Study of the Instability of the Sign of the Nonadditivity Index in a Choquet Integral Model

. This paper studies the instability of the sign of the nonaddi-tivity index between criteria in a Choquet integral model. Nonadditivity is an essential property of capacities defined on the sets of decision criteria and allows one to flexibly represent the phenomenon of interaction between criteria. In some cases, we show that the sign of the nonaddi-tivity index proposed in the literature depends on arbitrary choice of a numerical representation in the set of all numerical representations compatible with the strict preferential information given by the Decision Maker(DM). This makes its interpretation difficult. We illustrate ours results with examples.


Introduction
In Multiple Criteria Decision Making (MCDM), the theory of value functions aims to assign a real number to each alternative, so that the order on the alternatives induced by these reals number does not contradict the preferences of the DM.When the preferences of the DM satisfy preferential independence hypothesis, the value assigned to each alternative can be obtained from an additive model [1].Since this hypothesis is restrictive [9], the Choquet integral model, more general, was popularized by the work of Michel Grabisch [5,6].It is now considered as a central tool in MCDM when one wants to escape the independence hypothesis [8][9][10].
When a set of preferential information is not compatible with an additive model, it is common to deduce the existence of interaction between criteria.Interaction among multiple decision criteria can be measured by cardinal probabilistic interaction indices, in particular the Shapley interaction index [7].More details in the literature on axiomatic properties of some cardinal probabilistic interaction indices are given in [8,3].In [15], this lack of compatibility with an additive model is simply translated by the notion of nonadditivity index.This article deals with the use of this idea to capture the phenomena of synergy in the framework of nonadditivity.In [12], we used the Shapley interaction index to study the interaction between criteria in a Choquet integral model.In particular, in [12] we show that the positive interaction is always possible for any subset of criteria, but could not transpose this result with the negative interaction.Here, we solve this dual problem using the nonadditivity index.
We show that the sign of the nonadditivity index [15] proposed in the literature is not stable in the set of all capacities compatible with strict preferences of DM.Indeed, we prove that from a null nonadditivity index, we can build a strictly positive and a strictly negative nonadditivity indices while remaining in the set of all capacities compatible with strict preferences of DM.
Within the framework of binary alternatives, we show that it is always possible to represent the strict preferences of the DM with a Choquet integral model inducing the strictly positive nonadditivity indices, then with another inducing the strictly negative nonadditivity indices for all subsets of at least two criteria.
This paper is organized as follows.Section 2 recalls some basic elements on the model of the Choquet integral in MCDM. in Section 3 and 4, we present ours results.We illustrate each result with an example.

Framework
Let X be a set of alternatives evaluated on an index set of n criteria N = {1, 2, . . ., n} (n ≥ 2).Throughout this paper we use the notation A ⊆ N ≥2 if and only if A ⊆ N and |A| ≥ 2. The set of all alternatives X is assumed to be a subset of a Cartesian product X 1 × X 2 × . . .× X n , where X i is the set of possible levels on criterion i ∈ N .The criteria are recoded numerically using, for all i ∈ N, a function u i from X i into R.Using these functions, we assume that the various recoded criteria are commensurate, so we can use the Choquet integral model [11].

Choquet integral
A generalization of criteria weights consists of assigning weights to subsets of criteria.This can be achieved by a capacity [2] defined as a function µ from the power set 2 N into [0, 1] such that: For an alternative x = (x 1 , . . ., x n ) ∈ X, the expression of the Choquet integral [7][8][9] w.r.t. the capacity µ is given by: where σ is a permutation on N such that: In the next subsection, we recall the definition of the nonadditivity index.

Nonadditivity index
We work with the nonadditivity index introduced and studies in [15].Definition 1.The nonadditivity index w.r.t. a capacity µ is defined by: for all A ⊆ N ≥2 , For all A ⊆ N ≥2 , for each partition (B, A \ B) of A with ∅ ⊊ B ⊊ A, we compute the difference µ(A) − µ(B) + µ(A \ B) .Thus η µ A corresponds to the arithmetic mean of these differences.
Remark 1.We have η µ ij = µ ij − µ i − µ j , therefore the nonadditivity index coincides with the Shapley interaction index I µ ij , for pairs {i, j} ⊆ N .Remark 2 below gives two equivalent expressions of η µ A that we find in [15].Remark 2. Given a capacity µ on N and A ⊆ N ≥2 , Equation ( 1) is equivalent to each of Equations ( 2) and (3).
In the following section, we propose the concept of necessary and possible nonadditivity index similar to that of necessary and possible interaction introduced on [13] in the case of 2-additive Choquet integral model.
3 Necessary and possible nonadditivity index on X

Some definitions and notations
The DM compares some alternatives only in terms of strict preference P define as follow.Definition 2. A strict ordinal preference information P on X is given by: We denote by C Pref the set of all capacities compatible with P .The following definition of necessary and possible nonadditivity will be central in the rest of this text.It is inspired from [13] where it was given in the case of 2-additive Choquet integral model.Definition 4. Let A ⊆ N ≥2 and P a strict ordinal preference information.We say that: 1.There exists a possible positive (resp.null, negative) nonadditivity index between the elements of A if there exists µ ∈ C Pref such that η µ A > 0 (resp. There exists a necessary positive (resp.null, negative) nonadditivity index between the elements of The interpretation of the nonadditivity index is difficult in the case of a possible but not necessary, because it depends on the arbitrary choice of a capacity in C Pref .Indeed, the interpretation of the nonadditivity index really makes sense in the case of the necessary.Our results of the next subsection show that null nonadditivity is not necessary.

Results on X
Proposition 1 shows that from a null nonadditivity index, we can build a strictly positive nonadditivity index while remaining in C Pref .
Proposition 1.Let P be a strict ordinal preference information on X and A ⊆ N ≥2 .Assume that P is representable by a Choquet integral model using a capacity µ for which η µ A = 0. Then there exists a capacity β µ ∈ C Pref such that η β µ A > 0. Proof.Let A ⊆ N ≥2 , we suppose that P is representable by a Choquet integral model using a capacity µ such that η µ A = 0. Let us define a function β µ ε on power set 2 N into [0, 1] by: where ε is a strictly positive real number to be determined as follows.We show that β µ ε is a capacity for all ε > 0. Let ε > 0. It is obvious that µ is a capacity and S ⊆ T .
and µ is a capacity.Hence, In the both cases, we have We then have: We then have Thus for all (a, b) So in both cases we can choose ε = 1 2 min Moreover we have: η Thus there exists a possible positive nonadditivity index for A. Hence there is no null nonadditivity index for A.
Therefore the null nonadditivity index is never necessary.
Proposition 2 answers the dual problem.Indeed, it shows that from a null nonadditivity index, we can build a strictly negative nonadditivity index while remaining in C Pref .Note that this dual problem remains open in the case of the necessary and possible interaction with the Shapley interaction index [7].
Proposition 2. Let P be a strict ordinal preference information on X and A ⊆ N ≥2 .Assume that P is representable by a Choquet integral model using a capacity µ for which η µ A = 0. Then there exists a capacity γ µ ∈ C Pref such that η γ µ A < 0.
Proof.Let A ⊆ N ≥2 , we suppose that P is representable by a Choquet integral model using a capacity µ such that η µ A = 0. Let us define a function γ µ ε on power set 2 N into [0, 1] by: where ε is a strictly positive real number to be determined as follows.We show that γ µ ε is a capacity for all ε > 0. Let ε > 0. It is obvious that γ µ ε (∅) = 0 and γ µ ε (N ) = 1.Let S, T ⊆ N such that S ⊆ T .
Besides, we have We then have So in both cases we can choose ε = 1 2 min Thus there exists a possible negative nonadditivity index for A. Hence there is no null nonadditivity index for A. Therefore null nonadditivity index is never necessary.
The The strict preference P is representable by the capacity µ (with η µ 23 = 0) given by Table 1 and Choquet integral corresponding is given by Table 2.A capacity β µ ε ∈ C Pref such that η βε 23 > 0 and Choquet integral corresponding at β µ ε are respectively given by Table 3 and Table 4. Indeed, η We have so Ω = ∅ and we can choose any ε > 0. We take ε = 1.A capacity γ µ ε ∈ C Pref such that η γε 23 < 0 and Choquet integral corresponding γ µ ε are respectively given by Table 5  Indeed, η In the next section, we define the set of generalized binary alternatives, then we show that on this set, positive and negative nonadditivity index are always possible for all subsets A ⊆ N ≥2 .
4 Necessary and possible nonadditivity index with generalized binary alternatives

Framework of binary alternatives
We assume that the DM can identify two reference levels 0 i and 1 i on each criterion i ∈ N : • the level 0 i in X i is considered as a neutral level and we set u i (0 i ) = 0, • the level 1 i in X i is considered as a good level and we set u i (1 i ) = 1.
For all x = (x 1 , . . ., x n ) ∈ X and S ⊆ N, we will sometimes write u(x) as a shorthand for (u 1 (x 1 ), . . ., u n (x n )) and we define the alternatives a S = (1 S , 0 −S ) ∈ X such that a i = 1 i if i ∈ S and a i = 0 i otherwise.We work on the set B g which we define as follows.
Definition 5. We call the set of generalized binary alternatives, the set: We add to the strict preference P a binary relation M modeling the monotonicity relations between generalized binary alternatives, and allowing us to ensure the satisfaction of the monotonicity condition: S ⊆ T =⇒ µ(S) ≤ µ(T ) .
Definition 6.For all a S , a T ∈ B g , a S M a T if not(a S P a T ) and S ⊇ T .
In the sequel, we need the following basic definition in graph theory [14].

Results on binary alternatives
In [12] we find a necessary and sufficient condition for a strict ordinal preference information on B g to be representable by a Choquet integral model.Under this condition, Proposition 3 below shows that positive nonadditivity index is always possible for all subsets A ⊆ N ≥2 , in a Choquet integral model.In other words, negative and null nonadditivity index are not necessary.
Proposition 3. Let P be a strict ordinal preference information on B g such that (P ∪ M ) containing no strict cycle.Then there exists a capacity µ ∈ C Pref such that η µ A > 0 for all A ⊆ N ≥2 .
We have µ ∈ C Pref .Indeed, if a S P a T , then a S ∈ B q and a T ∈ B r with q > r.
Let A ⊆ N ≥2 and ∅ ⊊ B ⊊ A. There exists q, r, s ∈ {1, 2, . . ., m} such that a A ∈ B q , a B ∈ B r and a A\B ∈ B s with q > r and q > s.
In other words, positive and null nonadditivity index are not necessary.Note that this dual problem remains open in the case of the necessary and possible interaction with the interaction index [7].
Proposition 4. Let P be a strict ordinal preference information on B g such that (P ∪ M ) containing no strict cycle.Then there exists a capacity µ ∈ C Pref such that η µ A < 0 for all A ⊆ N ≥2 .Proof.We construct the partition {B 0 , B 1 , . . ., B m } and define the mapping ϕ as the proof of the Proposition 3 above.Now, the functions f and µ are defined as follows: for all S ⊆ N , for all ℓ ∈ {0, 1, . . ., m}, for all a S ∈ B ℓ , a T ∈ B g such that a S P a T .We show that C µ (u(a S )) > C µ (u(a T )).Since a S , a T ∈ B g and {B 0 , B 1 , . . ., B m } is a partition of B g , then there exists r, q ∈ {0, 1, . . ., m} such that a S ∈ B r , a T ∈ B q .As a S P a T , then r > q.We have since the sequence (f n ) n∈N is strictly increasing, where Let A ⊆ N ≥2 and ∅ ⊊ B ⊊ A. Then there exists q, r, s ∈ {1, 2, . . ., m} such that a A ∈ B q , a B ∈ B r and a A\B ∈ B s with q > r and q > s.Hence The following example illustrates Propositions 3 and 4.
The binary relation (P ∪ M ) contains no strict cycle, so P is representable by a Choquet integral model.A suitable topological sorting on (P ∪ M ) is given by: B 0 = {a 0 }; B We can see that ∀A ⊆ N ≥2 , we have η µ A > 0 and η α A < 0.

Conclusion
This article studies the nonadditivity index in the Choquet integral model.We make a restriction that the DM only gives strict preference information.The capacity to represent this strict preference information is not unique and the sign of the nonadditivity index can vary depending on the arbitrary choice of a capacity compatible with the strict preference of DM.Therefore we introduce the concept of necessary and possible nonadditivity index similar to that of necessary and possible interaction introduced in [13].Only necessary nonadditivity index are robust since their sign and, hence, interpretation, does not vary within the set of all representing capacities.
DM strictly prefers a to b} We note a P b or (a, b) ∈ P .The following definition tests if P is representable by a Choquet integral model.Definition 3. A strict ordinal preference information P on X is representable by a Choquet integral model if we can find a capacity µ such that: for all a, b ∈ X, a P b =⇒ C µ u(a) > C µ u(b) .

Table 8 :
A capacity α ∈ C Pref and the corresponding nonadditivity index.