[Paper Review] The arity gap of aggregation functions and further extensions

Miguel Couceiro, Erkko Lehtonen+1 more|arXiv (Cornell University)|Mar 10, 2010

Multi-Criteria Decision Making27 references3 citations

TL;DR

This paper provides a complete classification of aggregation functions using the concept of arity gap, offering explicit descriptions of the arity gap for Lovasz extensions of pseudo-Boolean functions and Choquet integrals. It further extends these results to order-preserving functions between arbitrary posets, establishing that similar structural characterizations hold in this broader class, thereby unifying and generalizing prior results in aggregation theory.

ABSTRACT

The aim of this paper is to completely classify all aggregation functions based on the notion of arity gap. We first establish explicit descriptions of the arity gap of the Lovasz extensions of pseudo-Boolean functions and, in particular, of the Choquet integrals. Then we consider the wider class of order-preserving functions between arbitrary, possibly different, posets, and show that similar explicit descriptions still hold for this function class which subsumes that of aggregation functions.

Motivation & Objective

• To fully classify aggregation functions based on the arity gap, a measure of how much a function's output changes when variables are removed.
• To derive explicit descriptions of the arity gap for Lovasz extensions of pseudo-Boolean functions.
• To extend these findings to Choquet integrals, which are fundamental in decision theory and fuzzy measures.
• To generalize the framework to order-preserving functions between arbitrary, possibly different, posets, broadening the scope beyond standard aggregation functions.

Proposed method

• The authors analyze the arity gap by examining the minimal number of variables whose removal reduces the function's output range.
• They derive explicit formulas for the arity gap of Lovasz extensions using the structure of the underlying set functions.
• The analysis leverages properties of pseudo-Boolean functions and their extensions to lattice-ordered structures.
• The framework is extended to order-preserving functions between arbitrary posets by exploiting order-theoretic properties and monotonicity.
• The authors use duality and representation theorems to characterize the arity gap in terms of the function's behavior on minimal and maximal elements of the domain posets.
• They establish that the arity gap remains well-defined and computable under these generalizations, preserving structural insights.

Experimental results

Research questions

• RQ1 What is the exact value and structure of the arity gap for Lovasz extensions of pseudo-Boolean functions?
• RQ2 How does the arity gap behave for Choquet integrals, and can it be explicitly characterized?
• RQ3 Can the notion of arity gap be meaningfully extended beyond standard aggregation functions to order-preserving functions between arbitrary posets?
• RQ4 What structural properties of the function class determine the behavior of the arity gap in the generalized setting?

Key findings

• The arity gap of Lovasz extensions of pseudo-Boolean functions is completely characterized in terms of the minimal number of variables whose removal reduces the function's output range.
• For Choquet integrals, the arity gap is explicitly described using the structure of the underlying capacity and the lattice of subsets.
• The arity gap remains well-defined and computable for order-preserving functions between arbitrary posets, generalizing results from standard aggregation functions.
• The explicit descriptions of the arity gap in the generalized setting preserve the same structural insights as in the classical case, demonstrating robustness across function classes.