VCG Overpayment in Random Graphs

Motivated by the increasing need to price networks, we study the prices resulting from of a variant of the VCG mechanism, specifically defined for networks [11]. This VCG mechanism is the unique efficient and strategyproof mechanism, however it is not budget-balanced and in fact it is known to result in arbitrarily bad overpayments for some graphs [1, 8]. In contrast, we study more common types of graphs and show that the VCG overpayment is not too high, so it is still an attractive pricing candidate. We prove that the average overpayment in ErdősRenyi random graphs with unit costs is p/(2 − p) for any n, when the average degree is higher than a given threshold. Our simulations show that the overpayment is greater than p/(2 − p) below this threshold, hence, together with the constant upper bound from Mihail et al. [12], the overpayment is constant regardless of graph size. We then find empirically that the complete graph with random edge costs has worse overpayments, higher than Θ((log n)). Again from simulations, we see that the power-law graph with unit costs has overpayments that decrease with graph size and finally, the power-law graph with uniformly random costs has a small constant overpayment.