Editing Nash equilibrium (section)

==History==
Nash equilibrium is named after American mathematician [[John Forbes Nash Jr.|John Forbes Nash Jr]]. The same idea was used in a particular application in 1838 by [[Antoine Augustin Cournot]] in his theory of [[oligopoly]].<ref>Cournot A. (1838) Researches on the Mathematical Principles of the Theory of Wealth</ref> In Cournot's theory, each of several firms choose how much output to produce to maximize its profit. The best output for one firm depends on the outputs of the others. A [[Cournot equilibrium]] occurs when each firm's output maximizes its profits given the output of the other firms, which is a [[pure strategy|pure-strategy]] Nash equilibrium. Cournot also introduced the concept of [[best response]] dynamics in his analysis of the stability of equilibrium. Cournot did not use the idea in any other applications, however, or define it generally.

The modern concept of Nash equilibrium is instead defined in terms of [[mixed strategy|mixed strategies]], where players choose a probability distribution over possible pure strategies (which might put 100% of the probability on one pure strategy; such pure strategies are a subset of mixed strategies). The concept of a mixed-strategy equilibrium was introduced by [[John von Neumann]] and [[Oskar Morgenstern]] in their 1944 book ''The Theory of Games and Economic Behavior'', but their analysis was restricted to the special case of [[zero-sum]] games. They showed that a mixed-strategy Nash equilibrium will exist for any zero-sum game with a finite set of actions.<ref>J. Von Neumann, O. Morgenstern, ''[https://archive.org/stream/theoryofgamesand030098mbp#page/n5/mode/2up Theory of Games and Economic Behavior]'', copyright 1944, 1953, Princeton University Press</ref> The contribution of Nash in his 1951 article "Non-Cooperative Games" was to define a mixed-strategy Nash equilibrium for any game with a finite set of actions and prove that at least one (mixed-strategy) Nash equilibrium must exist in such a game. The key to Nash's ability to prove existence far more generally than von Neumann lay in his definition of equilibrium. According to Nash, "an equilibrium point is an n-tuple such that each player's mixed strategy maximizes [their] payoff if the strategies of the others are held fixed. Thus each player's strategy is optimal against those of the others." Putting the problem in this framework allowed Nash to employ the [[Kakutani fixed-point theorem]] in his 1950 paper to prove existence of equilibria. His 1951 paper used the simpler [[Brouwer fixed-point theorem]] for the same purpose.<ref>{{Cite journal |last1=Carmona |first1=Guilherme |first2=Konrad |last2=Podczeck |year=2009 |title=On the Existence of Pure Strategy Nash Equilibria in Large Games |ssrn=882466 |journal=[[Journal of Economic Theory]] |volume=144 |issue=3 |pages=1300–1319 |doi=10.1016/j.jet.2008.11.009 |url=http://fesrvsd.fe.unl.pt/WPFEUNL/WP2008/wp531.pdf |archive-url=https://web.archive.org/web/20090521220332/http://fesrvsd.fe.unl.pt/WPFEUNL/WP2008/wp531.pdf |url-status=dead |archive-date=May 21, 2009 |hdl=10362/11577 |hdl-access=free }}</ref>

Game theorists have discovered that in some circumstances Nash equilibrium makes invalid predictions or fails to make a unique prediction. They have proposed many [[solution concept]]s ('refinements' of Nash equilibria) designed to rule out implausible Nash equilibria. One particularly important issue is that some Nash equilibria may be based on threats that are not '[[credibility|credible]]'. In 1965 [[Reinhard Selten]] proposed [[subgame perfect equilibrium]] as a refinement that eliminates equilibria which depend on [[non-credible threats]]. Other extensions of the Nash equilibrium concept have addressed what happens if a game is [[Repeated game|repeated]], or what happens if a game is played in the [[Global game|absence of complete information]]. However, subsequent refinements and extensions of Nash equilibrium share the main insight on which Nash's concept rests: the equilibrium is a set of strategies such that each player's strategy is optimal given the choices of the others.