Interaction in Choquet inegral model

Proof. Necessity. Assume that for all S, T ∈ 2 N , [ S ⊆ T =⇒ µ(S) ≤ µ(T ) ] .
Therefore ∀S N, ∀i ∈ N \ S, we have µ(S) ≤ µ(S ∪ {i}), since ∀S N, ∀i ∈ N \ S, we
have S S ∪ {i}.
Sufficiency. Suppose that for all S N, ∀i ∈ N \ S, µ(S) ≤ µ(S ∪ {i}).
Let S, T ∈ 2 N such that S ⊆ T , we show that µ(S) ≤ µ(T ).
- If S = T , then µ(S) = µ(T ) ≤ µ(T ).
- If S T , then T \ S = {i 1 , . . . , i r } ≠ ∅, with r ≥ 1. We have:
µ(T ) = µ ( S ∪ (T \ S) )
= µ ( S ∪ {i 1 , . . . , i r } )
≥ µ ( S ∪ {i 1 , . . . , i r−1 } )
.
≥ µ ( S ∪ {i 1 } )
≥ µ(S).
In both cases we have µ(S) ≤ µ(T ).
6. A LP model testing for necessary interaction
This section builds on (Mayag and Bouyssou, 2019). We drop the hypothesis that we only
ask preference information on binary alternatives. We show how to test for the existence
necessary positive and negative interactions on the basis of preference information given
on a subset of X that is not necessarily B g .
Assume that the DM provides a strict preference P and an indifference I relations on a
subset of X. Let A be a subset of at least two criteria. Our approach consists in testing
first, in two steps, the compatibility of this preference information with a general Choquet
integral model, and then, in the third step, the existence of necessary positive or negative
interaction into A.
6.1. The process
Step 1. The following linear program (P L 1 ) models each preference of {P, I} by introducing
two nonnegative slack variables α xy + and αxy − in the corresponding constraint
(Equation (1a) or (1b)). Equation (1c) (resp. (1d) ) ensures the normalization (resp.
monotonicity) of capacity µ. The objective function Z 1 minimizes all the nonnegative
variables introduced in (1a) and (1b).
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