Exact capacities and star-shaped distorted probabilities

In this work, we are interested in capacities which are deformations of probability i.e v = f ◦ P . We characterize balanced, totally balanced, exact and convex capacities, by properties concerning the probability transformation function, f . We also give the explicit expression, in the case of a convex capacity v = f ◦ P , of a probability in the core of v which coincides with v on a given finite chain of elements of the algebra A. As a dividend of our analysis, we obtain simple new characterizations of a large pattern of risk aversions, relevant to the Yaari rank-dependent expected utility model.