Combinatorial Auctions with Externalities: Basic Properties and Bidding Languages

Although combinatorial auctions have received a great deal of attention from the computer science community over the past decade, research in this domain has focused on settings in which a bidder only has preferences over the bundles of goods they themselves receive, and is indifferent about how other goods are allocated to other bidders. In general, however, bidders in combinatorial auctions will be subject to externalities: they care about how the goods they are not themselves allocated are allocated to others. Our aim in the present paper is to study such combinatorial auctions with externalities from a computational perspective. We first present our formal model, and then develop a classification scheme for the types of externalities that may be exhibited in a bidder’s valuation function. We then develop a bidding language for combinatorial auctions with externalities. The language uses weighted logical formulae to represent bidder valuation functions. We then investigate the properties of this representation: we study the complexity of the winner determination problem, and characterise the complexity of classifying the properties of valuation functions. We then present two approaches to winner determination for our bidding language: an exact approach based on integer linear programming, and approximation methods.

[1]  David H. Reiley,et al.  Northern exposure: a field experiment measuring externalities between search advertisements , 2010, EC '10.

[2]  Arpita Ghosh,et al.  Expressive auctions for externalities in online advertising , 2010, WWW '10.

[3]  Yann Chevaleyre,et al.  Representing Utility Functions via Weighted Goals , 2009, Math. Log. Q..

[4]  David Kempe,et al.  Auctions for Share-Averse Bidders , 2008, WINE.

[5]  Mohammad Mahdian,et al.  A Cascade Model for Externalities in Sponsored Search , 2008, WINE.

[6]  Anna R. Karlin,et al.  On the Equilibria and Efficiency of the GSP Mechanism in Keyword Auctions with Externalities , 2008, WINE.

[7]  Michael Wooldridge,et al.  A Tractable and Expressive Class of Marginal Contribution Nets and Its Applications , 2008, Math. Log. Q..

[8]  Mohammad Mahdian,et al.  Externalities in online advertising , 2008, WWW.

[9]  N. Nisan,et al.  Algorithmic Game Theory , 2007 .

[10]  Tuomas Sandholm,et al.  Expressive commerce and its application to sourcing: how we conducted $35 billion of generalized combinatorial auctions , 2007, AI Mag..

[11]  Yann Chevaleyre,et al.  Expressive Power of Weighted Propositional Formulas for Cardinal Preference Modeling , 2006, KR.

[12]  Acknowledgments , 2006, Molecular and Cellular Endocrinology.

[13]  Oded Schwartz,et al.  On the complexity of approximating k-set packing , 2006, computational complexity.

[14]  Tuomas Sandholm,et al.  Optimal Winner Determination Algorithms , 2005 .

[15]  B. Moldovanu,et al.  Allocative and Informational Externalities in Auctions and Related Mechanisms , 2005 .

[16]  Vincent Conitzer,et al.  Expressive Negotiation in Settings with Externalities , 2005, AAAI.

[17]  Yoav Shoham,et al.  Marginal contribution nets: a compact representation scheme for coalitional games , 2005, EC '05.

[18]  Avrim Blum,et al.  On polynomial-time preference elicitation with value queries , 2003, EC '03.

[19]  Craig Boutilier,et al.  Bidding Languages for Combinatorial Auctions , 2001, IJCAI.

[20]  J. Håstad.  Clique is hard to approximate withinn1−ε , 1999 .

[21]  Johan Håstad,et al.  Clique is hard to approximate within n/sup 1-/spl epsiv// , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[22]  E. Stacchetti,et al.  How (not) to sell nuclear weapons , 1996 .

[23]  Noga Alon,et al.  Derandomized graph products , 1995, computational complexity.

[24]  Michel Gendreau,et al.  Combinatorial auctions , 2007, Ann. Oper. Res..

[25]  Rudolf Müller,et al.  Tractable cases of the winner determination problem , 2006 .

[26]  Salil P. Vadhan,et al.  Computational Complexity , 2005, Encyclopedia of Cryptography and Security.

[27]  Giorgio Gambosi,et al.  Complexity and Approximation , 1999, Springer Berlin Heidelberg.

[28]  Johan Håstad,et al.  Clique is hard to approximate within n1-epsilon , 1996, Electron. Colloquium Comput. Complex..

[29]  Ingo Wegener,et al.  The complexity of Boolean functions , 1987 .

[30]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[31]  David Zuckerman,et al.  Electronic Colloquium on Computational Complexity, Report No. 100 (2005) Linear Degree Extractors and the Inapproximability of MAX CLIQUE and CHROMATIC NUMBER , 2005 .