More on the inclusion exclusion integral: benchmark interaction operators and equivalence concepts

Abstract

[EN]The inclusion-exclusion integral represents one of the numerous extensions of the Choquet integral introduced during the past decade. While other generalizations typically replace arithmetic operations within either the monotone measure-based or Möbius transform-based formulations of the discrete Choquet integral, the inclusion-exclusion integral incorporates a novel element (termed an interaction operator) into the computations. Just like the Möbius transform breaks down the monotone measure into interacting constituents, the interaction operator is responsible for decomposing the inputs, which are thereby distributed among negative and positive interactions too. In this work we revisit fundamental topics of the theory of inclusion-exclusion integrals, such as the properties of benchmark interaction operators, or the existence of unit elements. Then we carefully examine the case of two-component inputs. This case always produces non-negative inclusion–exclusion integrals. With general inputs, belief measures guarantee the production of non-negative inclusion–exclusion integrals, irrespectively of the interaction operator. We define two concepts of equivalence: one derived from the monotone measure, and the other based on the interaction operator. We give sufficient conditions for both properties. Finally, results from the founding literature are generalized. One conclusion is that the class of monotone measures defining inclusion–exclusion integrals that are independent of the interaction operator is characterized by additivity.