Shapley values and tolerance indices of the operators obtained with the Crescent Method

Abstract

Several operators have emerged in the framework of Choquet integral with the purpose of simultaneously generalizing weighted means and ordered weighted averaging (OWA) operators. However, on many occasions, not enough attention has been paid to whether the constructed operators behaved similarly to the weighted means and OWA operators that have been generalized. In this sense, it seems necessary that these new operators preserve the weights assigned to the information sources (which are established through the weighting vector associated with the weighted mean) and that they are able to rule out extreme values (which is an important characteristic of OWA operators).

In this paper we analyze a family of operators recently introduced in the literature through the Crescent Method. First, we introduce a broad class of weighting vectors that allow us to guarantee that the games generated with the Crescent Method are capacities. Next we analyze the conjunctive/disjuntive character of the Choquet integrals associated with these capacities and we also give closed-form expressions of some tolerance and importance indices such as $k$-conjunctiveness/disjunctiveness indices, the veto and favor indices, and the Shapley values. Finally, we give two examples to show the usefulness of the results obtained.