Paths and consistency in additive cost sharing

Using a new representation theorem for additive cost sharing methods as sums of path methods, we show that many of the standard additive cost sharing methods (Aumann-Shapley, Shapley Shubik, and Serial Cost) are consistent. These results follow directly from a simple suucient condition for consistency: being generated by associative paths, which can be used to show consistency for many other methods. We introduce a new axiom, dummy consistency, which is quite mild. Nonetheless there is an important relationship between dummy consistency and consistency. For example, we show that all additive cost sharing methods which are dummy consistent and demand monotonic are consistent. Using dummy consistency, we also show that the Aumann-Shapley and Serial Cost methods are the unique (additive) consistent extension of their restriction on all two agent problems, while the Shapley-Shubik method has multiple consistent extensions but a unique symmetric one. In fact, these results are unchanged when we replace consistency with dummy consistency. Our characterization of the set of dummy-consistent cost sharing methods provides a simple framework for analyzing consistent extensions and is useful for constructing nonsymmetric methods.