Abstract
This work obtains truthful mechanisms that aim at maximizing both the revenue and the economic efficiency (social welfare) of unit-demand auctions. In a unit-demand auction a set of k items is auctioned to a set of n consumers, and although each consumer bids on all items, no consumer can purchase more than one item.
We present a framework for devising polynomial-time randomized truthful mechanisms that are based on a new variant of the Vickrey-Clarke-Groves (VCG) mechanism. Instead of using reserve prices, this variant of VCG uses the number of objects that we wish to sell as a parameter. Our mechanisms differ in their selection of the number of items to be sold, and allow an interesting trade-off between revenue and economic efficiency, while improving upon the state-of-the-art results for the Unit-Demand Auctions problem (Guruswami et. al.[SODA 2005]).
Our probabilistic results depend on what we call the competitiveness of the auction, i.e., the minimum number of items that need to be sold in order to obtain a certain fraction of the maximum efficiency. We denote by \({\mathcal T}\) the optimal efficiency achieved by the VCG mechanism. Our efficiency-oriented mechanism achieves \(\Omega{(\mathcal T)}\) efficiency and \(\Omega({\mathcal T}/ln(min\{k,n\})\) revenue with probability that grows with the competitiveness of the auction. We also show that no truthful mechanism can obtain an \(\omega({\mathcal T}/ln(min\{k,n\})\) expected revenue on every set of bids. In fact, the revenue-oriented mechanism we present achieves \(\Omega({\mathcal T}/ln(min\{k,n\})\) efficiency and \(\Omega({\mathcal T}/ln(min\{k,n\})\) revenue, but the revenue can actually be much higher, even as large as \(\Omega({\mathcal T})\) for some bid distributions.
This work was partially supported by CNPq under grants 304093/2002-5, 300968/2003-5 and 476323/2004-5, and by FAPERJ under grant E-26/151.494/2005.
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Bornstein, C., Laber, E.S., Mas, M. (2006). On Behalf of the Seller and Society: Bicriteria Mechanisms for Unit-Demand Auctions. In: Correa, J.R., Hevia, A., Kiwi, M. (eds) LATIN 2006: Theoretical Informatics. LATIN 2006. Lecture Notes in Computer Science, vol 3887. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11682462_23
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DOI: https://doi.org/10.1007/11682462_23
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