Derivative of functions over lattices as a basis for the notion of interaction between attributes

Michel Grabisch (1) , Christophe Labreuche (2)

1 CES - Centre d'économie de la Sorbonne
2 Thales Research and Technology [Palaiseau]

The paper proposes a general notion of interaction between attributes, which can be applied to many fields in decision making and data analysis. It generalizes the notion of interaction defined for criteria modelled by capacities, by considering functions defined on lattices. For a given problem, the lattice contains for each attribute the partially ordered set of remarkable points or levels. The interaction is based on the notion of derivative of a function defined on a lattice, and appears as a generalization of the Shapley value or other probabilistic values.

Mots clés: interaction index; Shapley value; capacity; game; lattice; discrete derivative

Domaines: Mathématique discrète [cs.DM]; Informatique et théorie des jeux [cs.GT]

Michel Grabisch, Christophe Labreuche. Derivative of functions over lattices as a basis for the notion of interaction between attributes. Annals of Mathematics and Artificial Intelligence, 2007, 49 (1-4), pp.151-170. ⟨10.1007/s10472-007-9052-7⟩. ⟨hal-00187172⟩