Frugal Mechanism Design via Spectral Techniques
Abstract
We study the design of truthful mechanisms for set systems, i.e., scenarios where a customer needs to hire a team of agents to perform a complex task. In this setting, frugality [Archer&Tardos'02] provides a measure to evaluate the "cost of truthfulness", that is, the overpayment of a truthful mechanism relative to the "fair" payment. We propose a uniform scheme for designing frugal truthful mechanisms for general set systems. Our scheme is based on scaling the agents' bids using the eigenvector of a matrix that encodes the interdependencies between the agents. We demonstrate that the routofksystem mechanism and the \sqrtmechanism for buying a path in a graph [Karlin et. al'05] can be viewed as instantiations of our scheme. We then apply our scheme to two other classes of set systems, namely, vertex cover systems and kpath systems, in which a customer needs to purchase k edgedisjoint sourcesink paths. For both settings, we bound the frugality of our mechanism in terms of the largest eigenvalue of the respective interdependency matrix. We show that our mechanism is optimal for a large subclass of vertex cover systems satisfying a simple local sparsity condition. For kpath systems, while our mechanism is within a factor of k + 1 from optimal, we show that it is, in fact, optimal, when one uses a modified definition of frugality proposed in [Elkind et al.'07]. Our lower bound argument combines spectral techniques and Young's inequality, and is applicable to all set systems. As both routofk systems and single path systems can be viewed as special cases of kpath systems, our result improves the lower bounds of [Karlin et al.'05] and answers several open questions proposed in that paper.
 Publication:

arXiv eprints
 Pub Date:
 December 2009
 arXiv:
 arXiv:0912.3403
 Bibcode:
 2009arXiv0912.3403C
 Keywords:

 Computer Science  Computer Science and Game Theory
 EPrint:
 Proceedings of FOCS 2010