The Choquet integral for 2-additive bi-capacities
Michel Grabisch Christophe Labreuche Universit´e Paris I – Panth´eon-Sorbonne, LIP6 Thales Research & Technology 8 rue du Capitaine Scott, 75015 Paris, France Domaine de Corbeville, 91404 Orsay, France e-mail
[email protected] e-mail
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Abstract Bi-capacities have been presented recently by the authors as a natural generalization of capacities (fuzzy measures). Usual concepts as M¨obius transform, Shapley value and interaction index, Choquet integral, kadditivity can be generalized. We present formulas of the Choquet integral w.r.t. the M¨obius transform, and w.r.t. the interaction index for 2-additive bi-capacities. Keywords: fuzzy measure, bi-capacity, Choquet integral, interaction
Throughout the paper we consider a finite universal set N = {1, . . . , n}. We will often omit braces around elements, writing e.g. S ∪ i, i j, instead of S ∪ {i}, {i, j}.
2
Capacities and Choquet integrals
We recall only necessary material, see e.g. [5] for details. A capacity or fuzzy measure µ on N is a mono/ = 0. µ tonic set function µ : P −→ R+ such that µ(0) is normalized if µ(N) = 1. We consider here normalized capacities. The conjugate of capacity µ is defined as µ(A) := 1 − µ(Ac ). For positive real-valued functions, the Choquet integral of f : N −→ R+ w.r.t. µ is defined by
1
Introduction
The concept of bi-capacity appear as a natural generalization of capacities (or fuzzy measures) in several areas of decision making. For example, in multicriteria decision making, the authors define them as the overall value given to ternary alternatives, that is, taking 1, -1 or 0 as values on criteria [4, 3]. A similar definition has been proposed by Greco et al. [6] under the name bipolar capacity. In cooperative game theory, bi-cooperative games have been proposed by Bilbao et al. [1], based on the notion of ternary voting game [2]. In [4, 3], the authors have laid down the basis for the construction and main concepts around bi-capacities, among them the Choquet integral, the M¨obius transform and the interaction transform. In this paper, we continue along this line, and examine the expression of the Choquet integral w.r.t the M¨obius transform and interaction index. The latter will be done only for the 2-additive case.
n
Cµ ( f ) = ∑ [ f (σ(i)) − f (σ(i − 1))]µ(Aσ(i) ) i=1
with f (σ(1)) ≤ f (σ(2)) ≤ · · · ≤ f (σ(n)). The extension for real-valued functions can be done in 2 ways: the asymmetric integral Cµ (usual Choquet integral), and the symmetric Choquet integral Cˇ µ :
Cµ ( f ) = Cµ ( f + ) − Cµ ( f − ) Cˇ µ ( f ) = Cµ ( f + ) − Cµ ( f − ) where µ is the conjugate capacity, and f + = f ∧ 0, f − = (− f )+ .
3
Bi-capacities
(see [4, 3] for details) We denote Q (N) := {(A, B) ∈ / P (N) × P (N)|A ∩ B = 0}. Definition 1 A function v : Q (N) −→ R is a bicapacity if it satisfies:
/ 0) / =0 (i) v(0,
∀(K, L) ∈ Q (N \ (S ∪ T )).
(ii) A ⊂ B implies v(A, ·) ≤ v(B, ·) and v(·, A) ≥ v(·, B).
The Shapley value for bi-capacities can be defined axiomatically by introducing axioms which are straightforward extensions of the original axioms, plus some additional symmetry axioms. For any i ∈ N, it can be shown that the Shapley value of i for v is:
/ = 1 = −v(0, / N). In addition, v is normalized if v(N, 0) In the sequel, unless otherwise specified, we will consider that bi-capacities are normalized. The bicapacity is asymmetric if v(A, B) = µ(A) − µ(B), and symmetric if v(A, B) = µ(A) − µ(B) for some capacity µ.
∑
0
0
(−1)|A\B|+|B \A | v(B, B0 ).
i (n − s − 1)!s! h v(S∪i, N \(S∪i))−v(S, N \S) . n! S⊂N\i
∑
Note that the term into brackets is simply ∆i v(S, N \ (S ∪ i)). Interaction is defined with respect to elements of Q (N), and extends the notion of Shapley value.
The M¨obius transform of v is expressed by m(A, A0 ) =
φv (i) =
(1)
B⊂A A0 ⊂B0 ⊂Ac
Definition 3 Let (S, T ) ∈ Q (N). The bi-interaction index w.r.t (S, T ) is defined by:
The inverse equation is 0
v(A, A ) =
∑
0
m(B, B ).
I v (S, T ) :=
(2)
B⊂A,B0 ⊃A0
∆S,T v(K, N \ (K ∪ S ∪ T )).
The definition of the M¨obius transform permits us to introduce k-additive bi-capacities. Definition 2 A bi-capacity is said to be k-additive for some 1 ≤ k ≤ n − 1 iff m(A, B) = 0 whenever |B| < n − k. We extend the notion of derivative of a set function to bi-capacities (in fact to any function on Q (N)). As bi-capacities are defined on Q (N), so should be the variables used in derivation. For any i ∈ N, the leftderivative with respect to i of v at point (S, T ) is given by: ∆i,0/ v(S, T ) := v(S∪i, T )−v(S, T ),
(n − s − t − k)!k! (n − s − t + 1)! K⊂N\(S∪T )
∑
∀(S, T ) ∈ Q (N \i).
The Shapley value is recovered by / + I v (0, / i). φv (i) = I v (i, 0)
4
The Choquet integral w.r.t bi-capacities
Definition 4 Let v be a bi-capacity and f be a realvalued function on N. The Choquet integral of f w.r.t v is given by
Cv ( f ) := CµN+ (| f |) where µN + is a real-valued set function on N defined by µN + (C) := v(C ∩ N + ,C ∩ N − ),
Similarly, the right-derivative1 is given by:
and N + := {i ∈ N| fi ≥ 0}, N − = N \ N + .
∆0,i / v(S, T ) := v(S, T )−v(S, T ∪i),
Observe that we have Cv (1A , −1B ) = v(A, B) for any (A, B) ∈ Q (N). This definition amounts to coincide with the one of Greco et al. using bipolar capacities [6].
∀(S, T ) ∈ Q (N \i).
The monotonicity of v entails that these derivatives are non negative. One can also introduce the derivative w.r.t. i by ∆i v(S, T ) := ∆i,0/ v(S, T ) + ∆0,i / v(S, T ).
The following result expresses the Choquet integral in terms of the M¨obius transform.
Higher order derivatives can be defined recursively by
Proposition 1 For any bi-capacity v, any real valued function f on N,
∆S,T v(K, L) := ∆i,0/ (∆S\i,T v(K, L)) = ∆0,i / (∆S,T \i v(K, L)), 1 Our
definition is the opposite in sign of the one proposed in [3]. This impacts Def. 3. The present definition is more natural.
Cv ( f ) =
∑ m(0,/ B)
B⊂N
∑
(A,B)∈Q (N)A6=0/
^
i∈Bc ∩N −
fi +
h m(A, B)
^
i∈(A∪B)c ∩N −
fi +
^
i∈A
i fi ∨ 0
with the convention ∧0/ fi := 0. This result extends the well known result for the classical Choquet integral.
5
The 2-additive model
Proposition 2 For any 2-additive bi-capacity, we have: / m(i j, (i j)c ) = I(i j, 0) c / (i j) ) = I(0, / i j) m(0,
(4)
c
m(i, (i j) ) = I(i, j)
The 2-additive case is of particular interest, since it permits to keep a low complexity while benefiting of interaction. We begin by recalling classical results with 2-additive capacities (see e.g. [5]). 5.1
(3) (5)
/ − m(i, ic ) = I(i, 0)
1 / + I(i, j)] [I(i j, 0) 2∑ j6=i
(6)
/ ic ) = I(0, / i) − m(0,
1 / i j) + I( j, i)] [I(0, 2∑ j6=i
(7)
2-additive capacities Based on this, we obtain the following result.
A 2-additive capacity µ is such that its M¨obius transform m vanishes for subsets of more than 2 elements. Let us denote by Ii j the interaction index of i, j for µ, and φ its Shapley value. The relation between the interaction representation and the M¨obius representation is (omitting braces): m(i j) = Ii j ,
m(i) = φ(i) −
1 Ii j . 2∑ j6=i
Ii j >0
Ii j
