Note on the applicability of the VCG mechanism to capacitated assignment problems and extensions

For the allocation of heterogeneous items, it is known that the buyers-are-substitutes condition is necessary and sufficient to ensure that a pricing equilibrium can yield the same allocation and payments as the vcg mechanism. Furthermore, concavity of the corresponding transferable utility tu-game guarantees that this vcg outcome can also be achieved by an ascending price auction. We show that concavity, and hence the buyers-are-substitutes condition, holds for the tu-game of the assignment problem with general capacities. Therefore, the vcg mechanism is supported by a pricing equilibrium which can also be achieved by an ascending auction. We also show that the buyers-are-substitutes condition, and hence concavity, does not hold anymore for very natural and straightforward extensions of this problem. This shows that the necessity of the substitutes property is a considerable restriction on the applicability of the vcg mechanism.