Convergence of VCG mechanism to ex-post budget balance in a model of land acquisition

We explore the effect of competition among a large number of sellers in a model of Land Acquisition. Sellers with one unit of land each are located at the nodes of a graph. Two sellers are contiguous if they are connected by an edge in the graph. The buyer realizes a positive value only if he can purchase plots that constitute a path of given minimal length. We characterize conditions on the prior for different graph structures under which the VCG mechanism almost surely results in an ex-post budget surplus as the number of sellers becomes large. Our results show that such convergence depends on (a) the endpoints of the support of valuations and (b) whether the number of critical sellers, i.e., sellers who lie in every feasible path, is preserved as new sellers are added to the underlying graph.