Axiomatic representations for nonadditivity and nonmodularity indices: Describing interactions of fuzzy measures

Both the nonadditivity index and nonmodularity index have emerged as valuable indicators for characterizing the interaction phenomenon within the realm of fuzzy measures. The axiomatic representation plays a crucial role in distinguishing and elucidating the relationship and distinctions between these two interaction indices. In this paper, we employ a set of fundamental and intuitive properties related to interactions, such as equality, additivity, maximality, and minimality, to establish a comprehensive axiom system that facilitates a clear comprehension of the interaction indices. To clarify the impact of new elements' participation on the type and density of interactions within an initial coalition, we investigate and confirm the existence of proportional and linear effects in relation to null and dummy partnerships, specifically concerning the nonadditivity and nonmodularity indices. Furthermore, we propose the concept of the t-interaction index to depict a finer granularity for the interaction situations within a coalition, which involves subsets at different levels and takes the nonadditivity index and nonmodularity index as special cases. Finally, we establish and discuss the axiomatic theorems and empirical examples of this refined interaction index. In summary, the contributions of this work shed light on the axiomatic characteristics of the t-interaction indices, making it a useful reference for comprehending and selecting appropriate indices within this category of interactions.