Interaction in Choquet inegral model

(
)]
ε u σ(1) (x i+1
σ(1) ) − u γ(1)(x i γ(1) ) ∀i = 1, . . . , p − 1.
(
We are looking for ε such that C βε u(x i+1 ) ) (
− C βε u(x i ) ) > 0 for all i = 1, . . . , p − 1.
(
C βε u(x i+1 ) ) (
− C βε u(x i ) ) (
) (
> 0 ⇐⇒ ε u σ(1) (x i+1
σ(1) ) − u γ(1)(x i γ(1) ) (
> − C µ u(x i+1 ) ) −
C µ
(
u(x i ) )) .
We consider the set Ω = {i = 1, . . . , p − 1 : u σ(1) (x i+1
σ(1) ) − u γ(1)(x i γ(1) ) < 0}.
• If Ω = ∅, then for all i = 1, . . . , p − 1, we have u σ(1) (x i+1
σ(1) ) − u γ(1)(x i γ(1)
( ) ≥ 0.
Thus for all i = 1, . . . , p − 1, C βε u(x i+1 ) ) (
− C βε u(x i ) ) > 0 ∀ε > 0.
( (
Cµ u(x i ) ) (
− C µ u(x i+1 ) ) )
• If Ω ≠ ∅, we choose ε such that 0 < ε < min
i∈Ω u σ(1) (x i+1
σ(1) ) − u γ(1)(x i γ(1) ) in such a
(
way that C βε u(x i+1 ) ) (
− C βε u(x i ) ) > 0 for all i = 1, . . . , p − 1.
So in both cases we can choose ε = 1 ( (
2 min
Cµ u(x i ) ) (
− C µ u(x i+1 ) ) )
i∈Ω u σ(1) (x i+1
σ(1) ) − u γ(1)(x i γ(1) ) such that the
information {P, I} is representable by the Choquet integral model C βε .
Moreover we have:
I βε
A = ∑
=
K⊆N\A
(n − a)!
(n − a + 1)!
(n − k − a)!k!
(n − a + 1)!
∑
(−1) a−l β ε (K ∪ L)
L⊆A
∑ ( )
(−1) a−l β ε (N \ A) ∪ L +
L⊆A
∑
KN\A
(n − k − a)!k!
(n − a + 1)!
(n − a)!
=
(n − a + 1)! β (n − a)! ∑ ( )
ε(N) + (−1) a−l β ε (N \ A) ∪ L +
(n − a + 1)!
∑
KN\A
=
(n − k − a)!k!
(n − a + 1)!
LA
∑
(−1) a−l β ε (K ∪ L)
L⊆A
(n − a)!
(n − a + 1)! + 1
1 + ε
1
1 + ε
∑
KN\A
= 1
1 + ε Iµ A + ε
1 + ε
(n − k − a)!k!
(n − a + 1)!
(n − a)! ∑
(−1) a−l µ ( (N \ A) ∪ L ) +
(n − a + 1)!
LA
∑
(−1) a−l µ(K ∪ L)
L⊆A
1
n − a + 1
∑
(−1) a−l β ε (K ∪ L)
As I µ A
= 0, we have Iβε
A = ε 1
1 + ε n − a + 1 > 0.
Thus there exists a possible positive interaction into A. Hence there is no null interaction
into A.
Note that, if the modified capacity shows a possible null interaction into a set of criteria A,
L⊆A
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