Weak and strong consistency in additive cost sharing

In this paper we study consistency in the context of additive cost sharing mechanisms. We contrast an extremely weak notion of consistency with the standard definition, which we denote strong consistency. First we show that many well known CSMs are consistent in both senses: Aumann-Shapley, random order methods, Shapley-Shubik, Serial cost, and also the weighted versions of these. We also provide general conditions which characterize the different types of consistency - all methods generated by separable paths are weakly consistent, while those generated by associative paths are strongly consistent. Using this characterization, we show that any weakly consistent method which is demand monotonic is also strongly consistent.Next, we analyze the conditions under which a cost sharing method (CSM) for an arbitrary number of agents is uniquely defined by its behavior in the two agent case. We show that under weak (resp. strong) consistency all CSMs generated by a single separable (resp. associative) path are uniquely defined by their behavior on two agent problems. These include Aumann-Shapley, random order methods, Serial cost, and also the weighted versions of these. Shapley-Shubik, which is generated by multiple paths has a unique symmetric extensions, but also other nonsymmetric extensions as do many CSMs which are not generated by a single path.