On exact capacities

TL;DR

This paper explores the mathematical structure of capacity functions as monotone cooperative games, introducing topological analogues of exact games and analyzing their properties and representations.

Contribution

It defines topological analogues of exact capacities, shows they form subfunctors between known classes, and characterizes them via envelopes of probability measures.

Findings

Exact capacities are envelopes of convex sets of probability measures.

The functor of exact capacities is shown to be open.

The paper introduces strongly exact capacities and discusses their relation.

Abstract

We consider capacity (fuzzy measure, non-additive probability) on a compactum as a monotone cooperative normed game. We introduce topological analogues of well known class of exact games and show that these classes form subfunctors of the capacity functor which lie between known subfunctors of convex capacities and balanced capacities. It is natural to consider probability measures as elements of core of such games. We describe exact capacities as envelopes of the convex closed sets of probability measures. Using such representation we prove openness of the functor of exact capacities. We also consider strongly exact capacities and pose the problem of coincidence of these two classes.