K-order representative capacity

We propose a general notion of lower k-order representative capacity that is totally identified by the smallest k-order coefficients in one or another equivalent representation, and the k-additive, k-maxitive and k-tolerant capacities are just its special cases. We further construct its dual notion, the upper k-order representative capacity, which is accordingly determined by n-1 to n-k-order representation coefficients and takes k-minitive and k-intolerant capacities as its concrete instances. Some particular and important cases of the upper and lower k-order representative capacities based on nonadditivity index as well as simultaneous interaction indices are introduced and studied. The identification scheme of lower/upper k-order capacity is also given and illustrated. This general framework allows one to tailor and simplify capacities while matching and representing the decision makers preferences in terms of interaction index or other desired representation of capacity.