I have an updating scheme which always seem to converge to the same fixed point. Using Brouwer's fixed point theorem, I can proof the existence of a fixed point. However, I wasn't able to proof convergence of my algorithm to a fixed point. The problem is that I am not familiar with fixed-point theory.

Now I found out that my updating scheme is actually equivalent with a finite concave game in game theory (J.B. Rosen "Existence and uniqueness of equilibrium points for concave N-person games"). This means that the fixed point of my scheme is a Nash equilibrium.

I was wondering: if a (concave) finite game has at least one Nash equilibrium, will the game always converge to an equilibrium? If not, what are sufficient conditions for convergence?

Rosen gives a sufficient condition to have a unique equilibrium, but I am not able to use this in my scheme, since I'm not able to proof that my pay-off functions can be combined to a diagonally strictly concave function. Do you know if there are other sufficient conditions for uniqueness of the equilibrium of a concave game?

Thanks in advance!