A multifunction Γ is called a Kakutani multifunction if there exist two nonempty convex sets X and Y , each in a Hausdorff topological vector space, such that Γ : X → Y is upper semi-continuous with nonempty compact convex values. We prove the following extension of the Kakutani fixed point theorem : Let Γ : X → X be a multi-function from a simplex X into itself ; if Γ can be factorized by an arbitrary finite number of Kakutani multifunctions, then Γ has a fixed point. The proof relies on a simplicial approximation technique and the Brouwer fixed point theorem. Extensions to infinite-dimensional spaces and applications to game theory are given.