  Kakutani's Fixed Point theorem is a fundamental theorem in topology with
numerous applications in game theory and economics. Computational formulations
of Kakutani exist only in special cases and are too restrictive to be useful in
reductions. In this paper, we provide a general computational formulation of
Kakutani's Fixed Point Theorem and we prove that it is PPAD-complete. As an
application of our theorem we are able to characterize the computational
complexity of the following fundamental problems:
  (1) Concave Games. Introduced by the celebrated works of Debreu and Rosen in
the 1950s and 60s, concave $n$-person games have found many important
applications in Economics and Game Theory. We characterize the computational
complexity of finding an equilibrium in such games. We show that a general
formulation of this problem belongs to PPAD, and that finding an equilibrium is
PPAD-hard even for a rather restricted games of this kind: strongly-concave
utilities that can be expressed as multivariate polynomials of a constant
degree with axis aligned box constraints.
  (2) Walrasian Equilibrium. Using Kakutani's fixed point Arrow and Debreu we
resolve an open problem related to Walras's theorem on the existence of price
equilibria in general economies. There are many results about the PPAD-hardness
of Walrasian equilibria, but the inclusion in PPAD is only known for piecewise
linear utilities. We show that the problem with general convex utilities is in
PPAD.
  Along the way we provide a Lipschitz continuous version of Berge's maximum
theorem that may be of independent interest.
