Truthful Mechanisms for One-Parameter Agents

In this paper, we show how to design truthful (dominant strategy) mechanisms for several combinatorial problems where each agent’s secret data is naturally expressed by a single positive real number. The goal of the mechanisms we consider is to allocate loads placed on the agents, and an agent’s secret data is the cost she incurs per unit load. We give an exact characterization for the algorithms that can be used to design truthful mechanisms for such load balancing problems using appropriate side payments.We use our characterization to design polynomial time truthful mechanisms for several problems in combinatorial optimization to which the celebrated VCG mechanism does not apply. For scheduling related parallel machines (left. Q right|C_{max }), we give a 3-approximation mechanism based on randomized rounding of the optimal fractional solution. This problem is NP-complete, and the standard approximation algorithms (greedy load-balancing or the PTAS) cannot be used in truthful mechanisms. We show our mechanism to be frugal, in that the total payment needed is only a logarithmic factor more than the actual costs incurred by the machines, unless one machine dominates the total processing power. We also give truthful mechanisms for maximum flow, left. Q right|sum {C_j} (scheduling related machines to minimize the sum of completion times), optimizing an affine function over a fixed set, and special cases of uncapacitated facility location. In addition, for left. Q right|sum {w_j } C_j (minimizing the weighted sum of completion times), we prove a lower bound of frac{2}{{sqrt 3 }} for the best approximation ratio achievable by a truthful mechanism.