  The concept of Owen point, introduced in Guardiola et al. (2009), is an
appealing solution concept that for Production-Inventory games (PI-games)
always belongs to their core. The Owen point allows all the players in the game
to operate at minimum cost but it does not take into account the cost reduction
induced by essential players over their followers (fans). Thus, it may be seen
as an altruistic allocation for essential players what can be criticized. The
aim this paper is two-fold: to study the structure and complexity of the core
of PI-games and to introduce new core allocations for PI-games improving the
weaknesses of the Owen point. Regarding the first goal, we advance further on
the analysis of PI-games and we analyze its core structure and algorithmic
complexity. Specifically, we prove that the number of extreme points of the
core of PI-games is exponential on the number of players. On the other hand, we
propose and characterize a new core-allocation, the Omega point, which
compensates the essential players for their role on reducing the costs of their
fans. Moreover, we define another solution concept, the Quid Pro Quo set
(QPQ-set) of allocations, which is based on the Owen and Omega points. Among
all the allocations in this set, we emphasize what we call the Solomonic QPQ
allocation and we provide some necessary conditions for the coincidence of that
allocation with the Shapley value and the Nucleolus.
