Multiagent Systems

3. Decision Making (1) 2. Non-coop Games
Proof
We use Kakatuni’s theorem: Let X be a nonempty subset
of n-dimensional Euclidean space, and f : X −→ 2 X . The
following are sufficient conditions for f to have a fixed
point (i.e. an x ∗ ∈ X with x ∗ ∈ f(x ∗ )):
1 X is compact: any sequence in X has a limit in X.
2 X is convex: x, y ∈ X, α ∈ [0, 1] ⇒ αx + (1 − α)y ∈ X.
3 ∀x : f(x) is nonempty and convex.
4 For any sequence of pairs (x i , x ∗ i ) such that x i , x ∗ i ∈ X
and x ∗ i ∈ f(x i ), if lim i→∞ (x i , x ∗ i ) = (x, x ∗ ) then
x ∗ ∈ f(x).
Prof. Dr. Jürgen Dix · Department of Informatics, TUC Multiagent Systems, WS 06/07 214/731