Pre-print, Working paper: In coalitional games with side payments, the core predicts which coalitions form and how benefits are shared. The predictions however run into difficulties if the core is empty or if some coalitions benefit from not blocking truthfully. These difficulties are analyzed in games in which an a priori given collection of coalitions can form, as the collection of pairs of buyer-seller in an assignment game. The incentive properties of the core and of its selections are investigated in function of the collection. Furthermore the relationships with Vickrey-Clarke-Groves mechanisms are drawn.