In this paper we develop a unified framework for the study of additive cost sharing methods. We show that any additive cost sharing method satisfying the dummy axiom can be generated by a (possibly infinite) convex combination of path generated methods. We also show that the set of scale invariant cost sharing methods can be generated by the set of scale invariant paths and the set of demand monotonic methods by the set of demand monotonic paths, both of which we construct. We first apply these results to the study a strict version of marginality, and show that none of the standard methods satisfy this requirement. We construct two new methods, which are generated by infinite sums of paths, and show that these satisfy strict marginality. We then note that the minimum of any concave functional over the set of cost sharing methods, either general, scale invariant, or demand monotonic, must be path generated, and therefore can be computed using techniques from the theory of optimal control. This allows us to provide a new characterization of the Random Order methods as the methods which minimize a lexicographic function of agents' payments according for supermodular cost functions. It may also lead to new characterizations of other interesting methods.
