Interaction in Choquet inegral model

Let B i = {x ∈ A : A ∈ B ′ i} for all i = 0, 1, . . . , m. Therefore {B 0 , B 1 , . . . , B m } is a
suitable topological sorting on graph (B g , P ∪ M ∪ I) since {B ′ 0, B ′ 1, . . . , B ′ m} is a suitable
topological sorting on graph (B ′ , P ′ ∪ M ′ ).
Let us define the mapping f : B g −→ { R and µ : 2 N −→ [0, 1] as follows: for l ∈
{0, 1, . . . , m}, ∀x ∈ B l , f(φ(x)) =
0 if l = 0,
(2n) l if l ∈ {1, 2, . . . , m}.
µ(S) = f S
α
f S = f(φ(a S )) and α = f N = (2n) m .
Let a S , a T ∈ B g .
• Assume that a S Ia T , therefore there exists A ∈ B ′ such that a S , a T ∈ A. Since A ∈ B ′ ,
thus there exists l ∈ {0, 1, . . . , m} such that A ∈ B ′ l , then we have a S, a T ∈ B l .
– If l = 0, therefore C µ (u(a S )) = 0 = C µ (u(a T )).
– If l ∈ {1, . . . , m}, therefore C µ (u(a S )) = (2n)l
α = C µ(u(a T )).
In both cases, we have C µ (u(a S )) = C µ (u(a T )).
• Assume that a S P a T , then there exists A ∈ B ′ r, C ∈ B ′ q, such that a S ∈ A, a T ∈ C
with r, q ∈ {0, 1, ..., m} and r > q. We then have C µ (u(a S )) = µ(S) = f S
α = (2n)r
α .
– If q = 0, therefore C µ (u(a T )) = µ(T ) = 0 < (2n)r
α = µ(S) = C µ(u(a S )).
– If q ∈ {1, . . . , m}, therefore C µ (u(a T )) = µ(T ) = f T
α = (2n)q . Since r > q,
α
therefore (2n) r > (2n) q , then (2n)r
α
> (2n)q
α , i.e., C µ(u(a S )) > C µ (u(a T )).
Thus {P, I} is representable by C µ .
Remark 6. In the case where I = ∅, the previous Proposition 4 say that: {P, I} is
representable by a Choquet integral model if and only if the binary relation (P ∪ M)
contains no strict cycle. Which coincides with proposition 2, therefore Proposition 4 is a
generalization of Proposition 2.
The following proposition gives a sufficient condition on {P, I} such that positive
interaction is always possible within all subsets of criteria in general Choquet integral
model.
Proposition 5. Let {P, I} be an ordinal preference information on B g
such that (P ∪
M ∪ I) containing no strict cycle. If not(a S T C I∪M a N ) for all S N, then there exists a
capacity µ such that C µ represent {P, I} and I µ A
> 0 for all A ⊆ N such that |A| ≥ 2.
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