Abstract
We consider computationally-efficient truthful mechanisms that use the VCG payment scheme, and study how well they can approximate the social welfare in auction settings. We present a novel technique for setting lower bounds on the approximation ratio of this type of mechanisms. Our technique is based on setting lower bounds on the communication complexity by analyzing combinatorial properties of the algorithms. Specifically, for combinatorial auctions among submodular (and thus also subadditive) bidders we prove an \(\Omega \left( {m^{\tfrac{1} {6}} } \right)\) lower bound, which is close to the known upper bound of \({\rm O}\left( {m^{\tfrac{1} {2}} } \right)\), and qualitatively higher than the constant factor approximation possible from a purely computational point of view.
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References
A. Archer, C. Papadimitriou, K. Talwar and E. Tardos: An approximate truthful mechanism for combinatorial auctions with single parameter agent, in: SODA’03.
Sanjeev Arora: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems, Journal of the ACM 45(5) (1998), 753–782.
Moshe Babaioff and Liad Blumrosen: Computationally-feasible auctions for convex bundles, in: 7th. International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX). LNCS Vol. 3122. (2004), 27–38.
Yair Bartal, Rica Gonen and Noam Nisan: Incentive compatible multi unit combinatorial auctions, in: TARK 03.
Liad Blumrosen and Shahar Dobzinski: Welfare maximization in congestion games, in: IEEE Journal on Selected Areas in Communications, Preliminary version in EC’06.
Liad Blumrosen and Noam Nisan: Combinatorial Auctions (a survey), in: Algorithmic Game Theory, N. Nisan, T. Roughgarden, E. Tardos and V. Vazirani, editors.
E. H. Clarke: Multipart pricing of public goods, Public Choice (1971), 17–33.
Peerapong Dhangwatnotai, Shahar Dobzinski, Shaddin Dughmi and Tim Roughgarden: Truthful approximation schemes for single-parameter agents, in: FOCS’08.
Shahar Dobzinski and Noam Nisan: Limitations of vcg-based mechanisms, preliminary version in: STOC’07.
Shahar Dobzinski and Noam Nisan: Mechanisms for multi-unit auctions, in: EC’07.
Shahar Dobzinski, Noam Nisan and Michael Schapira: Approximation algorithms for combinatorial auctions with complement-free bidders, in: STOC’05.
Shahar Dobzinski, Noam Nisan and Michael Schapira: Truthful randomized mechanisms for combinatorial auctions, in: STOC’06.
Shahar Dobzinski and Michael Schapira: An improved approximation algorithm for combinatorial auctions with submodular bidders, in: SODA’06.
Uriel Feige: On maximizing welfare where the utility functions are subadditive, in: STOC’06.
Uriel Feige and Jan Vondrak: Approximation algorithms for allocation problems: Improving the factor of 1−1/e, in: FOCS’06.
T. Groves: Incentives in teams, Econometrica (1073), 617–631.
Ron Holzman, Noa Kfir-Dahav, Dov Monderer and Moshe Tennenholtz: Bundling equilibrium in combinatrial auctions, Games and Economic Behavior 47 (2004), 104–123.
Subhash Khot, Richard J. Lipton, Evangelos Markakis and Aranyak Mehta: Inapproximability results for combinatorial auctions with submodular utility functions, in: WINE’05, 2005.
Eyal Kushilevitz and Noam Nisan: Communication Complexity, Cambridge University Press, 1997.
Ron Lavi, Ahuva Mu’alem and Noam Nisan: Towards a characterization of truthful combinatorial auctions, in: FOCS’03.
Ron Lavi and Chaitanya Swamy: Truthful and near-optimal mechanism design via linear programming, in: FOCS 2005.
Benny Lehmann, Daniel Lehmann and Noam Nisan: Combinatorial auctions with decreasing marginal utilities, in: EC’01.
Daniel Lehmann, Liadan Ita O’Callaghan and Yoav Shoham: Truth revelation in approximately efficient combinatorial auctions, in: JACM 49(5) (Sept. 2002), 577–602.
A. Mas-Collel, W. Whinston and J. Green: Microeconomic Theory, Oxford university press, 1995.
Ahuva Mu’alem and Noam Nisan: Truthful approximation mechanisms for restricted combinatorial auctions, in: AAAI-02, 2002.
Noam Nisan: The communication complexity of approximate set packing and covering, in: ICALP, 2002.
Noam Nisan: Introduction to Mechanism Design (for Computer Scientists), in: Algorithmic Game Theory, N. Nisan, T. Roughgarden, E. Tardos and V. Vazirani, editors, 2007.
Noam Nisan and Amir Ronen: Computationally feasible vcg-based mechanisms, in: EC’00.
Noam Nisan and Ilya Segal: The communication requirements of efficient allocations and supporting prices, in: Journal of Economic Theory, 2006.
Kevin Roberts: The characterization of implementable choise rules, in: Jean-Jacques Laffont, editor, Aggregation and Revelation of Preferences. Papers presented at the first European Summer Workshop of the Economic Society, 321–349. North-Holland, 1979.
Tuomas Sandholm: Algorithm for optimal winner determination in combinatorial auctions, in: Artificial Intelligence 135 (2002), 1–54.
Vijay V. Vazirani: Approximation algorithms, Springer-Verlag New York, Inc., New York, NY, USA, 2001.
W. Vickrey: Counterspeculation, auctions and competitive sealed tenders, Journal of Finance (1961), 8–37.
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Dobzinski, S., Nisan, N. Limitations of VCG-based mechanisms. Combinatorica 31, 379–396 (2011). https://doi.org/10.1007/s00493-011-2528-4
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DOI: https://doi.org/10.1007/s00493-011-2528-4