Revisiting Monotonicity of the Inclusion–Exclusion Integral: The Case of the Arithmetic–Mean Interaction Operator

Abstract

[EN]The inclusion–exclusion (IE) integral is a flexible aggregation framework that extends classical nonlinear integrals by combining monotone measures with interaction operators. While monotonicity with respect to the input function is a fundamental requirement for aggregation models, a complete characterization of monotonicity for the IE integral is still lacking, and existing results rely on restrictive axioms imposed on the interaction operator. In order to analyze monotonicity in a precise manner, we consider interaction operators under mild assumptions, including non-negativity, and impose further structural conditions only when required. Within this framework, we systematically distinguish three types of monotonicity for the IE integral, namely monotonicity with respect to the input function, the underlying measure, and the interaction operator. We then revisit monotonicity of the IE integral by focusing on the interaction operator given by the arithmetic mean. Although this operator does not satisfy the classical sufficient conditions for monotonicity, we show that the corresponding IE integral nevertheless preserves input monotonicity, thereby demonstrating that these conditions are not necessary. We further prove that the arithmetic–mean IE integral unexpectedly admits an equivalent linear representation, whose coefficients coincide with the Shapley values of the underlying monotone measure. These results demonstrate that the known sufficient conditions for monotonicity of the IE integral are not necessary and can be substantially weakened, and they provide new insights into the structural origins of monotonicity in inclusion–exclusion based aggregation models.