Computing optimal outcomes under an expressive representation of settings with externalities

When a decision must be made based on the preferences of multiple agents, and the space of possible outcomes is combinatorial in nature, it becomes necessary to think about how preferences should be represented, and how this affects the complexity of finding an optimal (or at least a good) outcome. We study settings with externalities, where each agent controls one or more variables, and how these variables are set affects not only the agent herself, but also potentially the other agents. For example, one agent may decide to reduce her pollution, which will come at a cost to herself, but will result in a benefit for all other agents. We formalize how to represent such domains and show that in a number of key special cases, it is NP-complete to determine whether there exists a nontrivial feasible solution (and therefore the maximum social welfare is completely inapproximable). However, for one important special case, we give an algorithm that converges to the solution with the maximal concession by each agent (in a linear number of rounds for utility functions that additively decompose into piecewise constant functions). Maximizing social welfare, however, remains NP-hard even in this setting. We also demonstrate a special case that can be solved in polynomial time using linear programming.

[1]  William Vickrey,et al.  Counterspeculation, Auctions, And Competitive Sealed Tenders , 1961 .

[2]  Tuomas Sandholm Expressive Commerce and Its Application to Sourcing: How We Conducted $35 Billion of Generalized Combinatorial Auctions , 2007, AI Mag..

[3]  E. H. Clarke Multipart pricing of public goods , 1971 .

[4]  Yoav Shoham,et al.  Combinatorial Auctions , 2005, Encyclopedia of Wireless Networks.

[5]  Yann Chevaleyre,et al.  Multiagent Resource Allocation with K -additive Utility Functions , 2004 .

[6]  Jon Feldman,et al.  Sponsored Search Auctions with Markovian Users , 2008, WINE.

[7]  Georg Gottlob,et al.  Combinatorial auctions with tractable winner determination , 2007, SECO.

[8]  Ronald M. Harstad,et al.  Computationally Manageable Combinational Auctions , 1998 .

[9]  D. Parkes,et al.  Optimize-and-Dispatch Architecture for Expressive Ad Auctions , 2005 .

[10]  Ulrich Endriss,et al.  Bidding Languages and Winner Determination for Mixed Multi-unit Combinatorial Auctions , 2007, EUMAS.

[11]  Tuomas Sandholm,et al.  Preference elicitation in combinatorial auctions , 2002, EC '01.

[12]  Vincent Conitzer,et al.  Combinatorial Auctions with k-wise Dependent Valuations , 2005, AAAI.

[13]  E. Stacchetti,et al.  Multidimensional Mechanism Design for Auctions with Externalities , 1999 .

[14]  Theodore Groves,et al.  Incentives in Teams , 1973 .

[15]  A. Mas-Colell,et al.  Microeconomic Theory , 1995 .

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

[17]  Norman M. Sadeh,et al.  A Theory of Expressiveness in Mechanisms , 2008, AAAI.

[18]  David C. Parkes,et al.  On Expressing Value Externalities in Position Auctions , 2011, AAAI.

[19]  Georg Gottlob,et al.  On the complexity of combinatorial auctions: structured item graphs and hypertree decomposition , 2007, EC '07.

[20]  Tuomas Sandholm,et al.  Algorithm for optimal winner determination in combinatorial auctions , 2002, Artif. Intell..

[21]  David Levine,et al.  Winner determination in combinatorial auction generalizations , 2002, AAMAS '02.

[22]  Vincent Conitzer,et al.  Expressive negotiation over donations to charities , 2004, EC '04.

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

[24]  David C. Parkes,et al.  ICE: an iterative combinatorial exchange , 2005, EC '05.

[25]  Vincent Conitzer,et al.  Making decisions based on the preferences of multiple agents , 2010, CACM.

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

[27]  Alex Kulesza,et al.  TBBL: A Tree-Based Bidding Language for Iterative Combinatorial Exchanges , 2005 .

[28]  Tuomas Sandholm,et al.  Preference elicitation in combinatorial auctions , 2001, AAMAS '02.

[29]  Tomasz P. Michalak,et al.  Combinatorial Auctions with Externalities (Extended Abstract) , 2010 .

[30]  David Levine,et al.  CABOB: A Fast Optimal Algorithm for Winner Determination in Combinatorial Auctions , 2005, Manag. Sci..

[31]  Vincent Conitzer,et al.  Expressive markets for donating to charities , 2011, Artif. Intell..

[32]  M. Satterthwaite,et al.  Efficient Mechanisms for Bilateral Trading , 1983 .

[33]  Vincent Conitzer,et al.  Combinatorial Auctions with Structured Item Graphs , 2004, AAAI.

[34]  Tomasz P. Michalak,et al.  Combinatorial auctions with externalities , 2010, AAMAS.

[35]  David C. Parkes,et al.  Achieving Budget-Balance with Vickrey-Based Payment Schemes in Exchanges , 2001, IJCAI.

[36]  David C. Parkes,et al.  Iterative Combinatorial Auctions , 2006 .

[37]  Mohammad Mahdian,et al.  Charity auctions on social networks , 2008, SODA '08.

[38]  Yoav Shoham,et al.  Taming the Computational Complexity of Combinatorial Auctions: Optimal and Approximate Approaches , 1999, IJCAI.

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

[40]  Nicole Immorlica,et al.  Externalities in Keyword Auctions: An Empirical and Theoretical Assessment , 2009, WINE.

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

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

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