Editing Nash equilibrium (section)

== Existence ==

===Nash's existence theorem===
Nash proved that if [[strategy (game theory)#Pure and mixed strategies|mixed strategies]] (where a player chooses probabilities of using various pure strategies) are allowed, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium, which might be a pure strategy for each player or might be a probability distribution over strategies for each player.

Nash equilibria need not exist if the set of choices is infinite and non-compact. For example:

* A game where two players simultaneously name a number and the player naming the larger number wins does not have a NE, as the set of choices is not compact because it is unbounded.
* Each of two players chooses a real number strictly less than 5 and the winner is whoever has the biggest number; no biggest number strictly less than 5 exists (if the number could equal 5, the Nash equilibrium would have both players choosing 5 and tying the game). Here, the set of choices is not compact because it is not closed.

However, a Nash equilibrium exists if the set of choices is [[compact space|compact]] with each player's payoff continuous in the strategies of all the players.<ref>MIT OpenCourseWare. 6.254: Game Theory with Engineering Applications, Spring 2010. [https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-254-game-theory-with-engineering-applications-spring-2010/lecture-notes/MIT6_254S10_lec06.pdf Lecture 6: Continuous and Discontinuous Games].</ref>

=== Rosen's existence theorem ===
Rosen<ref>{{Cite journal |last=Rosen |first=J. B. |date=1965 |title=Existence and Uniqueness of Equilibrium Points for Concave N-Person Games |url=https://www.jstor.org/stable/1911749 |journal=Econometrica |volume=33 |issue=3 |pages=520–534 |doi=10.2307/1911749 |jstor=1911749 |issn=0012-9682|hdl=2060/19650010164 |hdl-access=free }}</ref> extended Nash's existence theorem in several ways. He considers an n-player game, in which the strategy of each player ''i'' is a vector ''s<sub>i</sub>'' in the Euclidean space R<sup>''mi''</sup><sub>.</sub> Denote ''m'':=''m''<sub>1</sub>+...+''m<sub>n</sub>''; so a strategy-tuple is a vector in R<sup>m</sup>. Part of the definition of a game is a subset ''S'' of R<sup>m</sup> such that the strategy-tuple must be in ''S''. This means that the actions of players may potentially be constrained based on actions of other players. A common special case of the model is when ''S'' is a Cartesian product of convex sets ''S''<sub>1</sub>,...,''S<sub>n</sub>'', such that the strategy of player ''i'' must be in ''S<sub>i</sub>''. This represents the case that the actions of each player ''i'' are constrained independently of other players' actions. If the following conditions hold:

* T is [[Convex set|convex]], closed and bounded;
* Each payoff function ''u<sub>i</sub>'' is continuous in the strategies of all players, and [[Concave function|concave]] in ''s<sub>i</sub>'' for every fixed value of ''s''<sub>−''i''</sub>.

Then a Nash equilibrium exists. The proof uses the [[Kakutani fixed-point theorem]]. Rosen also proves that, under certain technical conditions which include strict concavity, the equilibrium is unique.

Nash's result refers to the special case in which each ''S<sub>i</sub>'' is a [[simplex]] (representing all possible mixtures of pure strategies), and the payoff functions of all players are [[bilinear function]]s of the strategies.