Abstract
We study set-system auctions whereby a single buyer wants to purchase Q items of some commodity. There are multiple sellers, each of whom has some known number of items, and a private cost for supplying those items. Thus a “feasible set” of sellers (a set that is able to comprise the winning bidders) is any set of sellers whose total quantity sums to at least Q. We show that, even in a limited special case, VCG has a frugality ratio of at least n − 1 (with respect to the NTUmin benchmark) and that this matches the upper bound for any set-system auction. We show a lower bound on the frugality of any truthful mechanism of \(\sqrt{Q}\) in this setting and give a truthful mechanism with a frugality ratio of \(2\sqrt{Q}\). However, we show that similar types of ‘scaling’ mechanism, in the general (integer) case, give a frugality ratio of at least \({{4Qe^{-2}}\over{{\rm In}^2Q}}\) .
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Goldberg, P.W., McCabe, A. (2012). Commodity Auctions and Frugality Ratios. In: Serna, M. (eds) Algorithmic Game Theory. SAGT 2012. Lecture Notes in Computer Science, vol 7615. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33996-7_16
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DOI: https://doi.org/10.1007/978-3-642-33996-7_16
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