About partial probabilistic information

Suppose a decision maker (DM) has partial information about certain events of a σ-algebra A belonging to set ε and assesses their likelihood through a capacity v. When is this information probabilistic, i.e. compatible with a probability ? We consider three notions of compatibility with a probability in increasing degree of preciseness. The weakest requires the existence of a probability P on A such that P(E) ≥ v(E) for all E ∈ ε, we then say that v is a probability minorant. A stronger one is to ask that v be a lower probability, that is the infimum of a family of probabilities on A. The strongest notion of compatibility is for v to be an extendable probability, i.e. there exists a probability P on A which coincides with v on A. We give necessary and sufficient conditions on v in each case and, when ε is finite, we provide effective algorithms that check them in a finite number of steps.