Some Relations among Values, Interactions, and Decomposability of Non-Additive Measures

We will discuss some relations among values, i.e., global/overall importance of each single element of the object to be considered, interaction indices, i.e., simultaneous interaction among the elements in a coalition, and inclusion-exclusion coverings, i.e., decomposability of non-additive measures and/or of the Choquet integral. So, we show that inclusion-exclusion coverings are characterized by the Shapley interaction index. Then, inclusion-exclusion coverings are invariant under the transformations which characterize the Shapley value and/or the Shapley interaction index, and preserve the Shapley value.