Editing Nash equilibrium (section)

=== 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.