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.
Lok, R. B., Romero Morales, D., & Vermeulen, D. (2007). Note on the Applicability of the VCG Mechanism to Capacitated Assignment Problems and Extensions. Statistica Neerlandica, 61(1), 156–171. https://doi.org/10.1111/j.1467-9574.2007.00356.x