Interaction in Choquet inegral model

the above process can be repeated. This will lead to exhibit a capacity in which there are
only positive interactions. Hence, null interactions are never necessary when I = ∅.
Example 1. N = {1, 2, 3}, X = {x 1 , x 2 , x 3 , x 4 }, x 1 = (6, 11, 9), x 2 = (6, 13, 7),
x 3 = (16, 11, 9), x 4 = (16, 13, 7) and P = {(x 4 , x 3 ), (x 2 , x 1 )}.
The ordinal preference information {P, I} is representable by the capacity µ (such that I µ 13 =
0) given by Table 1 and Choquet integral corresponding is given by Table 2.
S {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3}
µ(S) 0 0.5 0 1 0 0.5 1
Table 1: A capacity µ ∈ C Pref such that I µ 13 = 0.
x x 4 x 3 x 2 x 1
( )
C µ u(x) 13 11 9.5 8.5
Table 2: Choquet integral corresponding at previous capacity µ.
The linear order induced by the Choquet integral is x 4 P x 3 P x 2 P x 1 .
We have Ω = {3} and ε = 1 2 × C (
µ u(x 3 ) ) (
− C µ u(x 4 ) )
u σ(1) (x 4 σ(1) ) − u γ(1)(x 3 γ(1) ) = 1 11 − 13
×
2 7 − 9 = 0.5
A capacity β µ ∈ C Pref such that I βµ
13 > 0 and Choquet integral corresponding at β µ are
respectively given by Table 3 and Table 4.
S {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3}
β µ (S) 0 1/3 0 2/3 0 1/3 1
Table 3: A capacity β µ ∈ C Pref such that I βµ
13 > 0.
C β µ
x x 4 x 3 x 2 x 1
( )
u(x) 11 10.33 8.33 7.66
Table 4: Choquet integral corresponding at previous capacity β µ .
Indeed, I βµ
13 = 1
1 + ε Iµ 13 + ε
1 + ε × 1
3 − 2 + 1 = 0.5
1.5 × 1 2 = 1 6 > 0.
Now we will restrict ourselves to the case of generalized binary alternatives.
following two propositions are still in (Kaldjob Kaldjob et al., 2020). We give the proof
again to facilitate the task for the reader. The following proposition gives a necessary and
sufficient condition for an ordinal preference information on B g
to be representable by a Choquet integral model.
10
The
containing no indifference