The Power of Lexicographic Maximization: Beyond Cross-Monotonicity∗

In a cost-sharing problem, finitely many players have an unknown preference for some non-rivalrous but excludable good (service). The task is to determine which set of players Q to serve and how to distribute the incurred cost C(Q). Typical constraints of cost-sharing problems demand for mechanisms that elicit truthful bids, balance the budget (i.e., recover the incurred cost with the prices charged), and are economically efficient (i.e., trade off the service cost and the excluded players’ valuations as good as possible). The only known general technique for the design of cost-sharing mechanisms that are truthful in the strong sense, i.e., resilient also against manipulation by coalitions (group-strategyproof, GSP), is due to Moulin (1999). Unfortunately, there are several natural cost-sharing problems for which any Moulin mechanism inevitably suffers poor budget-balance and economic efficiency. In this work, we make first important steps towards new general design techniques for GSP mechanisms: We devise a novel family of polynomial-time computable mechanisms that, for the class of subadditive symmetric cost functions, achieve provably better budget balance than any Moulin mechanism. We apply our findings to the problem of sharing the makespan cost when scheduling jobs on parallel related machines. Towards understanding the limitations imposed by GSP itself, we establish an impossibility result stating that for more than three players, exactly budget-balanced GSP mechanisms do not exist in general, even if the cost function is symmetric. ∗ This work was partially supported by the IST Program of the European Union under contract number IST-15964 (AEOLUS). The first, third, and fourth authors were supported by a fellowship by the International Graduate School of Dynamic Intelligent Systems, Paderborn. A preliminary version [6] of this work appeared in the Proceedings of the 32nd International Symposium on Mathematical Foundation of Computer Science, August 2007.