Axiomatization of an importance index for Generalized Additive Independence models

Transcription

1 Axiomatization of an importance index for Generalized Additive Independence models Mustapha Ridaoui 1, Michel Grabisch 1, Christophe Labreuche 2 1 Université Paris I - Panthéon-Sorbonne, Paris, France 2 Thales Research & Technology, Palaiseau, France ECSQARU July 12, 2017

2 Introduction MultiCriteria Decision Analysis (MCDA) Central question: to be able to quantify the importance of each attribute. Simple models : additive models (additive utility, WAM). Complex models : Choquet integral: importance index = the Shapley value. GAI (Fishburn 1967). Aim Define an importance index for discrete GAI models. Ridaoui, Grabisch, Labreuche Importance index for GAI models 2 / 15

3 Basic concepts N = {1,..., n}, n N in MCDA : attributes, criteria,... in cooperative game theory : players, voters,... X i : set of values of attribute i. X = X 1 X n : set of potential alternatives. Each x X is a vector (x 1,..., x n ) := (x i, x i ). GAI model (Fishburn 1967) U(x) = A S u A (x A ), with S 2 N \ { }, and u A : X A R Ridaoui, Grabisch, Labreuche Importance index for GAI models 3 / 15

4 Basic concepts A game is a function v : 2 N R, v( ) = 0. A capacity is a game v such that : S T implies v(s) v(t ). A Multichoice game (Hsiao and Raghavan, 1990): v : L = {0, 1,..., k} N R, v(0,..., 0) = 0. G(L) := {v : L R, v(0 N ) = 0} k-ary capacity (Grabisch and Labreuche 2003): v G(L) monotonic : x y v(x) v(y). x L, S(x) := {i N x i > 0}, K(x) := {i N x i = k i } Ridaoui, Grabisch, Labreuche Importance index for GAI models 4 / 15

5 Discrete GAI models are multichoice games We consider that attributes are discrete: X i = {x 0 i,..., xk i }. Any alternative x X is mapped to L by ϕ (x i 1 1,..., x in n ) ϕ(x i 1 1,..., x in n ) = (i 1,..., i n ). Given a GAI model U with discrete attributes Assumption : U(0,..., 0) = 0. We define v : L = {0, 1,..., k} N R by U(x) =: v(ϕ(x)), x X, and let v(z) := v(k,..., k) when z L \ ϕ(x). v is a multichoice game. Ridaoui, Grabisch, Labreuche Importance index for GAI models 5 / 15

6 Aim and related work Aim φ : G(L) R N vector of importance of the attributes. k = 1: the standard solution to give an interpretation of the model in terms of importance index : the Shapley value, several generalizations for multichoice games exist, e.g., Hsiao and Raghavan (1990), Peters and Zank (2005), Grabisch and Lange (2007), etc. axioms: linearity, null, symmetry, efficiency: i N φ i(v) = v(k N ), efficiency axiom has no justification in MCDA. to propose a value: for multichoice games, for non monotonone models in MCDA. Ridaoui, Grabisch, Labreuche Importance index for GAI models 6 / 15

7 The preferences are not necessary monotone Example: Level of comfort of humans N = {1, 2, 3}, X 1 = temperature of the air, X 2 = humidity of the air, X 3 = velocity of the air, v measures the comfort level. i N, x i fixed, l i L i, v increasing in x i < l i, decreasing in x i > l i, preferences are single-peaked. v(x 1, x 2, x 3 ) is not monotone in its three arguments. Ridaoui, Grabisch, Labreuche Importance index for GAI models 7 / 15

8 Example: efficiency does not make sense Example: simplest single-peaked function { 1, if x = y, δ y (x) = 0, otherwise. N = {1, 2, 3}, k = 2, y = (2, 1, 1), attribute x 1 has a nonzero importance since δ y (x 1,, ) is non-decreasing in x 1 : 0 = δ y (0, 1, 1) < δ y (2, 1, 1) = 1. However efficiency implies φ i (δ y ) = δ y (2, 2, 2) = 0. i N Ridaoui, Grabisch, Labreuche Importance index for GAI models 8 / 15

9 Axiomatisation: Importance index Linearity (L): φ is linear on G(L). Proposition Under axiom (L), i N, a i x R, x L, φ i (v) = a i xv(x), v G(L). x L i is said to be null for v G(L) if v(x + 1 i ) = v(x), x L, x i < k. Nullity (N): If a criterion i is null for v G(L), φ i (v) = 0. Proposition Under axioms (L) and (N), i N, b i x R, x L, x i < k, φ i (v) = x L x i <k b i x( v(x + 1i ) v(x) ), v G(L). Ridaoui, Grabisch, Labreuche Importance index for GAI models 9 / 15

10 Axiomatisation: Importance index Symmetry (S): σ on N, φ σ(i) (σ v) = φ i (v), i N. Proposition Under axioms (L), (N) and (S), v G(L), i N, φ i (v) = b xi ;n(x i )( v(x + 1i ) v(x) ), x L x i <k b xi ;n(x i ) R, n(x i ) = (n 0,..., n k ), n j = {l N \ {i}, x l = j}. Invariance (I): Let us consider v, w G(L) such that i N, v(x + 1 i ) v(x) = w(x) w(x 1 i ), x L, x i / {0, k}, v(x i, 1 i ) v(x i, 0 i ) = w(x i, k i ) w(x i, k i 1), x i L i, Proposition then φ i (v) = φ i (w). Under axioms (L), (N), (S) and (I), v G(L), i N, φ i (v) = b n(x i )( v(x + 1i ) v(x) ). x L x i <k Ridaoui, Grabisch, Labreuche Importance index for GAI models 10 / 15

11 Axiomatisation: Importance index Restricted Efficiency axiom φ i : overall added value from any x to (x + 1 i ). i N φ i(v) : overall added value when increasing simultaneously the value of all attributes of one unit. v G(L), i N φ i (v) = x L x j <k RE axiom: y L \ {0 N }, φ i (δ y ) = δ y (y i, k i ) δ y (y j, 0 j ), i N where, i = argmax y and j = argmin y. ( v(x + 1) v(x) ). +1, if k(y) 0 and s(y) = n, φ i (δ y ) = 1, if k(y) = 0 and s(y) < n, i N 0, else. Ridaoui, Grabisch, Labreuche Importance index for GAI models 11 / 15

12 Axiomatisation: Importance index Theorem Under axioms (L), (N), (I), (S) and (RE), v G(L), i N φ i (v) = (n s(x i) 1)!k(x i)! ( v(x i, k i ) v(x i, 0 i ) ). (n + k(x i ) s(x i ))! x i L i Average variation along the axis of the attribute. Ridaoui, Grabisch, Labreuche Importance index for GAI models 12 / 15

13 Interpretation of φ in continuous spaces Choquet integral w.r.t. v (Grabisch and Labreuche 2003) z [0, k] N, C v (z) = v(x) + C µx (z x), where, x = z, µ x (S) = v((x + 1) S, x S ) v(t), S N. Lemma For all k-ary capacity v, C v φ i (v) = (z) dz, i N, [0,k] n z i φ i (v) = x {0,...,k 1} N φ Sh i (µ x ), i N. Ridaoui, Grabisch, Labreuche Importance index for GAI models 13 / 15

14 Conclusion and future work Conclusion New importance index for MCDA models : Average variation along the axis of the attribute. The value on i N {x i, x i+1 } = usual Shapley value. φ i on continuous domain is the integrated local importance. Future works Use other values: Banzhaf value. φ i (v) = x L x i <k p i x v(x + 1 i ) v(x), p i x R. Ridaoui, Grabisch, Labreuche Importance index for GAI models 14 / 15

15 Thank you for your attention! Ridaoui, Grabisch, Labreuche Importance index for GAI models 15 / 15