Mechanisms with Verification for Any Finite Domain

In this paper we generalize the concept of verification introduced by Nisan and Ronen [STOC 1999]. We assume to have selfish agents with general valuation functions and we study mechanisms with verification for optimization problems with these selfish agents. We provide a technique for designing truthful mechanisms with verification that optimally solve the underlying optimization problem. Our technique can be applied to a rich class of problems that includes, as special cases, utilitarian problems and many others considered in literature for so called one-parameter agents (e.g., the makespan studied by Archer and Tardos [STOC 2001]). Our technique extends the one recently presented by Auletta et al as it works for any finite multi-dimensional valuation domain. No method was known to deal with any domain. As special case we give a different proof (w.r.t. to the one given by Nisan and Ronen) of the existence of exact truthful mechanisms with verification for Scheduling Unrelated Selfish Machines. Furthermore, our technique also applies to the case of compound agents (i.e., agents declaring more than a value). No solution was known for designing mechanisms (with verification) for problems involving such general kind of agents. As an application we provide the first optimal truthful mechanism with verification for Scheduling Unrelated Selfish Compound Machines in which every agent controls more than one (unrelated) machine. This mechanism does not run in polynomial time. We then turn our attention to efficient approximating truthful mechanisms and provide a technique that transforms any approximation algorithm into a mechanism with verification with no significant loss of approximation ratio. This technique works for smooth problems involving compound one-parameter agents. We apply this technique to Scheduling Related Compound Machines problem (i.e, agents control more related machines). If the number of machines is constant then our solution runs in polynomial-time. Finally, we give some considerations on the construction of mechanisms (with verification) for infinite domains.