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

TL;DR

This paper introduces a novel approach to quantify attribute interactions in decision making by defining derivatives of functions over lattices, extending existing concepts like the Shapley value.

Contribution

It generalizes the concept of attribute interaction using derivatives on lattice-structured functions, applicable across various decision analysis fields.

Findings

Defines a derivative-based interaction measure on lattices

Generalizes the Shapley value for attribute interaction

Applicable to diverse decision-making contexts

Abstract

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.