Multiagent Systems

Prof. Dr. Jürgen Dix · Department of Informatics, TUC Multiagent Systems, WS 06/07 216/731
3. Decision Making (1) 2. Non-coop Games
Theorem 3.23 (Brouwer’s Fixpoint Theorem)
Let D be the unit Euclidean ball in R n and let f : D → D be a
continuous mapping. Then there exists x ∈ D with f(x) = x.
Proof
Reduction to the C 1 -differentiable case: Let r : R n → R
be the function
r(x) = r(x 1 , x 2 , . . . , x n ) = √ x 2 1 + x 2 2 + · · · + x 2 n.
Let φ : R n → R given by:
φ(x) = a n (1/4r(x) 4 − 1/2r(x) 2 + 1/4)
on D and equal to 0 in the complement of D,
where the constant a n is chosen such that the
integral of φ over R n equals 1.