Extending Characterizations of Truthful Mechanisms from Subdomains to Domains

Abstract

The already extended literature in Combinatorial Auctions, Public Projects and Scheduling demands a more systematic classification of the domains and a clear comparison of the results known. Connecting characterization results for different settings and providing a characterization proof using another characterization result as a black box without having to repeat a tediously similar proof is not only elegant and desirable, but also greatly enhances our intuition and provides a classification of different results and a unified and deeper understanding. We consider whether one can extend a characterization of a subdomain to a domain in a black-box manner. We show that this is possible for n-player stable mechanisms if the only truthful mechanisms for the subdomain are the affine maximizers. We further show that if the characterization of the subdomain involves a combination of affine maximizers and threshold mechanisms, the threshold mechanisms for the subdomain cannot be extended to truthful mechanisms for the union of the subdomain with a (very slight) affine transformation of it. We also show that for every truthful mechanism in a domain there exists a corresponding truthful mechanism for any affine transformation of the domain. We finally plug in as a black box to our theorems the characterization of additive 2-player combinatorial auctions that are decisive and allocate all items (which essentially is the domain for scheduling unrelated machines) and show that the 2-player truthful mechanisms of any domain, which is strictly a superdomain of it are only the affine maximizers. This gives a unique characterization proof of the decisive 2-player mechanisms for: Combinatorial Public Projects, Unrestricted domains, as well as for Submodular and Subadditive Combinatorial Auctions that allocate all items.

This work has been funded by the Vienna Science and Technology Fund WWTF grant ICT10-002.