Editing Game theory (section)

==History==
===Earliest results===
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]].