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

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.

💡 Research Summary

The paper introduces a unified framework for measuring interaction among attributes (or criteria) by extending the classical notion of interaction defined for capacities to functions defined on arbitrary lattices. In traditional cooperative game theory, a capacity (a monotone set function) assigns a value to each subset of criteria, and interaction indices such as the Shapley interaction index quantify the synergy or redundancy between groups of criteria. However, this approach is limited to binary levels (0/1) for each attribute.

To overcome this limitation, the authors model each attribute as a partially ordered set of “reference levels” that capture meaningful qualitative states (e.g., unacceptable, neutral, satisfactory). The Cartesian product of these posets yields a product lattice (L = L_1 \times \dots \times L_n). A real‑valued function (v: L \to \mathbb{R}) assigns an overall score to any combination of reference levels. The paper’s central contribution is the definition of a discrete derivative (or difference) of (v) with respect to a set of attributes (S) (and optionally a set of negative attributes (T)) on this lattice. The basic derivative is \