I'm trying to understand why [John Nash's 1950 2-page paper](http://www.pnas.org/content/36/1/48.full) that was published in PNAS was such a big deal. Unless I'm mistaken, the 1928 paper by John von Neumann demonstrated that all n-player non-cooperative and zero-sum games possess an equilibrium solution in terms of pure or mixed strategies. 


From what I understand, Nash used fixed point iteration to prove that non-zero-sum games would also have the analogous result. Why was this such a big deal in light of the earlier work by von Neumann? 


There are two references I provide that are good: One is [this discussion on simple proofs of Nash's theorem](https://mathoverflow.net/questions/121542/simple-proof-of-the-existence-of-nash-equilibria-for-2-person-games) and [this one](http://www.pnas.org/site/classics/classics5.xhtml) is a very well done (readable and accurate) survey of the history in PNAS. 



Why was John Nash's 1950 Game Theory paper such a big deal?
I think von Neumann dealt with the case $n=2$, and it was by no means obvious how to extend the concept of equilibrium for the general case and prove that it always exists. More precisely, $n$ players before Nash were reduced to the $n=2$ case by partioning the players into two groups in all possible ways. Once you regard several players as a single player, they are meant to cooperate as they must act like a single player. Nash is very clear about this in his 1951 Annals paper:



> 
> Von Neumann and Morgenstern have developed a very fruitful theory of 
>  two-person zero-sum games in their book Theory of Games and Economic Behavior. 
>  This book also contains a theory of $n$-person games of a type which 
>  we would call cooperative. This theory is based on an analysis of the interrelationships 
>  of the various coalitions which can be formed by the players of the 
>  game. 
> 
> 
> Our theory, in contradistinction, is based on the absence of coalitions in that 
>  it is assumed that each participant acts independently, without collaboration or 
>  communication with any of the others. 
> 
> 
> The notion of an equilibrium point is the basic ingredient in our theory. This 
>  notion yields a generalization of the concept of the solution of a two-person zero-sum game. 
>  It turns out that the set of equilibrium points of a two-person zero-sum game is 
>  simply the set of all pairs of opposing "good strategies." 
>  In the immediately following sections we shall define equilibrium points and 
>  prove that a finite non-cooperative game always has at least one equilibrium 
>  point. We shall also introduce the notions of solvability and strong solvability 
>  of a non-cooperative game and prove a theorem on the geometrical structure of 
>  the set of equilibrium points of a solvable game. 
> 
> 
> 


