Discrete Choquet integral and some of its symmetric extensions

Classical extensions of the Choquet integral (defined on [0,1]) to [−1,1] are the asymmetric and the symmetric Choquet integral, the second one being called also the Šipoš integral. Recently, the balancing Choquet integral was introduced as another kind of a symmetric extension of the discrete Choquet integral. We introduce and discuss a new type of such extension, the fusion Choquet integral, and discuss its properties and relationship to the balancing and the symmetric Choquet integral. The symmetric maximum introduced by Grabisch is shown to be a special case of the fusion and the balancing Choquet integral. Several extensions of OWA operators are also discussed.