The Serial Cost Sharing Rule was originally conceived for situations where the demands of agents pertain to a homogeneous private good, produced by an unreplicable technology. In this context, it is endowed with a variety of desirable equity and coherency properties. This paper investigates the extension of this rule to the context where agents request many goods that may be public, private or specific to some of them, where the aggregation rule may be very general and where demands may have to be scaled in a non proportional way, more precisely along a path, in order to compute cost shares. It proposes the Path Serial Rule to address these general problems. It then shows which properties and characteristics are satisfied by this rule. Some of them are transposed directly from the single good context to the general one while other must be weakened. More precisely, they are required to hold only on the paths along which demands must be scaled if needed. Nevertheless, some of the characterisations of the serial rule in the single good case do not carry over to the general context.
