Exact capacities and star-shaped distorted probabilities

We are interested in capacities which are deformations of probability, i.e. v=f∘Pv=f∘P. We characterize balanced, totally balanced, and exact capacities by properties concerning the probability transformation function, ff. These results allow us to obtain simple new characterizations of a large pattern of risk aversions relevant to Yaari’s dual theory of choice under risk.