Multiagent Systems

Prof. Dr. Jürgen Dix · Department of Informatics, TUC Multiagent Systems, WS 06/07 218/731
3. Decision Making (1) 2. Non-coop Games
Proof for C 1 -differentiable mappings: The proof is by
contradiction.
Assume, that the continuously differential mapping
f : D → D has no fixed points. Let g : D → ∂D the
mapping, such that for every point x ∈ D the points
f(x), x, g(x) are in that order on a line of R n . The mapping
g is also continuously differentiable and satisfies g(x) = x
for x ∈ ∂D. We write g(x) = (g 1 (x), g 2 (x), . . . , g n (x)) and
get (for x ∈ ∂D and i = 1 . . . n) g i (x 1 , x 2 , . . . , x n ) = x i . Note
dg 1 ∧ dg 2 ∧ · · · ∧ dg n = 0 since g 2 1 + g 2 2 + · · · + g 2 n = 1. Then:
0 ≠ ∫ D dx 1 ∧ dx 2 ∧ · · · ∧ dx n = ∫ ∂D x 1 ∧ dx 2 ∧ · · · ∧ dx n
= ∫ ∂D g 1 ∧ dg 2 ∧ · · · ∧ dg n = ∫ D dg 1 ∧ dg 2 ∧ · · · ∧ dg n
= ∫ D 0 = 0