On the Structure of the k-Additive Fuzzy Measures

The family of k-additive measures has been introduced as a midterm between probabilities and general fuzzy measures and finds a wide number of applications in practice. However, its structure is different from other families of fuzzy measures and is certainly more complex (for instance, its vertices are not always {0, 1}valued), so it has not been yet fully studied. In this paper we present some results concerning the extreme points of the k-additive fuzzy measures. We give a characterization of these vertices as well as an algorithm to compute them. We show some examples of the results of this algorithm and provide lower bounds on the number of vertices of the n − 1-additive measures, proving that it grows much faster than the number of vertices of the general fuzzy measures. This suggests that k-additive measures might not be a good choice in modeling certain decision problems when the value of k is high but not equal to n. Keywords— Fuzzy measures, k-additive measures, polytope vertices.