A fair procedure in insurance

We consider a random variable R, representing a risk that has to be insured by n companies and which a premium φ is assigned to. How should they share the risk and the premium in order to be better off? Under suitable hypotheses, there exists an optimal decomposition of the risk. According to this division, we develop a procedure that results in an allocation of (φ,R) that is efficient, individually rational and envy-free.