Editing Nash equilibrium (section)

===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>