but merely serves to provide a roll of the dice where required by the game.

For some problems, different approaches to modeling stochastic outcomes may lead to different solutions. For example, the difference in approach between MDPs and the [[Minimax|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.<ref name="McMahan">{{cite thesis |first=Hugh Brendan |last=McMahan |date=2006 |title=Robust Planning in Domains with Stochastic Outcomes, Adversaries, and Partial Observability |type=PhD dissertation |publisher=Carnegie Mellon University |url=https://www.cs.cmu.edu/~mcmahan/research/mcmahan_thesis.pdf |archive-url=https://web.archive.org/web/20110401124804/http://www.cs.cmu.edu/~mcmahan/research/mcmahan_thesis.pdf |archive-date=2011-04-01 |url-status=live |pages=3–4}}</ref> (See [[Black swan theory]] for more discussion on this kind of modeling issue, particularly as it relates to predicting and limiting losses in investment banking.)

General models that include all elements of stochastic outcomes, adversaries, and partial or noisy observability (of moves by other players) have also been studied. The "[[gold standard]]" is considered to be partially observable [[stochastic game]] (POSG), but few realistic problems are computationally feasible in POSG representation.<ref name="McMahan" />

===Metagames===
These are games the play of which is the development of the rules for another game, the target or subject game. [[Metagame]]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]] is also used to refer to a practical approach developed by Nigel Howard,{{sfnp|Howard|1971}} 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]].

===Mean field game theory===
{{main|Mean field game theory}}
Mean field game theory 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]] and [[Robert W. Rosenthal]], in the engineering literature by [[Peter E. Caines]], and by mathematicians [[Pierre-Louis Lions]] 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 [[Utility|''payoffs'']] for each outcome. (Eric Rasmusen refers to these four "essential elements" by the acronym "PAPI".)<ref name="r7a"/><ref name="r7b"/><ref name="r7c"/><ref name="r7d"/> A game theorist typically uses these elements, along with a [[solution concept]] of their choosing, to deduce a set of equilibrium [[Strategy (game theory)|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 [[Economic equilibrium|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===
{{main|Extensive form game}}
[[File:Ultimatum Game Extensive Form.svg|thumb|right|An extensive form game]]
The extensive form can be used to formalize games with a time sequencing of moves. Extensive form games can be visualized using game [[Tree (graph theory)|trees]] (as pictured here). Here each [[Graph (discrete mathematics)|vertex]] (or node) represents a point of choice for a player. The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree. The extensive form can be viewed as a multi-player generalization of a [[decision tree]].<ref>{{cite book |last1=Fudenberg |first1=Drew |last2=Tirole |first2=Jean |title=Game Theory |date=1991 |publisher=MIT Press |isbn=978-0-262-06141-4 |page=67 }}</ref> To solve any extensive form game, [[backward induction]] 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.<ref>{{Cite book |title=Security Studies: an Introduction |last=Williams |first=Paul D. |publisher=Routledge |year=2013 |location=[[Abingdon-on-Thames|Abingdon]] |pages=55–56 |edition=second}}</ref>

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 {{var|F}} or {{var|U}} (fair or unfair). Next in the sequence, ''Player 2'', who has now observed ''Player 1''{{'}}s move, can choose to play either {{var|A}} or {{var|R}}  (accept or reject). Once ''Player 2'' has made their choice, the game is considered finished and each player gets their respective payoff, represented in the image as two numbers, where the first number represents Player 1's payoff, and the second number represents Player 2's payoff.  Suppose that ''Player 1'' chooses {{var|U}} and then ''Player 2'' chooses {{var|A}}: ''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 [[#Perfect information and imperfect information|imperfect information section]].)

===Normal form===
<!----- {| align=right border="1" cellpadding="4" cellspacing="0" style="margin: 1em 1em 1em 1em; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse; font-size: 95%;"
|+ align=bottom |''A normal form game''
|-
|
! scope="col" style="color: #900;width: 90px;" | ''Player 2 chooses left''
! scope="col" style="color: #900;width: 90px;" | ''Player 2 chooses right''
|-
! scope="col" style="color: #009;width: 90px;" | ''Player 1 chooses top''
| align=center | <span style="color: #009">4</span>, <span style="color: #900">3</span>
| align=center | <span style="color: #009">-1</span>, <span style="color: #900">-1</span>
|-
! scope="col" style="color: #009;width: 100px;" | ''Player 1 chooses bottom''
| align=center | <span style="color: #009">0</span>, <span style="color: #900">0</span>
| align=center | <span style="color: #009">3</span>, <span style="color: #900">4</span>
|} ----->
{{Payoff matrix |Float=right |Width=330
| Name = Normal form or payoff matrix of a 2-player, 2-strategy game
| 2L = {{color|#900|Player 2<br />chooses ''Left''}}
| 2R = {{color|#900|Player 2<br />chooses ''Right''}}
| 1U = {{color|#009|Player 1<br />chooses ''Up''}}
| 1D = {{color|#009|Player 1<br />chooses&nbsp;''Down''}}
| UL = {{color|#009|'''4'''}}, {{color|#900|'''3'''}}
| UR = {{color|#009|'''–1'''}}, {{color|#900|'''–1'''}}
| DL = {{color|#009|'''0'''}}, {{color|#900|'''0'''}}
| DR = {{color|#009|'''3'''}}, {{color|#900|'''4'''}}
}}
{{main|Normal-form game}}
The normal (or strategic form) game is usually represented by a [[Matrix (mathematics)|matrix]] which shows the players, strategies, and payoffs (see the example to the right). More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions. In the accompanying example there are two players; one chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in our example). Suppose that Player 1 plays ''Up'' and that Player 2 plays ''Left''. Then Player 1 gets a payoff of 4, and Player 2 gets 3.

When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices of other players, the game is usually presented in extensive form.

Every extensive-form game has an equivalent normal-form game, however, the transformation to normal form may result in an exponential blowup in the size of the representation, making it computationally impractical.{{sfnp|Shoham|Leyton-Brown|2008|p=35}}

===Characteristic function form===
{{main|Cooperative game theory}}
In cooperative game theory the characteristic function lists the payoff of each coalition. The origin of this formulation is in John von Neumann and Oskar Morgenstern's book.<ref>{{Cite web |date=2025-02-12 |title=Game theory - Von Neumann, Morgenstern, Theory {{!}} Britannica |url=https://www.britannica.com/science/game-theory/The-von-Neumann-Morgenstern-theory |access-date=2025-03-19 |website=Britannica |language=en}}</ref>

Formally, a characteristic function is a function <math> v : 2^N \to \mathbb{R} </math><ref><math>2^N</math> denotes the [[power set]] of <math>N</math>.</ref> from the set of all possible coalitions of players to a set of payments, and also satisfies <math> v( \emptyset ) = 0 </math>. The function describes how much collective payoff a set of players can gain by forming a coalition.

===Alternative game representations===
{{see also|Succinct game}}

Alternative game representation forms are used for some subclasses of games or adjusted to the needs of interdisciplinary research.<ref>{{cite arXiv |last1=Tagiew |first1=Rustam |title=If more than Analytical Modeling is Needed to Predict Real Agents' Strategic Interaction |date=3 May 2011 |eprint=1105.0558 |class=cs.GT }}</ref> In addition to classical game representations, some of the alternative representations also encode time related aspects.
{| class="wikitable sortable"
|-
!Name
!Year
!Means
![[#Game_types|Type of games]]
!Time
|-
|[[Congestion game]]<ref>{{cite journal |last1=Rosenthal |first1=Robert W. | author-link1=Robert W. Rosenthal |title=A class of games possessing pure-strategy Nash equilibria |journal=International Journal of Game Theory |date=December 1973 |volume=2 |issue=1 |pages=65–67 |doi=10.1007/BF01737559|s2cid=121904640 }}</ref>
|1973
|functions
|subset of n-person games, simultaneous moves
|No
|-
|Sequential form<ref>{{cite book |last1=Koller |first1=Daphne | author-link1=Daphne Koller |last2=Megiddo |first2=Nimrod | author-link2=Nimrod Megiddo |last3=von Stengel |first3=Bernhard |title=Proceedings of the twenty-sixth annual ACM symposium on Theory of computing – STOC '94 |chapter=Fast algorithms for finding randomized strategies in game trees |pages=750–759 |date=1994 |doi=10.1145/195058.195451 |isbn=0-89791-663-8 |s2cid=1893272 }}</ref>
|1994
|matrices
|2-person games of imperfect information
|No
|-
|Timed games<ref>{{cite journal |last1=Alur |first1=Rajeev |last2=Dill |first2=David L. |title=A theory of timed automata |journal=Theoretical Computer Science |date=April 1994 |volume=126 |issue=2 |pages=183–235 |doi=10.1016/0304-3975(94)90010-8|doi-access=free }}</ref><ref>{{cite journal |last1=Tomlin |first1=C.J. |last2=Lygeros |first2=J. |last3=Shankar Sastry |first3=S. |title=A game theoretic approach to controller design for hybrid systems |journal=Proceedings of the IEEE |date=July 2000 |volume=88 |issue=7 |pages=949–970 |doi=10.1109/5.871303|bibcode=2000IEEEP..88..949T |s2cid=1844682 }}</ref>
|1994
|functions
|2-person games
|Yes
|-
|Gala<ref>{{cite journal |last1=Koller |first1=Daphne |last2=Pfeffer |first2=Avi |title=Representations and solutions for game-theoretic problems |journal=Artificial Intelligence |date=July 1997 |volume=94 |issue=1–2 |pages=167–215 |doi=10.1016/S0004-3702(97)00023-4 }}</ref>
|1997
|[[First-order logic|logic]]
|n-person games of imperfect information
|No
|-
|[[Graphical game theory|Graphical games]]<ref>{{cite journal |last1=Michael |first1=Michael Kearns |last2=Littman |first2=Michael L. |title=Graphical Models for Game Theory |journal=In UAI |date=2001 |pages=253–260 |citeseerx=10.1.1.22.5705 }}</ref><ref>{{cite arXiv |last1=Kearns |first1=Michael |last2=Littman |first2=Michael L. |last3=Singh |first3=Satinder |title=Graphical Models for Game Theory |date=7 March 2011 |eprint=1301.2281 |class=cs.GT}}</ref>
|2001
|graphs, functions
|n-person games, simultaneous moves
|No
|-
|Local effect games<ref>{{cite conference |last1=Leyton-Brown |first1=Kevin |last2=Tennenholtz |first2=Moshe |title=Local-Effect Games |publisher=Schloss Dagstuhl-Leibniz-Zentrum für Informatik |conference=Dagstuhl Seminar Proceedings |date=2005 |url=https://drops.dagstuhl.de/volltexte/2005/219/pdf/05011.LeytonBrownKevin.Paper.219.pdf |access-date=February 3, 2023 |archive-date=3 February 2023 |archive-url=https://web.archive.org/web/20230203174844/https://drops.dagstuhl.de/volltexte/2005/219/pdf/05011.LeytonBrownKevin.Paper.219.pdf |url-status=dead }}</ref>
|2003
|functions
|subset of n-person games, simultaneous moves
|No
|-
|[[Game description language|GDL]]<ref>{{cite journal |last1=Genesereth |first1=Michael |last2=Love |first2=Nathaniel |last3=Pell |first3=Barney |title=General Game Playing: Overview of the AAAI Competition |journal=AI Magazine |date=15 June 2005 |volume=26 |issue=2 |pages=62 |doi=10.1609/aimag.v26i2.1813 }}</ref>
|2005
|[[First-order logic|logic]]
|deterministic n-person games, simultaneous moves
|No
|-
|Game Petri-nets<ref>{{cite journal |last1=Clempner |first1=Julio |title=Modeling shortest path games with Petri nets: a Lyapunov based theory |journal=International Journal of Applied Mathematics and Computer Science |date=2006 |volume=16 |issue=3 |pages=387–397 |url=https://eudml.org/doc/207801 }}</ref>
|2006
|[[Petri net]]
|deterministic n-person games, simultaneous moves
|No
|-
|Continuous games<ref>{{cite journal |last1=Sannikov |first1=Yuliy |title=Games with Imperfectly Observable Actions in Continuous Time |journal=Econometrica |date=September 2007 |volume=75 |issue=5 |pages=1285–1329 |doi=10.1111/j.1468-0262.2007.00795.x |url=http://www.dklevine.com/archive/sannikov_games.pdf }}</ref>
|2007
|functions
|subset of 2-person games of imperfect information
|Yes
|-
|PNSI<ref>{{cite book |last1=Tagiew |first1=Rustam |title=2008 International Conference on Computational Intelligence for Modelling Control & Automation |chapter=Multi-Agent Petri-Games |date=December 2008 |pages=130–135 |doi=10.1109/CIMCA.2008.15 |isbn=978-0-7695-3514-2 |s2cid=16679934 }}</ref><ref>{{cite book |last1=Tagiew |first1=Rustam |title=New Challenges in Computational Collective Intelligence |chapter=On Multi-agent Petri Net Models for Computing Extensive Finite Games |volume=244 |date=2009 |pages=243–254 |doi=10.1007/978-3-642-03958-4_21 |publisher=Springer |series=Studies in Computational Intelligence |isbn=978-3-642-03957-7 }}</ref>
|2008
|[[Petri net]]
|n-person games of imperfect information
|Yes
|-
|Action graph games<ref>{{cite arXiv |last1=Bhat |first1=Navin |last2=Leyton-Brown |first2=Kevin |title=Computing Nash Equilibria of Action-Graph Games |date=11 July 2012 |eprint=1207.4128 |class=cs.GT}}</ref>
|2012
|graphs, functions
|n-person games, simultaneous moves
|No
|}

==General and applied uses==
As a method of [[applied mathematics]], game theory has been used to study a wide variety of human and animal behaviors. It was initially developed in [[economics]] to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers. The first use of game-theoretic analysis was by [[Antoine Augustin Cournot]] in 1838 with his solution of the [[Cournot competition|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.<ref name="Larson">{{cite journal |last1=Larson |first1=Jennifer M. |title=Networks of Conflict and Cooperation |journal=Annual Review of Political Science |date=11 May 2021 |volume=24 |issue=1 |pages=89–107 |doi=10.1146/annurev-polisci-041719-102523 |doi-access=free }}</ref>

Although pre-twentieth-century [[Natural history|naturalists]] such as [[Charles Darwin]] 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]]''.<ref>{{Cite journal |last=Friedman |first=Daniel |date=1998 |title=On economic applications of evolutionary game theory |url=https://leeps.ucsc.edu/media/papers/EconAppEvolGameTheory3-1-98.pdf |archive-url=https://web.archive.org/web/20140211210314/http://leeps.ucsc.edu/media/papers/EconAppEvolGameTheory3-1-98.pdf |archive-date=2014-02-11 |url-status=live |journal=Journal of Evolutionary Economics |volume=8 |pages=14–53}}</ref>

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 [[Decision theory#Normative and descriptive decision theory|prescribe]] such behavior.<ref name="Camerer2003">{{cite book |author-link=Colin F. Camerer |first=Colin F. |last=Camerer |date=2003 |title=Behavioral Game Theory: Experiments in Strategic Interaction |pages=5–7 |url=http://press.princeton.edu/chapters/i7517.html |chapter=1.1 What Is Game Theory Good For? |archive-url=https://web.archive.org/web/20110514201411/http://press.princeton.edu/chapters/i7517.html |archive-date=14 May 2011}}</ref> In [[economics and philosophy]], scholars have applied game theory to help in the understanding of good or proper behavior. Game-theoretic approaches have also been suggested in the [[philosophy of language]] and [[philosophy of science]].<ref>{{cite journal |last1=Bruin |first1=Boudewijn de |title=Game Theory in Philosophy |journal=Topoi |date=September 2005 |volume=24 |issue=2 |pages=197–208 |doi=10.1007/s11245-005-5055-3 }}</ref> Game-theoretic arguments of this type can be found as far back as [[Plato]].<ref>{{cite encyclopedia |url=http://plato.stanford.edu/archives/spr2008/entries/game-theory/ |title=Game Theory |access-date=21 August 2008 |last=Ross |first=Don |encyclopedia=Stanford Encyclopedia of Philosophy |date=10 March 2006 |publisher=Stanford University |editor-first=Edward N. |editor-last=Zalta }}</ref> An alternative version of game theory, called chemical game theory, represents the player's choices as metaphorical chemical reactant molecules called "knowlecules".<ref>{{cite journal |last1=Velegol |first1=Darrell |last2=Suhey |first2=Paul |last3=Connolly |first3=John |last4=Morrissey |first4=Natalie |last5=Cook |first5=Laura |title=Chemical Game Theory |journal=Industrial & Engineering Chemistry Research |date=17 October 2018 |volume=57 |issue=41 |pages=13593–13607 |doi=10.1021/acs.iecr.8b03835 |s2cid=105204747 }}</ref> &nbsp;Chemical game theory then calculates the outcomes as equilibrium solutions to a system of chemical reactions.

===Description and modeling===
[[File:Centipede game.svg|thumb|upright=1.25|right|A four-stage [[centipede game]]]]
The primary use of game theory is to describe and [[Conceptual model#Economic models|model]] how human populations behave.{{Citation needed|date=November 2019}} Some{{Who|date=July 2012}} 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 rationality and/or behavior often deviates from the model of rationality as used in game theory. Game theorists respond by comparing their assumptions to those used in [[physics]]. Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific [[Idealization (science philosophy)|ideal]] akin to the models used by [[physicist]]s. However, empirical work has shown that in some classic games, such as the [[centipede game]], [[guess 2/3 of the average]] game, and the [[dictator game]], people regularly do not play Nash equilibria. There is an ongoing debate regarding the importance of these experiments and whether the analysis of the experiments fully captures all aspects of the relevant situation.{{efn|Experimental work in game theory goes by many names, [[experimental economics]], [[behavioral economics]], and [[behavioural game theory]] are several.<ref>{{cite book |first=Colin F. |last=Camerer |date=2003 |title=Behavioral Game Theory: Experiments in Strategic Interaction |url=http://press.princeton.edu/chapters/i7517.html |chapter=Introduction |archive-url=https://web.archive.org/web/20110514201411/http://press.princeton.edu/chapters/i7517.html |archive-date=14 May 2011 |pages=1–25 }}</ref>}}

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

===Prescriptive or normative analysis===
{{Payoff matrix |Name=The prisoner's dilemma | 2L=Cooperate |2R=Defect |1U=Cooperate |UL=-1, −1 |UR=-10, 0 |1D=Defect |DL=0, −10 |DR=-5, −5}}
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 a Nash equilibrium of a game constitutes one's [[best response]] to the actions of the other players – provided they are in (the same) Nash equilibrium – playing a strategy that is part of a Nash equilibrium seems appropriate. This normative use of game theory has also come under criticism.<ref>{{cite journal |last1=Kadane |first1=Joseph B. |last2=Larkey |first2=Patrick D. |title=The Confusion of Is and Ought in Game Theoretic Contexts |journal=Management Science |date=December 1983 |volume=29 |issue=12 |pages=1365–1379 |doi=10.1287/mnsc.29.12.1365 }}</ref>

{{Anchor|Economics}}
=== Economics ===<!-- This Anchor tag serves to provide a permanent target for incoming section links. Please do not remove it, nor modify it, except to add another appropriate anchor. If you modify the section title, please anchor the old title. It is always best to anchor an old section header that has been changed so that links to it will not be broken. See [[Template:Anchor]] for details. This template is {{subst:Anchor comment}} -->
Game theory is a major method used in mathematical economics and business for [[Economic model|modeling]] competing behaviors of interacting [[Agent (economics)|agents]].{{efn|At [[JEL classification codes#Mathematical and quantitative methods JEL: C Subcategories|JEL:C7]] of the ''[[Journal of Economic Literature]]'' classification codes.}}<ref>{{cite book |author-link= Robert Aumann |first=Robert J. |last=Aumann |date=2008 |chapter=game theory |title=[[The New Palgrave Dictionary of Economics]] |edition=2nd |chapter-url= http://www.dictionaryofeconomics.com/article?id=pde2008_G000007&edition=current&q=game%20theory&topicid=&result_number=4 |access-date=22 August 2011 |archive-url=https://web.archive.org/web/20110515034120/http://www.dictionaryofeconomics.com/article?id=pde2008_G000007&edition=current&q=game%20theory&topicid=&result_number=4 |archive-date=15 May 2011 |url-status=dead }}</ref><ref>{{cite book |author-link= Martin Shubik |first=Martin |last=Shubik |date=1981 |chapter=Game Theory Models and Methods in Political Economy |editor-link1=Kenneth Arrow |editor-first1=Kenneth |editor-last1=Arrow |editor-first2= Michael |editor-last2=Intriligator |title=Handbook of Mathematical Economics, v.&nbsp;1 |volume=1 |version=1 |pages=285–330 |doi=10.1016/S1573-4382(81)01011-4|isbn=978-0-444-86126-9 }}</ref><ref>{{cite journal |last= Shapiro|first=Carl|author-link=Carl Shapiro|date=Spring 1989|title=The Theory of Business Strategy|journal=The RAND Journal of Economics|volume=20|number=1|pages=125–137|jstor=2555656 |publisher= [[Wiley (publisher)|Wiley]] |pmid= 10296625 }}.</ref> Applications include a wide array of economic phenomena and approaches, such as [[auction]]s, [[bargaining]], [[mergers and acquisitions]] pricing,<ref name="GT-A-E-00">{{cite conference|last1=Agarwal|first1=N.|last2=Zeephongsekul|first2=P.|url=http://www.mssanz.org.au/modsim2011/D6/agarwal.pdf |title=Psychological Pricing in Mergers & Acquisitions using Game Theory|conference=19th International Congress on Modelling and Simulation|conference-url=https://mssanz.org.au/modsim2011/ |location=[[Perth]]|access-date=February 3, 2023|date=December 11–12, 2011}}</ref> [[fair division]], [[duopoly|duopolies]], [[oligopoly|oligopolies]], [[social network]] formation, [[agent-based computational economics]],<ref>{{cite book |doi=10.1016/S1574-0021(05)02016-2 |title=Agent-Based Computational Economics: A Constructive Approach to Economic Theory |series=Handbook of Computational Economics |date=2006 |last1=Tesfatsion |first1=Leigh |volume=2 |pages=831–880 |isbn=978-0-444-51253-6 }}</ref><ref>{{cite book |date=2008 |title=The New Palgrave Dictionary of Economics |chapter=computer science and game theory |author=[[Joseph Y. Halpern]] |chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000566&edition=current&topicid=&result_number=1 }}</ref> [[general equilibrium]], mechanism design,<ref>{{cite book |date=2008 |title=The New Palgrave Dictionary of Economics |chapter=mechanism design |author-link=Roger B. Myerson |first=Roger B. |last=Myerson |chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_M000132&edition=current&q=mechanism%20design&topicid=&result_number=3 |access-date=4 August 2011 |archive-url=https://web.archive.org/web/20111123042038/http://www.dictionaryofeconomics.com/article?id=pde2008_M000132&edition=current&q=mechanism%20design&topicid=&result_number=3 |archive-date=23 November 2011 |url-status=dead }}</ref><ref>{{cite book |date=2008 |title=The New Palgrave Dictionary of Economics |chapter=revelation principle |author-link=Roger B. Myerson |first=Roger B. |last=Myerson |chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_R000137&edition=current&q=moral&topicid=&result_number=1 |access-date=4 August 2011 |archive-date=16 May 2013 |archive-url=https://web.archive.org/web/20130516143359/http://www.dictionaryofeconomics.com/article?id=pde2008_R000137&edition=current&q=moral&topicid=&result_number=1 |url-status=live }}</ref><ref>{{cite book |date=2008 |title=The New Palgrave Dictionary of Economics |chapter=computing in mechanism design |first=Tuomas |last=Sandholm |chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000563&edition=&field=keyword&q=algorithmic%20mechanism%20design&topicid=&result_number=1 |access-date=5 December 2011 |archive-url=https://web.archive.org/web/20111123042038/http://www.dictionaryofeconomics.com/article?id=pde2008_C000563&edition=&field=keyword&q=algorithmic%20mechanism%20design&topicid=&result_number=1 |archive-date=23 November 2011 |url-status=dead }}</ref><ref name= "nisanronen2001">{{cite journal |last1=Nisan |first1=Noam |last2=Ronen |first2=Amir |title=Algorithmic Mechanism Design |journal=Games and Economic Behavior |date=April 2001 |volume=35 |issue=1–2 |pages=166–196 |doi=10.1006/game.1999.0790 }}</ref><ref name="nisan2007">{{cite book|editor-link1=Noam Nisan |editor-first1=Noam |editor-last1=Nisan|editor-last2=Roughgarden|editor-first2=Tim|editor-last3=Tardos|editor-first3= Eva |editor-last4= Vazirani|editor-first4=Vijay V.|year=2007|title=Algorithmic Game Theory |publisher=[[Cambridge University Press]]|isbn=9780521872829|lccn=2007014231}}</ref> and [[voting system]]s;<ref>{{cite book|doi=10.1016/S1574-0005(05)80062-1|title=Chapter 30 Voting procedures|volume=2|pages=1055–1089|series=Handbook of Game Theory with Economic Applications |year= 1994 |last1= Brams|first1=Steven J.|isbn=978-0-444-89427-4}} and {{cite book|doi=10.1016/S1574-0005(05)80063-3|title=Chapter 31 Social choice|volume=2|pages=1091–1125|series=Handbook of Game Theory with Economic Applications|year=1994|last1=Moulin|first1=Hervé|isbn=978-0-444-89427-4}}</ref> and across such broad areas as experimental economics,<ref>{{cite journal |last1=Smith |first1=Vernon L. |title=Game Theory and Experimental Economics: Beginnings and Early Influences |journal=History of Political Economy |date=December 1992 |volume=24 |issue=Supplement |pages=241–282 |doi=10.1215/00182702-24-Supplement-241 }}</ref><ref>{{cite book|doi= 10.1016/B0-08-043076-7/02232-4 |chapter=Experimental Economics|title=International Encyclopedia of the Social & Behavioral Sciences|pages=5100–5108|year=2001|last=Smith|first=Vernon L.|author-link= Vernon L. Smith |isbn= 978-0-08-043076-8}}</ref><ref>{{cite book |editor1-last=Plott |editor1-first=Charles R. |editor2-last=Smith |editor2-first=Vernon L. |title=Handbook of Experimental Economics Results |date=2008 |publisher=Elsevier |isbn=978-0-08-088796-8 }}{{page needed|date=July 2024}}</ref><ref>Vincent P. Crawford (1997). "Theory and Experiment in the Analysis of Strategic Interaction," in ''Advances in Economics and Econometrics: Theory and Applications'', pp. [http://weber.ucsd.edu/~vcrawfor/CrawfordThExp97.pdf 206–242] {{Webarchive|url=https://web.archive.org/web/20120401234518/http://weber.ucsd.edu/~vcrawfor/CrawfordThExp97.pdf |date=1 April 2012 }}. Cambridge. Reprinted in Colin F. Camerer ''et al''., ed. (2003). ''Advances in Behavioral Economics'', Princeton. 1986–2003 papers. [http://press.princeton.edu/titles/8437.html Description] {{Webarchive|url=https://web.archive.org/web/20120118024451/http://press.princeton.edu/titles/8437.html |date=18 January 2012 }}, [https://books.google.com/books?id=sA4jJOjwCW4C&pg=PR7 preview], Princeton, ch. 12</ref><ref>{{cite book|doi=10.1016/S1574-0005(02)03025-4|chapter=Chapter 62 Game theory and experimental gaming |title=Handbook of Game Theory with Economic Applications Volume 3|volume=3|pages=2327–2351|year=2002|last1=Shubik|first1=Martin|isbn=978-0-444-89428-1}}</ref> [[Behavioral game theory|behavioral economics]],<ref>{{cite book |date=2008 |title=The New Palgrave Dictionary of Economics }}[[Faruk Gul]]. "behavioural economics and game theory." [http://www.dictionaryofeconomics.com/article?id=pde2008_G000210&q=Behavioral%20economics%20&topicid=&result_number=2 Abstract.] {{Webarchive|url=https://web.archive.org/web/20170807112808/http://www.dictionaryofeconomics.com/article?id=pde2008_G000210&q=Behavioral%20economics%20&topicid=&result_number=2 |date=7 August 2017 }}</ref><ref>{{cite book |date=2008 |title=The New Palgrave Dictionary of Economics |chapter=behavioral game theory |author-link=Colin F. Camerer |first=Colin F. |last=Camerer |chapter-url=http://www.dictionaryofeconomics.com/article?id=pde2008_B000302&q=Behavioral%20economics%20&topicid=&result_number=13 |access-date=4 August 2011 |archive-url=https://web.archive.org/web/20111123034346/http://www.dictionaryofeconomics.com/article?id=pde2008_B000302&q=Behavioral%20economics%20&topicid=&result_number=13 |archive-date=23 November 2011 |url-status=dead }}</ref><ref>{{cite journal |author-link=Colin F. Camerer |first=Colin F. |last=Camerer |date=1997 |title=Progress in Behavioral Game Theory |journal=Journal of Economic Perspectives |volume=11 |issue=4 |pages=172 |doi=10.1257/jep.11.4.167 |url=https://resolver.caltech.edu/CaltechAUTHORS:20110211-074641387 }}</ref><ref>{{cite book |author-link=Colin F. Camerer |first=Colin F. |last=Camerer |date=2003 |title=Behavioral Game Theory |publisher= Princeton}} [http://press.princeton.edu/chapters/i7517.html Description] {{Webarchive|url=https://web.archive.org/web/20110514201411/http://press.princeton.edu/chapters/i7517.html |date= 14 May 2011 }}, [https://books.google.com/books?id=cr_Xg7cRvdcC&pg=PR7 preview] {{Webarchive|url=https://web.archive.org/web/20230326164811/https://books.google.com/books?id=cr_Xg7cRvdcC&pg=PR7 |date=26 March 2023 }} ([ctrl]+), and ch. 1 [http://press.princeton.edu/chapters/i7517.pdf link] {{Webarchive|url=https://web.archive.org/web/20130704185133/http://press.princeton.edu/chapters/i7517.pdf |date=4 July 2013 }}.</ref><ref>{{cite book |editor1-last=Camerer |editor1-first=Colin F. |editor2-last=Loewenstein |editor2-first=George |editor3-last=Rabin |editor3-first=Matthew |title=Advances in Behavioral Economics |date=2011 |publisher=Princeton University Press |isbn=978-1-4008-2911-8 }}{{page needed|date=July 2024}}</ref><ref>{{cite journal |first=Drew |last=Fudenberg |author-link=Drew Fudenberg |date=2006 |title=Advancing Beyond Advances in Behavioral Economics |journal=Journal of Economic Literature |volume=44 |issue=3 |pages=694–711 |doi=10.1257/jel.44.3.694 |jstor=30032349 |s2cid=3490729 |url=http://nrs.harvard.edu/urn-3:HUL.InstRepos:3208222 }}</ref> [[information economics]],<ref name=r7a>{{cite book |first=Eric |last=Rasmusen |date=2007 |title=Games and Information |publisher= Wiley |isbn=978-1-4051-3666-2 |edition=4th |url=https://books.google.com/books?id=5XEMuJwnBmUC&pg=PR5}}</ref><ref name=r7b>{{cite book |last1=Kreps |first1=David M. |author-link=David M. Kreps |title=Game Theory and Economic Modelling |date=1990 |publisher=Oxford University Press |isbn=978-0-19-828381-2 |doi=10.1093/0198283814.001.0001 }}{{page needed|date=July 2024}}</ref><ref name=r7c>{{cite book |editor1-last=Aumann |editor1-first=R. J. |editor2-last=Hart |editor2-first=S. |title=Handbook of Game Theory with Economic Applications |date=1992 |publisher=Elsevier |isbn=978-0-444-89427-4 }}{{page needed|date=July 2024}}</ref><ref name=r7d>{{cite book|doi=10.1016/S1574-0005(02)03006-0|chapter=Chapter 43 Incomplete information |title=Handbook of Game Theory with Economic Applications Volume 3|volume=3|pages=1665–1686|year=2002|last1=Aumann |first1=Robert J.|last2=Heifetz|first2=Aviad|isbn= 978-0-444-89428-1}}</ref> [[industrial organization]],<ref>{{cite book |author-link=Jean Tirole |first=Jean |last=Tirole |date=1988 |title=The Theory of Industrial Organization |publisher=MIT Press}} [https://archive.org/details/theoryofindustri00jean Description] and chapter-preview links, pp. [https://archive.org/details/theoryofindustri00jean vii–ix], "General Organization," pp. [https://archive.org/details/theoryofindustri00jean/page/5 5–6], and "Non-Cooperative Game Theory: A User's Guide Manual,' " ch. 11, pp. [https://archive.org/details/theoryofindustri00jean/page/423 423–59].</ref><ref>{{cite book |doi=10.1016/S1574-0005(02)03012-6 |chapter=Game theory and industrial organization |title=Handbook of Game Theory with Economic Applications Volume 3 |date=2002 |last1=Bagwell |first1=Kyle |last2=Wolinsky |first2=Asher |volume=3 |pages=1851–1895 |isbn=978-0-444-89428-1 }}</ref><ref>{{cite journal |last1=Fels |first1=E. M. |title=Review of Strategy and Market Structure: Competition, Oligopoly, and the Theory of Games |journal=Weltwirtschaftliches Archiv |date=1961 |volume=87 |pages=12–14 |jstor=40434883 }}</ref><ref>{{cite journal |last1=Reid |first1=Gavin C. |title=Review of Market Structure and Behavior |journal=The Economic Journal |date=1982 |volume=92 |issue=365 |pages=200–202 |doi=10.2307/2232276 |jstor=2232276 }}</ref> and [[political economy]].<ref>Martin Shubik (1981). "Game Theory Models and Methods in Political Economy," in ''Handbook of Mathematical Economics'', v.&nbsp;1, pp. 285–330 {{doi|10.1016/S1573-4382(81)01011-4}}.</ref><ref>Martin Shubik (1987). ''A Game-Theoretic Approach to Political Economy''. MIT Press. [http://mitpress.mit.edu/catalog/item/default.asp?tid=5086&ttype=2 Description]. {{webarchive |url=https://web.archive.org/web/20110629151809/http://mitpress.mit.edu/catalog/item/default.asp?tid=5086&ttype=2 |date=29 June 2011 }}</ref><ref>Martin Shubik (1978). "Game Theory: Economic Applications," in W. Kruskal and J.M. Tanur, ed., ''International Encyclopedia of Statistics'', v.&nbsp;2, pp.&nbsp;372–78.</ref><ref name="r7c"/>

This research usually focuses on particular sets of strategies known as [[Solution concept|"solution concepts" or "equilibria"]]. A common assumption is that players act rationally. In non-cooperative games, the most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. 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.<ref name="GT-F-R-09">{{cite web|url=http://www.insead.edu/facultyresearch/research/doc.cfm?did=46503 |title=Game-theoretic model to examine the two tradeoffs in the acquisition of information for a careful balancing act |archive-url=https://web.archive.org/web/20130524231021/http://www.insead.edu/facultyresearch/research/doc.cfm?did=46503 |archive-date=24 May 2013 |publisher=[[INSEAD]] |first=Markus |last=Christen |date=1 July 1998 |url-status=dead |access-date=1 July 2012 }}</ref><ref name="GT-F-R-10">{{cite web |url=http://www.europeanfinancialreview.com/?p=4645 |url-status=dead |title=Options Games: Balancing the trade-off between flexibility and commitment |archive-url=https://web.archive.org/web/20130620053305/http://www.europeanfinancialreview.com/?p=4645 |archive-date=20 June 2013 |website=The European Financial Review |date=15 February 2012 |access-date=2013-01-03 |first1=Benoît |last1=Chevalier-Roignant |first2=Lenos |last2=Trigeorgis }}</ref>

The payoffs of the game are generally taken to represent the [[utility]] 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 ''[[Decision theory#Normative and descriptive decision theory|prescriptive]]''.<ref name="Camerer2003" />

==== Managerial economics ====
Game theory also has an extensive use in a specific branch or stream of economics – [[Managerial economics|Managerial Economics]]. One important usage of it in the field of managerial economics is in analyzing strategic interactions between firms.<ref>{{cite book |doi=10.1017/CBO9780511810534.015 |chapter=Game theory |title=Managerial Economics |date=2005 |pages=331–381 |isbn=978-0-521-81993-0 |first1=Nick |last1=Wilkinson }}</ref> For example, firms may be competing in a market with limited resources, and game theory can help managers understand how their decisions impact their competitors and the overall market outcomes. Game theory can also be used to analyze cooperation between firms, such as in forming strategic alliances or joint ventures. Another use of game theory in managerial economics is in analyzing pricing strategies. For example, firms may use game theory to determine the optimal [[pricing strategy]] based on how they expect their competitors to respond to their pricing decisions. Overall, game theory serves as a useful tool for analyzing strategic interactions and decision making in the context of managerial economics.

=== Business ===
The [[Chartered Institute of Procurement & Supply]] (CIPS) promotes knowledge and use of game theory within the context of business [[procurement]].<ref>{{Cite web |date=2020-11-27 |title=CIPS and TWS Partners promote game theory on the global stage |url=https://www.cips.org/who-we-are/news/cips-and-tws-partners-promote-game-theory-on-the-global-stage/ |access-date=2023-04-20 |archive-url=https://web.archive.org/web/20201127230832/https://www.cips.org/who-we-are/news/cips-and-tws-partners-promote-game-theory-on-the-global-stage/ |archive-date=27 November 2020 }}</ref> CIPS and TWS Partners have conducted a series of surveys designed to explore the understanding, awareness and application of game theory among [[procurement]] 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.<ref>CIPS (2021), [https://www.cips.org/knowledge/procurement-topics-and-skills/game-theory/game-theory/#tabs-3 Game Theory] {{Webarchive|url=https://web.archive.org/web/20210411050812/https://www.cips.org/knowledge/procurement-topics-and-skills/game-theory/game-theory/#tabs-3 |date=11 April 2021 }}, CIPS in conjunction with TWS Partners, accessed 11 April 2021</ref>

===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)<ref name="Piraveenan 2019">{{cite journal |last1=Piraveenan |first1=Mahendra |title=Applications of Game Theory in Project Management: A Structured Review and Analysis |journal=Mathematics |date=2019 |volume=7 |issue=9 |page=858 |doi=10.3390/math7090858 |doi-access=free }}</ref> 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<ref name="Piraveenan 2019" /> summarizes 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]]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===
{{Conflict resolution sidebar}}
The application of game theory to [[political science]] is focused in the overlapping areas of [[fair division]], [[political economy]], [[public choice]], [[war's inefficiency puzzle|war bargaining]], [[positive political theory]], and [[social choice theory]]. In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians.<ref>{{Cite web |title=What game theory tells us about politics and society |url=https://news.mit.edu/2018/game-theory-politics-alexander-wolitzky-1204 |access-date=2023-04-23 |website=MIT News {{!}} Massachusetts Institute of Technology |date=4 December 2018 |archive-date=23 April 2023 |archive-url=https://web.archive.org/web/20230423144157/https://news.mit.edu/2018/game-theory-politics-alexander-wolitzky-1204 |url-status=live }}</ref>

Early examples of game theory applied to political science are provided by [[Anthony Downs]]. In his 1957 book ''[[An Economic Theory of Democracy]]'',{{sfnp|Downs|1957}} he applies the [[Hotelling's law|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]] during the presidency of John F. Kennedy.<ref>{{cite web |first=Steven J. |last=Brams |author-link=Steven Brams |url=https://plus.maths.org/content/game-theory-and-cuban-missile-crisis |title=Game theory and the Cuban missile crisis |website=Plus Magazine |date=1 January 2001 |access-date=31 January 2016 |archive-date=24 April 2015 |archive-url=https://web.archive.org/web/20150424010528/https://plus.maths.org/content/game-theory-and-cuban-missile-crisis |url-status=live }}</ref>

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.<ref>{{Cite web |title=How game theory explains 'irrational' behavior |url=https://mitsloan.mit.edu/ideas-made-to-matter/how-game-theory-explains-irrational-behavior |access-date=2023-04-23 |website=MIT Sloan |date=5 April 2022 |archive-date=23 April 2023 |archive-url=https://web.archive.org/web/20230423144158/https://mitsloan.mit.edu/ideas-made-to-matter/how-game-theory-explains-irrational-behavior |url-status=live }}</ref>  Thus, in a process that can be modeled by variants of the prisoner's dilemma, during periods of stability no citizen will find it rational to move to replace the sovereign, even if all the citizens know they would be better off if they were all to act collectively.{{citation needed|date=July 2024}}

A game-theoretic explanation for [[democratic peace theory|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.<ref>{{cite journal |last1=Levy |first1=Gilat |last2=Razin |first2=Ronny |title=It Takes Two: An Explanation for the Democratic Peace |journal=Journal of the European Economic Association |date=March 2004 |volume=2 |issue=1 |pages=1–29 |doi=10.1162/154247604323015463 |url=http://eprints.lse.ac.uk/539/ }}</ref>

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.<ref>{{Cite journal |last=Fearon |first=James D. |date=1 January 1995 |title=Rationalist Explanations for War |journal=International Organization |volume=49 |issue=3 |pages=379–414 |doi=10.1017/s0020818300033324 |jstor=2706903 |s2cid=38573183 }}</ref>

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 [[greenhouse gas emissions]]. However, he concluded that this idea could not work because it would create a prisoner's dilemma for the nations.<ref>{{cite journal |last1=Wood |first1=Peter John |title=Climate change and game theory |journal=Annals of the New York Academy of Sciences |date=February 2011 |volume=1219 |issue=1 |pages=153–170 |doi=10.1111/j.1749-6632.2010.05891.x |pmid=21332497 |bibcode=2011NYASA1219..153W |url=http://ageconsearch.umn.edu/record/95061 }}</ref>

=== Defence science and technology ===

Game theory has been used extensively to model decision-making scenarios relevant to defence applications.<ref name=":3">{{cite journal |last1=Ho |first1=Edwin |last2=Rajagopalan |first2=Arvind |last3=Skvortsov |first3=Alex |last4=Arulampalam |first4=Sanjeev |last5=Piraveenan |first5=Mahendra |title=Game Theory in Defence Applications: A Review |journal=Sensors |date=28 January 2022 |volume=22 |issue=3 |pages=1032 |doi=10.3390/s22031032 |doi-access=free |pmid=35161778 |pmc=8838118 |arxiv=2111.01876 |bibcode=2022Senso..22.1032H }}</ref>  Most studies that has applied game theory in defence settings are concerned with Command and Control Warfare, and can be further classified into studies dealing with (i) Resource Allocation Warfare (ii) Information Warfare (iii) Weapons Control Warfare, and (iv) Adversary Monitoring Warfare.<ref name=":3" />  Many of the problems studied are concerned with sensing and tracking, for example a surface ship trying to track a hostile submarine and the submarine trying to evade being tracked, and the interdependent decision making that takes place with regards to bearing, speed, and the sensor technology activated by both vessels.

The tool,<ref name=":4">{{cite conference |last1=Phetmanee |first1=Surasak |last2=Sevegnani |first2=Michele |last3=Andrei |first3=Oana |title=StEVe: A Rational Verification Tool for Stackelberg Security Games |book-title=Integrated Formal Methods: 19th International Conference, IFM 2024 |date=2024 |pages=267–275 |publisher=Springer-Verlag |location=Manchester, United Kingdom |doi=10.1007/978-3-031-76554-4_15 |url=https://doi.org/10.1007/978-3-031-76554-4_15 }}</ref> for example, automates the transformation of public vulnerability data into models, allowing defenders to synthesize optimal defence strategies through Stackelberg equilibrium analysis. This approach enhances cyber resilience by enabling defenders to anticipate and counteract attackers’ best responses, making game theory increasingly relevant in adversarial cybersecurity environments.

Ho et al. provide a broad summary of game theory applications in defence, highlighting its advantages and limitations across both physical and cyber domains.

=== Biology ===
{{Payoff matrix |Name=The [[Chicken (game)#Hawk-Dove|hawk-dove]] game |2L=Hawk |2R=Dove |1U=Hawk |UL=20, 20 |UR=80, 40 |1D=Dove |DL=40, 80 |DR=60, 60}}
{{main|Evolutionary game theory}}

Unlike those in economics, the payoffs for games in [[biology]] are often interpreted as corresponding to [[Fitness (biology)|fitness]]. In addition, the focus has been less on equilibria that correspond to a notion of rationality and more on ones that would be maintained by evolutionary forces. The best-known equilibrium in biology is known as the ''[[evolutionarily stable strategy]]'' (ESS), first introduced in {{harv|Maynard Smith|Price|1973}}. Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium.

In biology, game theory has been used as a model to understand many different phenomena. It was first used to explain the evolution (and stability) of the approximate 1:1 [[sex ratio]]s. {{harv|Fisher|1930}} suggested that the 1:1 sex ratios are a result of evolutionary forces acting on individuals who could be seen as trying to maximize their number of grandchildren.

Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of [[animal communication]].{{sfnp|Harper|Maynard Smith|2003}} The analysis of [[signaling games]] and [[Cheap talk|other communication games]] has provided insight into the evolution of communication among animals. For example, the [[mobbing behavior]] of many species, in which a large number of prey animals attack a larger predator, seems to be an example of spontaneous emergent organization. Ants have also been shown to exhibit feed-forward behavior akin to fashion (see [[Paul Ormerod]]'s ''[[Butterfly Economics]]'').

Biologists have used the [[Chicken (game)|game of chicken]] to analyze fighting behavior and territoriality.<ref>{{Cite journal |last1=Maynard Smith |first1=John |author-link=John Maynard Smith |title=The theory of games and the evolution of animal conflicts |doi=10.1016/0022-5193(74)90110-6 |journal=Journal of Theoretical Biology |volume=47 |issue=1 |pages=209–221 |year=1974 |pmid=4459582 |bibcode=1974JThBi..47..209M |url=http://www.dklevine.com/archive/refs4448.pdf }}</ref>

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.<ref name="stan-egt">{{cite encyclopedia |editor-link=Edward N. Zalta |editor-first=Edward N. |editor-last=Zalta |url=http://plato.stanford.edu/entries/game-evolutionary/ |title=Evolutionary Game Theory |encyclopedia=Stanford Encyclopedia of Philosophy |publisher=Stanford University |access-date=3 January 2013 |date=19 July 2009 |first=J. McKenzie |last=Alexander}}</ref>

One such phenomenon is known as [[Altruism in animals|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]]s that warn group members of a predator's approach, even when it endangers that individual's chance of survival.<ref name="qudzyh">{{cite encyclopedia |editor-first=Edward N. |editor-last=Zalta |url=https://plato.stanford.edu/entries/altruism-biological/ |title=Biological Altruism |encyclopedia=Stanford Encyclopedia of Philosophy |date=3 June 2003 |access-date=3 January 2013 |publisher=Stanford University |editor-link=Edward N. Zalta |first=Samir |last=Okasha}}</ref> All of these actions increase the overall fitness of a group, but occur at a cost to the individual.

Evolutionary game theory explains this altruism with the idea of [[kin selection]]. Altruists discriminate between the individuals they help and favor relatives. [[Kin selection#Hamilton's rule|Hamilton's rule]] explains the evolutionary rationale behind this selection with the equation {{Math|c < b &times; r}}, where the cost {{varserif|c}} to the altruist must be less than the benefit {{varserif|b}} to the recipient multiplied by the coefficient of relatedness {{varserif|r}}. The more closely related two organisms are causes the incidences of altruism to increase because they share many of the same alleles. This means that the altruistic individual, by ensuring that the alleles of its close relative are passed on through survival of its offspring, can forgo the option of having offspring itself because the same number of alleles are passed on. For example, helping a sibling (in diploid animals) has a coefficient of {{frac|1|2}}, 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.<ref name="qudzyh" /> 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 {{frac|1|2}} in the smaller field becomes 0.995. Similarly if it is considered that information other than that of a genetic nature (e.g. epigenetics, religion, science, etc.) persisted through time the playing field becomes larger still, and the discrepancies smaller.

===Computer science and logic===
Game theory has come to play an increasingly important role in [[logic]] and in [[computer science]]. Several logical theories have a basis in [[game semantics]]. In addition, computer scientists have used games to model [[interactive computation]]s. Also, game theory provides a theoretical basis to the field of [[multi-agent system]]s.<ref name="ShohamLeyton-Brown2008">{{cite book |last1=Shoham |first1=Yoav |last2=Leyton-Brown |first2=Kevin |title=Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations |date=2008 |publisher=Cambridge University Press |isbn=978-1-139-47524-2 }}{{page needed|date=July 2024}}</ref>

Separately, game theory has played a role in [[online algorithm]]s; in particular, the [[k-server problem|{{var|k}}-server problem]], which has in the past been referred to as ''games with moving costs'' and ''request-answer games''.{{sfnp|Ben-David|Borodin|Karp|Tardos|1994}} [[Yao's principle]] is a game-theoretic technique for proving [[Upper and lower bounds|lower bounds]] on the [[Analysis of algorithms|computational complexity]] of [[randomized algorithm]]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]]<ref name="nisan2007"/> and within it [[algorithmic mechanism design]]<ref name="nisanronen2001"/> combine computational [[algorithm design]] and analysis of [[complex system]]s with economic theory.<ref>{{cite book |author-link=Joseph Y. Halpern |first=Joseph Y. |last=Halpern |date=2008 |chapter=Computer science and game theory |title=The New Palgrave Dictionary of Economics |edition=2nd |url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000566&edition=current&topicid=&result_number=1}}</ref><ref>{{cite journal |last1=Shoham |first1=Yoav |title=Computer science and game theory |journal=Communications of the ACM |date=August 2008 |volume=51 |issue=8 |pages=74–79 |doi=10.1145/1378704.1378721 }}</ref><ref>{{cite journal|first1=Amy |last1=Littman |author-link2=Michael L. Littman |first2=Michael L. |year=2007 |title=Introduction to the Special Issue on Learning and Computational Game Theory |journal=Machine Learning |volume=67 |issue=1–2 |pages=3–6 |doi=10.1007/s10994-007-0770-1 |last2=Littman|s2cid=22635389 |doi-access=free }}</ref>

Game theory has multiple applications in the field of artificial intelligence and machine learning. It is often used in developing autonomous systems that can make complex decisions in uncertain environment.<ref>{{cite journal |last1=Hanley |first1=John T. |date=14 December 2021 |title=GAMES, game theory and artificial intelligence |journal=Journal of Defense Analytics and Logistics |volume=5 |issue=2 |pages=114–130 |doi=10.1108/JDAL-10-2021-0011 |doi-access=free}}</ref> Some other areas of application of game theory in AI/ML context are as follows - multi-agent system formation, reinforcement learning,<ref>{{cite book |last1=Albrecht |first1=Stefano V. |title=Multi-Agent Reinforcement Learning: Foundations and Modern Approaches |last2=Christianos |first2=Filippos |last3=Schäfer |first3=Lukas |date=2024 |publisher=MIT Press |isbn=978-0-262-04937-5}}{{page needed|date=July 2024}}</ref> mechanism design etc.<ref>{{cite news |last1=Parashar |first1=Nilesh |date=15 August 2022 |title=What is Game Theory in AI? |url=https://medium.com/@niitwork0921/what-is-game-theory-in-ai-6b7c4c383f03 |work=Medium}}</ref> By using game theory to model the behavior of other agents and anticipate their actions, AI/ML systems can make better decisions and operate more effectively.<ref>{{cite journal |last1=Hazra |first1=Tanmoy |last2=Anjaria |first2=Kushal |date=March 2022 |title=Applications of game theory in deep learning: a survey |journal=Multimedia Tools and Applications |volume=81 |issue=6 |pages=8963–8994 |doi=10.1007/s11042-022-12153-2 |pmc=9039031 |pmid=35496996}}</ref>

===Philosophy===
{{Payoff matrix |Name=Stag hunt | 2L=Stag |2R=Hare |1U=Stag |UL=3, 3 |UR=0, 2 |1D=Hare |DL=2, 0 |DR=2, 2}}
Game theory has been put to several uses in [[philosophy]]. Responding to two papers by {{harvard citations|txt=yes|first=W.V.O.|last=Quine|author1-link=Willard Van Orman Quine|year=1960|year2=1967}}, {{Harvtxt|Lewis|1969}} used game theory to develop a philosophical account of [[Convention (norm)|convention]]. In so doing, he provided the first analysis of [[Common knowledge (logic)|common knowledge]] and employed it in analyzing play in [[coordination game]]s. In addition, he first suggested that one can understand [[Meaning (semiotics)|meaning]] in terms of [[signaling games]]. This later suggestion has been pursued by several philosophers since Lewis.<ref>{{Harvtxt|Skyrms|1996}}</ref>{{sfnp|Grim|Kokalis|Alai-Tafti|Kilb|2004}}<!--<ref>{{harvard citations |txt=yes |last1=Grim |last2=Kokalis |last3=Alai-Tafti |last4=Kilb |last5=St Denis |year=2004}}.</ref>--> Following {{Harvtxt|Lewis|1969}} game-theoretic account of conventions, [[Edna Ullmann-Margalit]] (1977) and [[Cristina Bicchieri|Bicchieri]] (2006) have developed theories of [[social norms]] that define them as Nash equilibria that result from transforming a mixed-motive game into a coordination game.<ref>{{citation |first=E. |last=Ullmann-Margalit |title=The Emergence of Norms |publisher=Oxford University Press |year=1977 |isbn=978-0-19-824411-0 |url=https://archive.org/details/emergenceofnorms0024ullm }}{{page needed|date=July 2024}}</ref><ref>{{citation |first=Cristina |last=Bicchieri |author-link=Cristina Bicchieri |title=The Grammar of Society: the Nature and Dynamics of Social Norms |publisher=Cambridge University Press |year=2006 |isbn=978-0-521-57372-6 }}{{page needed|date=July 2024}}</ref>

Game theory has also challenged philosophers to think in terms of interactive [[epistemology]]: what it means for a collective to have common beliefs or knowledge, and what are the consequences of this knowledge for the social outcomes resulting from the interactions of agents. Philosophers who have worked in this area include Bicchieri (1989, 1993),<ref>{{cite journal|title=Self-Refuting Theories of Strategic Interaction: A Paradox of Common Knowledge |journal=Erkenntnis |volume=30 |issue=1–2 |year=1989 |pages=69–85 |doi=10.1007/BF00184816 |last1=Bicchieri |first1=Cristina |s2cid=120848181 |author-link=Cristina Bicchieri}}</ref><ref>{{Citation |last1=Bicchieri |first1=Cristina |author1-link=Cristina Bicchieri |title=Rationality and Coordination |publisher=[[Cambridge University Press]] |isbn=978-0-521-57444-0 |year=1993}}</ref> [[Brian Skyrms|Skyrms]] (1990),<ref>{{citation |first=Brian |last=Skyrms |author-link=Brian Skyrms |title=The Dynamics of Rational Deliberation |publisher=Harvard University Press |year=1990 |isbn=978-0-674-21885-7}}</ref> and [[Robert Stalnaker|Stalnaker]] (1999).<ref>{{cite journal |last1=Stalnaker |first1=Robert |title=Knowledge, Belief and Counterfactual Reasoning in Games |journal=Economics and Philosophy |date=October 1996 |volume=12 |issue=2 |pages=133–163 |doi=10.1017/S0266267100004132 }}</ref>

The synthesis of game theory with [[ethics]] was championed by [[R. B. Braithwaite]].<ref>{{cite book |last1=Braithwaite |first1=Richard Bevan |title=Theory of Games as a Tool for the Moral Philosopher. An Inaugural Lecture Delivered in Cambridge on 2 December 1954 |date=1955 |publisher=University Press |isbn=978-0-521-11351-9 }}{{page needed|date=July 2024}}</ref> The hope was that rigorous mathematical analysis of game theory might help formalize the more imprecise philosophical discussions. However, this expectation was only materialized to a limited extent.<ref>{{cite journal |last1=Kuhn |first1=Steven T. |title=Reflections on Ethics and Game Theory |journal=Synthese |date=July 2004 |volume=141 |issue=1 |pages=1–44 |doi=10.1023/B:SYNT.0000035846.91195.cb }}</ref>

In [[ethics]], some (most notably David Gauthier, Gregory Kavka, and Jean Hampton) {{Who|date=July 2012}} authors have attempted to pursue [[Thomas Hobbes]]' project of deriving morality from self-interest. Since games like the [[prisoner's dilemma]] present an apparent conflict between morality and self-interest, explaining why cooperation is required by self-interest is an important component of this project. This general strategy is a component of the general [[social contract]] view in [[political philosophy]] (for examples, see {{Harvtxt|Gauthier|1986}} and {{Harvtxt|Kavka |1986}}).{{efn|For a more detailed discussion of the use of game theory in ethics, see the Stanford Encyclopedia of Philosophy's entry [http://plato.stanford.edu/entries/game-ethics/ game theory and ethics].}}

Other authors have attempted to use evolutionary game theory in order to explain the emergence of human attitudes about morality and corresponding animal behaviors. These authors look at several games including the prisoner's dilemma, [[stag hunt]], and the [[Nash bargaining game]] as providing an explanation for the emergence of attitudes about morality (see, e.g., {{harvard citations|txt=yes|last=Skyrms|year=1996|year2=2004}} and {{harvard citations|txt=yes|last1=Sober|last2=Wilson|year=1998}}).

===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.<ref>{{cite journal |last1=Chang |first1=Sheryl L. |last2=Piraveenan |first2=Mahendra |last3=Pattison |first3=Philippa |last4=Prokopenko |first4=Mikhail |title=Game theoretic modelling of infectious disease dynamics and intervention methods: a review |journal=Journal of Biological Dynamics |date=2020 |volume=14 |issue=1 |pages=57–89 |doi=10.1080/17513758.2020.1720322 |pmid=31996099 |arxiv=1901.04143 |bibcode=2020JBioD..14...57C }}</ref><ref>{{cite news |last1=Roberts |first1=Siobhan |title='The Pandemic Is a Prisoner's Dilemma Game' |url=https://www.nytimes.com/2020/12/20/health/virus-vaccine-game-theory.html |work=The New York Times |date=20 December 2020 }}</ref>

==Well known examples of games==
===Prisoner's dilemma===
{{main|Prisoner's dilemma}}
{| class="wikitable floatright"
|+ Standard prisoner's dilemma payoff matrix
! {{diagonal split header|A|B}}
! B stays<br />silent
! B<br />betrays
|-
! A stays<br />silent
| {{diagonal split header|−2|−2|transparent}}
| {{diagonal split header|−10|0|transparent}}
|-
! A<br />betrays
| {{diagonal split header|0|−10|transparent}}
| {{diagonal split header|−5|−5|transparent}}
|}
[[William Poundstone]] described the game in his 1993 book Prisoner's Dilemma:{{sfn|Poundstone|1993|pp=8, 117}}

Two members of a criminal gang, A and B, are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communication with their partner. The principal charge would lead to a sentence of ten years in prison; however, the police do not have the evidence for a conviction. They plan to sentence both to two years in prison on a lesser charge but offer each prisoner a Faustian bargain: If one of them confesses to the crime of the principal charge, betraying the other, they will be pardoned and free to leave while the other must serve the entirety of the sentence instead of just two years for the lesser charge.

The [[dominant strategy]] (and therefore the best response to any possible opponent strategy), is to betray the other, which aligns with the [[sure-thing principle]].<ref>{{cite book |doi=10.1057/978-1-349-95121-5_1850-1 |chapter=Prisoner's Dilemma |title=The New Palgrave Dictionary of Economics |date=1987 |last1=Rapoport |first1=Anatol |pages=1–5 |isbn=978-1-349-95121-5 }}</ref> However, both prisoners staying silent would yield a greater reward for both of them than mutual betrayal.

===Battle of the sexes ===
{{main|Battle of the sexes (game theory)}}

The "battle of the sexes" is a term used to describe the perceived conflict between men and women in various areas of life, such as relationships, careers, and social roles. This conflict is often portrayed in popular culture, such as movies and television shows, as a humorous or dramatic competition between the genders. This conflict can be depicted in a game theory framework. This is an example of non-cooperative games.

An example of the "battle of the sexes" can be seen in the portrayal of relationships in popular media, where men and women are often depicted as being fundamentally different and in conflict with each other. For instance, in some romantic comedies, the male and female protagonists are shown as having opposing views on love and relationships, and they have to overcome these differences in order to be together.<ref>{{Cite web |title=Battle of the Sexes {{!}} History, Participants, & Facts {{!}} Britannica |url=https://www.britannica.com/topic/Battle-of-the-Sexes-tennis |access-date=2023-04-23 |website=www.britannica.com |archive-date=23 April 2023 |archive-url=https://web.archive.org/web/20230423131644/https://www.britannica.com/topic/Battle-of-the-Sexes-tennis |url-status=live }}</ref>

In this game, there are two pure strategy Nash equilibria: one where both the players choose the same strategy and the other where the players choose different options. If the game is played in mixed strategies, where each player chooses their strategy randomly, then there is an infinite number of Nash equilibria. However, in the context of the "battle of the sexes" game, the assumption is usually made that the game is played in pure strategies.<ref>{{Cite web |last=Athenarium |date=2020-08-12 |title=Battle of the Sexes – Nash equilibrium in mixed strategies for coordination |url=https://athenarium.com/battle-of-the-sexes-mixed-strategies/ |access-date=2023-04-23 |website=Athenarium |archive-date=23 April 2023 |archive-url=https://web.archive.org/web/20230423144200/https://athenarium.com/battle-of-the-sexes-mixed-strategies/ |url-status=live }}</ref>

===Ultimatum game===
{{main|Ultimatum game}}
The ultimatum game is a game that has become a popular instrument of [[experimental economics|economic experiments]]. An early description is by Nobel laureate [[John Harsanyi]] in 1961.<ref>{{cite journal |last1=Harsanyi |first1=John C. |title=On the rationality postulates underlying the theory of cooperative games |journal=Journal of Conflict Resolution |date=June 1961 |volume=5 |issue=2 |pages=179–196 |doi=10.1177/002200276100500205 }}</ref>

One player, the proposer, is endowed with a sum of money. The proposer is tasked with splitting it with another player, the responder (who knows what the total sum is). Once the proposer communicates his decision, the responder may accept it or reject it. If the responder accepts, the money is split per the proposal; if the responder rejects, both players receive nothing.  Both players know in advance the consequences of the responder accepting or rejecting the offer. The game demonstrates how social acceptance, fairness, and generosity influence the players decisions.<ref>{{cite journal |last1=Aoki |first1=Ryuta |last2=Yomogida |first2=Yukihito |last3=Matsumoto |first3=Kenji |title=The neural bases for valuing social equality |journal=Neuroscience Research |date=January 2015 |volume=90 |pages=33–40 |doi=10.1016/j.neures.2014.10.020 |pmid=25452125 }}</ref>

Ultimatum game has a variant, that is the dictator game. They are mostly identical, except in dictator game the responder has no power to reject the proposer's offer.

===Trust game===
The Trust Game is an experiment designed to measure trust in economic decisions. It is also called "the investment game" and is designed to investigate trust and demonstrate its importance rather than "rationality" of self-interest. The game was designed by Berg Joyce, John Dickhaut and Kevin McCabe in 1995.<ref>{{cite journal |last1=Berg |first1=Joyce |last2=Dickhaut |first2=John |last3=McCabe |first3=Kevin |title=Trust, Reciprocity, and Social History |journal=Games and Economic Behavior |date=July 1995 |volume=10 |issue=1 |pages=122–142 |doi=10.1006/game.1995.1027 }}</ref>

In the game, one player (the investor) is given a sum of money and must decide how much of it to give to another player (the trustee). The amount given is then tripled by the experimenter. The trustee then decides how much of the tripled amount to return to the investor. If the recipient is completely self interested, then he/she should return nothing. However that is not true as the experiment conduct. The outcome suggest that people are willing to place a trust, by risking some amount of money, in the belief that there would be reciprocity.<ref>{{cite journal |last1=Johnson |first1=Noel D. |last2=Mislin |first2=Alexandra A. |title=Trust games: A meta-analysis |journal=Journal of Economic Psychology |date=October 2011 |volume=32 |issue=5 |pages=865–889 |doi=10.1016/j.joep.2011.05.007 }}</ref>

=== Cournot Competition ===
{{main|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-maximizing 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.<ref name=":2">{{Cite book|last=Gibbons|first=Robert|title=Game Theory for Applied Economists|publisher=Princeton University Press|year=1992|isbn=0-691-04308-6|location=Princeton, New Jersey|pages=14–17}}</ref> 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 maximize  profit.<ref name=":2" /> However this option does not provide the highest payoff, as a firm's ability to maximize profits depends on its market share and the elasticity of the market demand.<ref>{{cite web|date=18 April 2013|title=Cournot (Nash) Equilibrium|url=https://stats.oecd.org/glossary/detail.asp?ID=3183|access-date=20 April 2021|website=OECD|archive-date=23 May 2021|archive-url=https://web.archive.org/web/20210523222914/https://stats.oecd.org/glossary/detail.asp?ID=3183|url-status=live}}</ref> 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.<ref name=":2" /> Within the game, firms reach the Nash equilibrium when the Cournot equilibrium is achieved.   
[[File:Cournot Duopoly Nash equilibrium.png|thumb|Equilibrium for Cournot quantity competition]]

=== Bertrand Competition ===
{{main|Bertrand competition}}
The Bertrand competition assumes homogenous products and a constant marginal cost and players choose the prices.<ref name=":2" /> 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.<ref>{{cite journal |last1=Spulber |first1=Daniel F. |title=Bertrand Competition when Rivals' Costs are Unknown |journal=The Journal of Industrial Economics |date=1995 |volume=43 |issue=1 |pages=1–11 |doi=10.2307/2950422 |jstor=2950422 }}</ref>

==In popular culture==
* Based on [[A Beautiful Mind (book)|the 1998 book]] by [[Sylvia Nasar]],<ref>Nasar, Sylvia (1998) ''A Beautiful Mind'', Simon & Schuster. {{isbn|0-684-81906-6}}.</ref> the life story of game theorist and mathematician [[John Forbes Nash Jr.|John Nash]] was turned into the 2001 [[biopic]] ''[[A Beautiful Mind (film)|A Beautiful Mind]]'', starring [[Russell Crowe]] as Nash.<ref>{{cite news |last1=Singh |first1=Simon |title=Between Genius and Madness |url=https://archive.nytimes.com/www.nytimes.com/books/98/06/14/reviews/980614.14singht.html |work=The New York Times |date=14 June 1998 }}</ref>
* The 1959 [[military science fiction]] novel ''[[Starship Troopers]]'' by [[Robert A. Heinlein]] mentioned "games theory" and "theory of games".<ref>{{citation |last=Heinlein |first=Robert A. |title=Starship Troopers |year=1959}}</ref> In the 1997 film [[Starship Troopers (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.<ref>{{Cite AV media|url=https://archive.org/details/DRStrangelove_20130616|title=Dr. Strangelove Or How I Learned to Stop Worrying and Love the Bomb|date=1964-01-29|minutes=51|quote=... is that the whole point of the doomsday machine is lost, if you keep it a secret!}}</ref>
* The 1980s [[power pop]] band [[Game Theory (band)|Game Theory]] was founded by singer/songwriter [[Scott Miller (pop musician)|Scott Miller]], who described the band's name as alluding to "the study of calculating the most appropriate action given an adversary{{nbsp}}... to give yourself the minimum amount of failure".<ref name=intpacsun>{{cite news|last=Guzman|first=Rafer|title=Star on hold: Faithful following, meager sales|date=6 March 1996|work=[[Pacific Sun (newspaper)|Pacific Sun]]|url=http://www.loudfamily.com/old/intpacsun.html|access-date=25 July 2018|archive-url=https://web.archive.org/web/20131106223809/http://www.loudfamily.com/old/intpacsun.html|archive-date=6 November 2013|url-status=dead|df=dmy-all}}.{{cbignore}}</ref>
* ''[[Liar Game]]'', a 2005 Japanese [[manga]] 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.<ref>{{Cite web |title=Liar Game (manga) – Anime News Network |url=https://www.animenewsnetwork.com/encyclopedia/manga.php?id=8178 |access-date=2022-11-25 |website=www.animenewsnetwork.com |archive-date=25 November 2022 |archive-url=https://web.archive.org/web/20221125140852/https://www.animenewsnetwork.com/encyclopedia/manga.php?id=8178 |url-status=live }}</ref>
* The 1974 novel ''[[Spy Story (novel)|Spy Story]]'' by [[Len Deighton]] explores elements of game theory in regard to cold war army exercises.
* The 2008 novel ''[[The Dark Forest]]'' by [[Liu Cixin]] explores the relationship between extraterrestrial life, humanity, and game theory.
* Joker, the prime antagonist in the 2008 film ''[[The Dark Knight]]'' presents game theory concepts—notably the [[prisoner's dilemma]] in a scene where he asks passengers in two different ferries to bomb the other one to save their own.
* In the 2018 film ''[[Crazy Rich Asians (film)|Crazy Rich Asians]]'', the female lead Rachel Chu is a professor of economics and game theory at [[New York University]]. At the beginning of the film she is seen in her NYU classroom playing a game of poker with her teaching assistant and wins the game by [[Glossary of card game terms#bluffing|bluffing]];<ref>{{cite web |url=https://www.pokernews.com/news/2018/08/poker-and-game-theory-featured-in-hit-film-crazy-rich-asians-31847.htm |title=Poker and Game Theory Featured in Hit Film 'Crazy Rich Asians' |last1=Chaffin |first1=Sean |date=August 20, 2018 |publisher=PokerNews.com |access-date=5 November 2022 |archive-date=5 November 2022 |archive-url=https://web.archive.org/web/20221105190628/https://www.pokernews.com/news/2018/08/poker-and-game-theory-featured-in-hit-film-crazy-rich-asians-31847.htm |url-status=live }}</ref> then in the [[Climax (narrative)|climax]] of the film, she plays a game of [[mahjong]] with her boyfriend's disapproving mother Eleanor, losing the game to Eleanor on purpose but winning her approval as a result.<ref>{{cite web |url=https://filmcolossus.com/crazy-rich-asians-game-theory/ |title=Game theory in Crazy Rich Asians: explaining the Mahjong showdown between Rachel and Eleanor |last1=Bean |first1=Travis |date=8 February 2019 |publisher=Colossus |access-date=5 November 2022 |archive-date=5 November 2022 |archive-url=https://web.archive.org/web/20221105190628/https://filmcolossus.com/crazy-rich-asians-game-theory/ |url-status=live }}</ref>
* In the 2017 film [[Molly's Game]], Brad, an inexperienced poker player, makes an irrational betting decision without realizing and causes his opponent Harlan to deviate from his Nash Equilibrium strategy, resulting in a significant loss when Harlan loses the hand.<ref>{{Cite web |title=An Analysis of the Applications of Networks in "Molly's Game" : Networks Course blog for INFO 2040/CS 2850/Econ 2040/SOC 2090 |url=https://blogs.cornell.edu/info2040/2022/10/25/an-analysis-of-the-applications-of-networks-in-mollys-game/ |access-date=2023-04-08 |archive-date=8 April 2023 |archive-url=https://web.archive.org/web/20230408025922/https://blogs.cornell.edu/info2040/2022/10/25/an-analysis-of-the-applications-of-networks-in-mollys-game/ |url-status=live }}</ref>

==See also==
* {{Annotated link|Applied ethics}}
* {{Annotated link|Bandwidth-sharing game}}
* {{Annotated link|Chainstore paradox}}
* {{Annotated link|Collective intentionality}}
* {{Annotated link|Core (game theory)}}
* {{Annotated link|Glossary of game theory}}
* {{Annotated link|Intra-household bargaining}}
* {{Annotated link|Kingmaker scenario}}
* {{Annotated link|Law and economics}}
* {{Annotated link|Mutual assured destruction}}
* {{Annotated link|Outline of artificial intelligence}}
* {{Annotated link|Parrondo's paradox}}
* {{Annotated link|Precautionary principle}}
* {{Annotated link|Quantum refereed game}}
* {{Annotated link|Risk management}}
* {{Annotated link|Self-confirming equilibrium}}
* {{Annotated link|Tragedy of the commons}}
* {{Annotated link|Traveler's dilemma}}
* {{Annotated link|Wilson doctrine (economics)}}
* [[Compositional game theory]]

'''Lists'''
* [[List of cognitive biases]]
* [[List of emerging technologies]]
* [[List of games in game theory]]

==Notes==
{{notelist}}

==References==
{{Reflist}}

==Further reading==
{{Wikiquote}}
{{Commons category}}
* {{cite journal |last1=Ben-David |first1=S. |last2=Borodin |first2=A. |last3=Karp |first3=R. |last4=Tardos |first4=G. |last5=Wigderson |first5=A. |title=On the power of randomization in on-line algorithms |journal=Algorithmica |date=January 1994 |volume=11 |issue=1 |pages=2–14 |doi=10.1007/BF01294260 |s2cid=26771869 }}
* {{Citation |last1=Downs |first1=Anthony |author1-link=Anthony Downs |title=An Economic theory of Democracy |title-link=An Economic Theory of Democracy |publisher=Harper |location=New York |year=1957}}
* {{cite book |last1=Fisher |first1=Sir Ronald Aylmer |title=The Genetical Theory of Natural Selection |date=1930 |publisher=Clarendon Press |url=https://books.google.com/books?id=t3jwAAAAMAAJ }}
* {{Citation |last1=Gauthier |first1=David |author1-link=David Gauthier |title=Morals by agreement |publisher=[[Oxford University Press]] |isbn=978-0-19-824992-4 |year=1986}}
* {{Citation |last1=Grim |first1=Patrick |last2=Kokalis |first2=Trina |last3=Alai-Tafti |first3=Ali |last4=Kilb |first4=Nicholas |last5=St Denis |first5=Paul |title=Making meaning happen |year=2004 |journal=Journal of Experimental & Theoretical Artificial Intelligence |volume=16 |issue=4 |pages=209–243 |doi=10.1080/09528130412331294715 |bibcode=2004JETAI..16..209G |s2cid=5737352 }}
* {{Citation |last1=Harper |first1=David |author1-link=David Harper (biologist) |last2=Maynard Smith |first2=John |author2-link=John Maynard Smith |title=Animal signals |publisher=Oxford University Press |isbn=978-0-19-852685-8 |year=2003}}
* {{Citation |last=Howard |first=Nigel |title=Paradoxes of Rationality: Games, Metagames, and Political Behavior |publisher=The MIT Press |year=1971 |location=Cambridge, [[Massachusetts|MA]] |isbn=978-0-262-58237-7}}
* {{cite book |last1=Kavka |first1=Gregory S. |title=Hobbesian Moral and Political Theory |date=1986 |publisher=Princeton University Press |isbn=978-0-691-02765-4 }}
* {{Citation |last1=Lewis |first1=David |author1-link=David Kellogg Lewis |title=Convention: A Philosophical Study |year=1969}}, {{isbn|978-0-631-23257-5}} (2002 edition)
* {{Citation |last1=Maynard Smith |first1=John |author1-link=John Maynard Smith |first2=George R. |last2=Price |s2cid=4224989 |author2-link=George R. Price |title=The logic of animal conflict |year=1973 |journal=Nature |volume=246 |issue=5427 |pages=15–18 |doi=10.1038/246015a0 |bibcode=1973Natur.246...15S}}
* {{Citation |last2=Rubinstein |first2=Ariel |author2-link=Ariel Rubinstein |last1=Osborne |first1=Martin J. |title=A course in game theory |publisher=MIT Press |isbn=978-0-262-65040-3 |year=1994}}. A modern introduction at the graduate level.
* {{cite book |last1=Poundstone |first1=William |title=Prisoner's Dilemma |date=1993 |publisher=Anchor |location=New York |isbn=0-385-41580-X |edition=1st Anchor Books |url=https://archive.org/details/prisonersdilemma00poun}}
* {{Citation |last1=Quine |first1=W.v.O |author1-link=Willard Van Orman Quine |title=Philosophica Essays for A.N. Whitehead |publisher=Russel and Russel Publishers |isbn=978-0-8462-0970-6 |year=1967 |chapter=Truth by Convention}}
* {{Citation |last1=Quine |first1=W.v.O |s2cid=46979744 |author1-link=Willard Van Orman Quine |title=Carnap and Logical Truth |year=1960 |journal=Synthese |volume=12 |issue=4 |pages=350–374 |doi=10.1007/BF00485423}}
* {{Citation |last1=Skyrms |first1=Brian |title=Evolution of the social contract |publisher=Cambridge University Press |isbn=978-0-521-55583-8 |year=1996}}
* {{Citation |last1=Skyrms |first1=Brian |title=The stag hunt and the evolution of social structure |publisher=Cambridge University Press |isbn=978-0-521-53392-8 |year=2004}}
* {{Citation |last1=Sober |first1=Elliott |last2=Wilson |first2=David Sloan |title=Unto others: the evolution and psychology of unselfish behavior |publisher=Harvard University Press |isbn=978-0-674-93047-6 |year=1998}}
* {{Citation |last=Webb |first=James N. |year=2007 |title=Game theory: decisions, interaction and evolution |series=Undergraduate mathematics |publisher=Springer |isbn=978-1-84628-423-6}} Consistent treatment of game types usually claimed by different applied fields, e.g. [[Markov decision process]]es.

===Textbooks and general literature===
* {{citation |first=Robert J |last=Aumann |author-link=Robert Aumann |year=1987 |contribution=game theory |title=The New Palgrave: A Dictionary of Economics |volume=2 |pages=460–82|title-link=The New Palgrave: A Dictionary of Economics }}.
* {{Citation |last=Camerer |first=Colin |author-link=Colin Camerer |title=Behavioral Game Theory: Experiments in Strategic Interaction |publisher=Russell Sage Foundation |isbn=978-0-691-09039-9 |year=2003 |chapter-url=http://press.princeton.edu/chapters/i7517.html |chapter=Introduction |pages=1–25 |access-date=9 February 2011 |archive-date=14 May 2011 |archive-url=https://web.archive.org/web/20110514201411/http://press.princeton.edu/chapters/i7517.html |url-status=dead }}, [http://press.princeton.edu/titles/7517.html Description].
* {{Citation |last=Dutta |first=Prajit K. |title=Strategies and games: theory and practice |publisher=[[MIT Press]] |isbn=978-0-262-04169-0 |year=1999}}. Suitable for undergraduate and business students.
* {{Citation |last1=Fernandez |first1=L F. |last2=Bierman |first2=H S. |title=Game theory with economic applications |publisher=[[Addison-Wesley]] |isbn=978-0-201-84758-1 |year=1998}}. Suitable for upper-level undergraduates.
* {{cite book |last1=Gaffal |first1=Margit |last2=Padilla Gálvez |first2=Jesús |title=Dynamics of Rational Negotiation: Game Theory, Language Games and Forms of Life |date=2014 |publisher=Springer }}
* {{Citation |last=Gibbons |first=Robert D. |title=Game theory for applied economists |publisher=Princeton University Press |isbn=978-0-691-00395-5 |year=1992}}. Suitable for advanced undergraduates.
** Published in Europe as {{Citation |title=A Primer in Game Theory |publisher=Harvester Wheatsheaf |isbn=978-0-7450-1159-2 |location=London |first=Robert |last=Gibbons |year=2001}}.
* {{Citation |last=Gintis |first=Herbert |title=Game theory evolving: a problem-centered introduction to modeling strategic behavior |publisher=Princeton University Press |isbn=978-0-691-00943-8 |year=2000}}
* {{Citation |last1=Green |first1=Jerry R. |last2=Mas-Colell |first2=Andreu |author2-link=Andreu Mas-Colell |last3=Whinston |first3=Michael D. |title=Microeconomic theory |publisher=[[Oxford University Press]] |isbn=978-0-19-507340-9 |year=1995}}. Presents game theory in formal way suitable for graduate level.
* Joseph E. Harrington (2008) ''Games, strategies, and decision making'', Worth, {{isbn|0-7167-6630-2}}. Textbook suitable for undergraduates in applied fields; numerous examples, fewer formalisms in concept presentation.
* {{Citation |last=Isaacs |first=Rufus |author-link=Rufus Isaacs (game theorist) |title=Differential Games: A Mathematical Theory With Applications to Warfare and Pursuit, Control and Optimization |publisher=[[Dover Publications]] |location=New York |isbn=978-0-486-40682-4 |year=1999}}
*[[Michael Maschler]]; [[Eilon Solan]]; Shmuel Zamir (2013), ''Game Theory'', Cambridge University Press, {{ISBN|978-1-108-49345-1}}. Undergraduate textbook.
* {{Citation |last=Miller |first=James H. |title=Game theory at work: how to use game theory to outthink and outmaneuver your competition |publisher=[[McGraw-Hill]] |location=New York |isbn=978-0-07-140020-6 |year=2003}}. Suitable for a general audience.
* {{Citation |last1=Shoham |first1=Yoav |last2=Leyton-Brown |first2=Kevin |title=Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations |publisher=Cambridge University Press |isbn=978-0-521-89943-7 |url=http://www.masfoundations.org/download.html |year=2009 |location=New York |access-date=8 March 2016}}
* {{Citation |last1=Watson |first1=Joel |title=Strategy: An Introduction to Game Theory (3rd edition) |publisher=W.W. Norton and Co. |isbn=978-0-393-91838-0 |url=https://wwnorton.com/books/Strategy/ |year=2013 |location=New York}}. A leading textbook at the advanced undergraduate level.
* {{cite book |last1=McCain |first1=Roger A. |title=Game Theory: A Nontechnical Introduction to the Analysis of Strategy |date=2010 |publisher=World Scientific |isbn=978-981-4289-65-8 }}

===Historically important texts===
* {{Citation |author-link1=Robert Aumann |last1=Aumann |first1=R. J. |author-link2=Lloyd Shapley |last2=Shapley |first2=L. S. |year=1974 |title=Values of Non-Atomic Games |publisher=Princeton University Press}}
* {{Citation |last1=Cournot |first1=A. Augustin |author1-link=Antoine Augustin Cournot |title=Recherches sur les principles mathematiques de la théorie des richesses |year=1838 |journal=Libraire des Sciences Politiques et Sociales}}
* {{Citation |last1=Edgeworth |first1=Francis Y. |author1-link=Francis Ysidro Edgeworth |title=Mathematical Psychics |publisher=Kegan Paul |location=London |year=1881|url=https://books.google.com/books?id=CElYAAAAcAAJ}}
* {{Citation |title=Theory of Voting |first=Robin |last=Farquharson |author-link=Robin Farquharson |publisher=Blackwell (Yale U.P. in the U.S.) |year=1969 |isbn=978-0-631-12460-3}}
<!-- * {{Citation |last1=Fisher |first1=Ronald |author1-link=Ronald Fisher |title=[[The Genetical Theory of Natural Selection]] |publisher=Clarendon Press |location=Oxford |year=1930}}
:*reprinted edition: {{Citation |title=The Genetical Theory of Natural Selection: A Complete Variorum Edition |publisher=[[Oxford University Press]] |isbn=978-0-19-850440-5 |year=1999 |author=R.A. Fisher; edited with a foreword and notes by J. H. Bennett.}} -->
* {{Citation |last1=Luce |first1=R. Duncan |author1-link=R. Duncan Luce |last2=Raiffa |first2=Howard |author2-link=Howard Raiffa |title=Games and decisions: introduction and critical survey |publisher=[[John Wiley & Sons|Wiley]] |location=New York |year=1957}}
:*reprinted edition: {{Citation |title=Games and decisions: introduction and critical survey |publisher=[[Dover Publications]] |location=New York |isbn=978-0-486-65943-5 |year=1989 |author1=R. Duncan Luce |author2=Howard Raiffa}}
* {{Citation |last1=Maynard Smith |first1=John |author1-link=John Maynard Smith |title=Evolution and the theory of games |publisher=Cambridge University Press |isbn=978-0-521-28884-2 |year=1982|title-link=Evolution and the Theory of Games }}
* {{Citation |last1=Nash |first1=John |author1-link=John Forbes Nash |title=Equilibrium points in n-person games |year=1950 |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=36 |issue=1 |pages=48–49 |doi=10.1073/pnas.36.1.48 |pmc=1063129 |pmid=16588946 |bibcode=1950PNAS...36...48N|doi-access=free }}
* [[Lloyd Shapley|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.)
* {{cite journal |last1=Shapley |first1=L. S. |authorlink1=Lloyd Shapley |title=Stochastic Games |journal=Proceedings of the National Academy of Sciences |date=October 1953 |volume=39 |issue=10 |pages=1095–1100 |doi=10.1073/pnas.39.10.1095 |doi-access=free |pmid=16589380 |pmc=1063912 |bibcode=1953PNAS...39.1095S }}
* {{Citation |last1=von Neumann |first1=John |title=Zur Theorie der Gesellschaftsspiele |year=1928 |journal=[[Mathematische Annalen]] |volume=100 |issue=1 |pages=295–320 |doi=10.1007/bf01448847|s2cid=122961988 }} 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.&nbsp;4, p. [https://books.google.com/books?id=9lSVFzsTGWsC&pg=PA42 42.] Princeton University Press.
* {{Citation |last1=von Neumann |first1=John |author1-link=John von Neumann |last2=Morgenstern |first2=Oskar |author2-link=Oskar Morgenstern |title=Theory of games and economic behavior |journal=Nature |volume=157 |issue=3981 |pages=172 |publisher=Princeton University Press |year=1944|title-link=Theory of games and economic behavior |bibcode=1946Natur.157..172R |doi=10.1038/157172a0 |s2cid=29754824 }}
* {{Citation |last1=Zermelo |first1=Ernst |author1-link=Ernst Zermelo |title=Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels |year=1913 |journal=Proceedings of the Fifth International Congress of Mathematicians |volume=2 |pages=501–4}}

===Other material===

* [[Allan Gibbard]], "Manipulation of voting schemes: a general result", ''Econometrica'', Vol. 41, No. 4 (1973), pp.&nbsp;587–601.
* {{Citation |last1=McDonald |first1=John |title=Strategy in Poker, Business & War |publisher=[[W. W. Norton]] |isbn=978-0-393-31457-1 |year=1950–1996}}. A layman's introduction.
* {{Citation |last1=Papayoanou |first1=Paul |title=Game Theory for Business: A Primer in Strategic Gaming |publisher=[[Probabilistic]] |isbn=978-0-9647938-7-3 |year=2010}}. 
* {{cite journal |last1=Satterthwaite |first1=Mark Allen |title=Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions |journal=Journal of Economic Theory |date=April 1975 |volume=10 |issue=2 |pages=187–217 |doi=10.1016/0022-0531(75)90050-2 |url=http://www.kellogg.northwestern.edu/research/math/papers/122.pdf }}
* {{Citation |last1=Siegfried |first1=Tom |title=A Beautiful Math |publisher=Joseph Henry Press |isbn=978-0-309-10192-9 |year=2006}}
* {{Citation |last1=Skyrms |first1=Brian |author1-link=Brian Skyrms |title=The Dynamics of Rational Deliberation |publisher=[[Harvard University Press]] |isbn=978-0-674-21885-7 |year=1990}}
* {{Citation |last1=Thrall |first1=Robert M. |author-link=Robert M. Thrall |last2=Lucas |first2=William F. |title=<math>n</math>-person games in partition function form |journal=Naval Research Logistics Quarterly |volume=10 |issue=4 |pages=281–298 |year=1963 |doi=10.1002/nav.3800100126}}
* {{cite book |doi=10.1145/1993806.1993858 |chapter=Rationality authority for provable rational behavior |title=Proceedings of the 30th annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing |date=2011 |last1=Dolev |first1=Shlomi |last2=Panagopoulou |first2=Panagiota N. |last3=Rabie |first3=Mikaël |last4=Schiller |first4=Elad M. |last5=Spirakis |first5=Paul G. |pages=289–290 |isbn=978-1-4503-0719-2 |chapter-url=http://publications.lib.chalmers.se/publication/230780-rationality-authority-for-provable-rational-behavior }}
* {{citation | last1 = Chastain | first1 = Erick | last2 = Livnat | first2 = Adi | last3 = Papadimitriou | first3 = Christos | author3-link = Christos Papadimitriou | last4 = Vazirani | first4 = Umesh | author4-link = Umesh Vazirani | bibcode = 2014PNAS..11110620C | date = June 2014 | doi = 10.1073/pnas.1406556111 | doi-access = free | issue = 29 | journal = [[Proceedings of the National Academy of Sciences of the United States of America]] | pages = 10620–10623 | pmc = 4115542 | title = Algorithms, games, and evolution | volume = 111| pmid = 24979793 }}

==External links==
{{Wikiquote}}
{{Wiktionary|game theory}}
{{Wikiversity|Game Theory}}
{{Wikibooks|Introduction to Game Theory}}
* James Miller (2015): [https://www.youtube.com/playlist?list=PLqekkRyYeow3cR9U4c4wkIekm2pXxORPn Introductory Game Theory Videos].
* {{springer|title=Games, theory of|id=p/g043280}}
* Paul Walker: [https://web.archive.org/web/20000815223335/http://www.economics.harvard.edu/~aroth/alroth.html History of Game Theory Page].
* David Levine: [http://dklevine.com/ Game Theory. Papers, Lecture Notes and much more stuff.]
* Alvin Roth:{{cite web |url=http://www.economics.harvard.edu/~aroth/alroth.html |title=Game Theory and Experimental Economics page |access-date=13 September 2003 |url-status=dead |archive-url=https://web.archive.org/web/20000815223335/http://www.economics.harvard.edu/~aroth/alroth.html |archive-date=15 August 2000 |df=dmy}}  — Comprehensive list of links to game theory information on the Web
* Adam Kalai: [https://web.archive.org/web/20080416035356/http://wiki.cc.gatech.edu/theory/index.php/CS_8803_-_Game_Theory_and_Computer_Science._Spring_2008 Game Theory and Computer Science] — Lecture notes on Game Theory and Computer Science
* Mike Shor: [http://www.gametheory.net/ GameTheory.net] — Lecture notes, interactive illustrations and other information.
* Jim Ratliff's [http://virtualperfection.com/gametheory/ Graduate Course in Game Theory] {{Webarchive|url=https://web.archive.org/web/20100329222934/http://virtualperfection.com/gametheory/ |date=29 March 2010 }} (lecture notes).
* Don Ross: [http://plato.stanford.edu/entries/game-theory/ Review Of Game Theory] in the ''Stanford Encyclopedia of Philosophy''.
* Bruno Verbeek and Christopher Morris: [http://plato.stanford.edu/entries/game-ethics/ Game Theory and Ethics]
* Elmer G. Wiens: [http://www.egwald.ca/operationsresearch/gameintroduction.php Game Theory] — Introduction, worked examples, play online two-person zero-sum games.
* Marek M. Kaminski: [http://webfiles.uci.edu/mkaminsk/www/courses.html Game Theory and Politics] {{Webarchive|url=https://web.archive.org/web/20061020072523/http://webfiles.uci.edu/mkaminsk/www/courses.html |date=20 October 2006 }} — Syllabuses and lecture notes for game theory and political science.
* [http://www.socialcapitalgateway.org/eng-gametheory.htm Websites on game theory and social interactions]
* Kesten Green's {{webarchive |url=https://web.archive.org/web/20110411143752/http://conflictforecasting.com/ |date=11 April 2011 |title=Conflict Forecasting }} — See [http://www.forecastingprinciples.com/paperpdf/Greenforecastinginconflict.pdf Papers] for [http://www.decisionworkshops.com/#/graphs-of-findings/4553562008 evidence on the accuracy of forecasts from game theory and other methods] {{Webarchive|url=https://web.archive.org/web/20190915052113/http://www.decisionworkshops.com/#/graphs-of-findings/4553562008 |date=15 September 2019 }}.
* McKelvey, Richard D., McLennan, Andrew M., and Turocy, Theodore L. (2007) ''[https://gambit.sourceforge.net/ Gambit: Software Tools for Game Theory]''.
* Benjamin Polak: [http://oyc.yale.edu/economics/game-theory Open Course on Game Theory at Yale] {{Webarchive|url=https://web.archive.org/web/20100803224320/http://oyc.yale.edu/economics/game-theory |date=3 August 2010 }} [https://www.youtube.com/playlist?list=PL6EF60E1027E1A10B videos of the course]
* Benjamin Moritz, Bernhard Könsgen, Danny Bures, Ronni Wiersch, (2007) ''[https://web.archive.org/web/20130718133141/http://www.spieltheorie-software.de/ Spieltheorie-Software.de: An application for Game Theory implemented in JAVA]''.
* Antonin Kucera: [http://slideslive.com/38891162/stochastic-twoplayer-games Stochastic Two-Player Games].
* Yu-Chi Ho: [http://blog.sciencenet.cn/home.php?mod=space&uid=1565&do=blog&id=17894 What is Mathematical Game Theory]; [http://blog.sciencenet.cn/home.php?mod=space&uid=1565&do=blog&id=20337 What is Mathematical Game Theory (#2)]; [http://blog.sciencenet.cn/home.php?mod=space&uid=1565&do=blog&id=21210 What is Mathematical Game Theory (#3)]; [http://blog.sciencenet.cn/home.php?mod=space&uid=1565&do=blog&id=21735 What is Mathematical Game Theory (#4)-Many person game theory]; [http://blog.sciencenet.cn/home.php?mod=space&uid=1565&do=blog&id=21891 What is Mathematical Game Theory ?( #5) – Finale, summing up, and my own view]

{{Use dmy dates|date=July 2017}}

{{Game theory}}
{{Areas of mathematics |collapsed}}
{{Microeconomics}}
{{Property navbox}}

{{Authority control}}

{{DEFAULTSORT:Game Theory}}
[[Category:Game theory| ]]
[[Category:Artificial intelligence]]
[[Category:Formal sciences]]
[[Category:Mathematical economics]]
[[Category:John von Neumann]]