On balanced capacities

TL;DR

This paper explores the properties of balanced capacities, a type of non-additive probability measure, establishing a core non-emptiness criterion and examining their categorical properties and fuzzy integrals.

Contribution

It proves an analogue of the Bondareva-Shapley theorem for capacities and characterizes fuzzy integrals of balanced capacities.

Findings

Core non-emptiness is equivalent to capacity balancedness.

Categorical properties of balanced capacities are characterized.

Fuzzy integrals of balanced capacities are characterized.

Abstract

We consider capacity (fuzzy measure, non-additive probability) on a compactum as a monotone cooperative normed game. Then it is naturally to consider probability measures as elements of core of such game. We prove an analogue of Bondareva-Shapley theorem that non-emptiness of the core is equivalent to balancedness of the capacity. We investigate categorical properties of balanced capacities and give characterizations of some fuzzy integrals of balanced capacities.