How do you compute the fixed point of a best-response function efficiently? I have a polynomial time best-response function that has the same properties as a game-theory game (convexity, compactness, set-valued).  I don't know that much topology, but my understanding is that the Kakutani fixed-point theorem confirms that this function has a fixed point, i.e., there exists a strategy profile whose best response is itself.

The "game" I have is not really a game, in the sense that there are no payoffs; there is only a best-response function.

My question is:  Given such a function, can one find a fixed-point quickly?  Ordinarily, there are polytime approximation algorithms for this task (finding a Nash equilibrium), but here I do not have the game in normal form, or an easy way to put it in normal form given the lack of payoffs specified.

Edit:  More specific information:  Given a fixed input, find a polytime machine bounded by n^8+8 s.t. the machine outputs its description on the input.  (There are ways of making this convex by allowing mixing over the edges of the machine.)
A fixed point of a best response function is a Nash equilibrium -- the fact that you do not have the payoff matrix cannot make the problem easier (since if you know the payoff matrix, you also know the best response function, but not vice versa). Unfortunately, there are not in general good algorithms for computing approximate Nash equilibria in multi-player games. It is known that in general settings, computing a fixed point of a function is equivalent in difficulty to computing a Nash equilibrium -- see for instance http://homepages.inf.ed.ac.uk/kousha/nash_focs07_full_j_spec_issue_sub.pdf

There are not believed to be efficient algorithms for computing fixed points for general (best response) functions, and it is open whether they can be approximated well in general.