A core-allocation family for generalized holding cost games

Inventory situations, introduced in Meca et al. (2004), study how a collective of firms can minimize its joint inventory cost by means of cooperation. Depending on the information revealed by the individual firms, they analyze two related cooperative TU games: inventory cost games and holding cost games, and focus on proportional division mechanisms to share the joint cost. It turns out that holding cost games are permutationally concave, hence balanced. Moreover, the proportional rule leads to a core allocation of the corresponding game that can even be sustained as a population monotonic allocation scheme (Sprumont, 1990). In this paper, we extend the class of holding cost games to the generalized holding cost games. We then focus on the study of the core structure for the new class of balanced games. We introduce a core-allocation family which is called N-rational solution family. It is proved that there exists a particular relation of inclusion between the former and the core. In addition, a new proportional rule called minimum square proportional rule, which is an N-rational solution, is studied.