Ozan Candogan
Pablo Parrilo
Abstract
We study pricing equilibria for graphical valuations, whichare a class of valuations that admit a compact representation. These valuations are associated with a value graph, whose nodes correspond to items, and edges encode (pairwise) complementarities/substitutabilities between items. It is known that for graphical valuations a Walrasian equilibrium (a pricing equilibrium that relies on anonymous item prices) does not exist in general. On the other hand, a pricing equilibrium exists when the seller uses an agent-specific graphical pricing rule that involves prices for each item and markups/discounts for pairs of items. We study the existence of pricing equilibria with simpler pricing rules which either (i) require anonymity (so that prices are identical for all agents) while allowing for pairwise markups/discounts or (ii) involve offering prices only for items. We show that a pricing equilibrium with the latter pricing rule exists if and only if a Walrasian equilibrium exists, whereas the former pricing rule may guarantee the existence of a pricing equilibrium even for graphical valuations that do not admit a Walrasian equilibrium. Interestingly, by exploiting a novel connection between the existence of a pricing equilibrium and the partitioning polytope associated with the underlying graph, we also establish that for simple (series-parallel) value graphs, a pricing equilibrium with anonymous graphical pricing rule exists if and only if a Walrasian equilibrium exists. These equivalence results imply that simpler pricing rules (i) and (ii) do not guarantee the existence of a pricing equilibrium for all graphical valuations.
- I. Abraham, M. Babaioff, S. Dughmi, and T. Roughgarden. 2012. Combinatorial auctions with restricted complements. In Proceedings of the 13th ACM Conference on Electronic Commerce. ACM, 3--16. Google Scholar
Digital Library
- Na An, Wedad Elmaghraby, and Pinar Keskinocak. 2005. Bidding strategies and their impact on revenues in combinatorial auctions. J. Revenue Pricing Manag. 3, 4 (2005), 337--357.Google ScholarCross Ref
- L. M. Ausubel. 2004. An efficient ascending-bid auction for multiple objects. Am. Econ. Revi. 94, 5 (2004), 1452--1475.Google ScholarCross Ref
- L. M. Ausubel. 2006. An efficient dynamic auction for heterogeneous commodities. Am. Econ. Rev. 96, 3 (2006), 602--629.Google ScholarCross Ref
- L. M. Ausubel and P. R. Milgrom. 2002. Ascending auctions with package bidding. BE J. Theor. Econ. 1, 1 (2002), 1--42.Google ScholarCross Ref
- S. Bikhchandani, S. de Vries, J. Schummer, and R. V. Vohra. 2002. Linear programming and Vickrey auctions. In Mathematics of the Internet: E-auction and Markets, B. Dietrich and R. Vohra (Eds.). New York: Springer Verlag, 75--116.Google Scholar
- S. Bikhchandani and J. W. Mamer. 1997. Competitive equilibrium in an exchange economy with indivisibilities. J. Econ. Theor. 74, 2 (1997), 385--413.Google ScholarCross Ref
- S. Bikhchandani and J. M. Ostroy. 2002. The package assignment model. J. Econ. Theor. 107, 2 (2002), 377--406.Google ScholarCross Ref
- Liad Blumrosen and Noam Nisan. 2007. Combinatorial auctions. Cambridge University Press, 267--300.Google Scholar
- Liad Blumrosen and Noam Nisan. 2009. On the computational power of demand queries. SIAM J. Comput. 39, 4 (2009), 1372--1391.Google ScholarDigital Library
- Eric Budish, Gérard P. Cachon, Judd B. Kessler, and Abraham Othman. 2016. Course match: A large-scale implementation of approximate competitive equilibrium from equal incomes for combinatorial allocation. Operations Res. (2016), to appear.Google Scholar
- Eric Budish and Judd B. Kessler. 2016. Bringing Real Market Participants’ Real Preferences into the Lab: An Experiment that Changed the Course Allocation Mechanism at Wharton. Technical Report. National Bureau of Economic Research.Google Scholar
- O. Candogan. 2013. Dynamic Strategic Interactions: Analysis and Mechanism Design. Ph.D. Dissertation. Massachusetts Institute of Technology.Google Scholar
- Ozan Candogan, Asuman Ozdaglar, and Pablo A. Parrilo. 2015. Iterative auction design for tree valuations. Operations Res. 63, 4 (2015), 751--771.Google ScholarDigital Library
- Sunil Chopra. 1994. The graph partitioning polytope on series-parallel and 4-wheel free graphs. SIAM J. Discrete Math. 7, 1 (1994), 16--31. Google ScholarDigital Library
- Vincent Conitzer, Tuomas Sandholm, and Paolo Santi. 2005. Combinatorial auctions with k-wise dependent valuations. In AAAI, Vol. 5. 248--254. Google ScholarDigital Library
- P. Cramton, L. M. Ausubel, R. P. McAfee, and J. McMillan. 1997. Synergies in wireless telephony: Evidence from the broadband PCS auctions. J. Econ. Manag. Strategy 6, 3 (1997), 497--527.Google ScholarCross Ref
- Peter Cramton, Yoav Shoham, and Richard Steinberg. 2006. Combinatorial Auctions. MIT Press. Google ScholarDigital Library
- S. De Vries, J. Schummer, and R. V. Vohra. 2007. On ascending Vickrey auctions for heterogeneous objects. J. Econ. Theor. 132, 1 (2007), 95--118.Google ScholarCross Ref
- S. De Vries and R. V. Vohra. 2003. Combinatorial auctions: A survey. INFORMS J. Comput. 15, 3 (2003), 284--309. Google ScholarDigital Library
- Reinhard Diestel. 2010. Graph Theory. Springer-Verlag, Berlin. Google ScholarDigital Library
- R. J. Duffin. 1965. Topology of series-parallel networks. J. Math. Anal. Appl. 10, 2 (1965), 303--318.Google ScholarCross Ref
- F. Gul and E. Stacchetti. 1999. Walrasian equilibrium with gross substitutes. J. Econ. Theor. 87, 1 (1999), 95--124.Google ScholarCross Ref
- F. Gul and E. Stacchetti. 2000. The English auction with differentiated commodities. J. Econ. Theor. 92, 1 (2000), 66--95.Google ScholarCross Ref
- Tommy R. Jensen and Bjarne Toft. 2011. Graph Coloring Problems, Vol. 39. John Wiley 8 Sons.Google Scholar
- A. S. Kelso and V. P. Crawford. 1982. Job matching, coalition formation, and gross substitutes. Econom.: J. Econom. Soc. (1982), 1483--1504.Google Scholar
- Sébastien Lahaie and David C. Parkes. 2009. Fair package assignment. In Proceedings of the 1st International ICST Conference, AMMA. Springer-Verlag Berlin and Heidelberg.Google Scholar
- B. Lehmann, D. Lehmann, and N. Nisan. 2006. Combinatorial auctions with decreasing marginal utilities. Games Econ. Behav. 55, 2 (2006), 270--296.Google ScholarCross Ref
- Paul Milgrom. 2000. Putting auction theory to work: The simultaneous ascending auction. J. Political Econ. 108, 2 (2000), 245--272.Google ScholarCross Ref
- D. Mishra and D. C. Parkes. 2007. Ascending price Vickrey auctions for general valuations. J. Econ. Theor. 132, 1 (2007), 335--366.Google ScholarCross Ref
- N. Nisan and I. Segal. 2006. The communication requirements of efficient allocations and supporting prices. J. Econ. Theor. 129, 1 (2006), 192--224.Google ScholarCross Ref
- Abraham Othman, Tuomas Sandholm, and Eric Budish. 2010. Finding approximate competitive equilibria: Efficient and fair course allocation. In Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems: Volume 1. International Foundation for Autonomous Agents and Multiagent Systems, 873--880. Google ScholarDigital Library
- D. C. Parkes. 1999. i-Bundle: An efficient ascending price bundle auction. In Proceedings of the First ACM Conference on Electronic Commerce. ACM, 148--157. Google ScholarDigital Library
- D. C. Parkes. 2006. Iterative combinatorial auctions. In Combinatorial Auctions, P. Cramton, Y. Shoham, and R. Steinberg (Eds.). MIT Press, 41--77.Google Scholar
- Neil Robertson and P. D. Seymour. 1986. Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7, 3 (1986), 309--322.Google ScholarCross Ref
- Tim Roughgarden and Inbal Talgam-Cohen. 2015. Why prices need algorithms. In Proceedings of the 16th ACM Conference on Economics and Computation. ACM, New York, NY, 19--36. Google ScholarDigital Library
- N. Sun and Z. Yang. 2006. Equilibria and indivisibilities: Gross substitutes and complements. Econometrica 74, 5 (2006), 1385--1402.Google ScholarCross Ref
- N. Sun and Z. Yang. 2009. A double-track adjustment process for discrete markets with substitutes and complements. Econometrica 77, 3 (2009), 933--952.Google ScholarCross Ref
- Ning Sun and Zaifu Yang. 2014. An efficient and incentive compatible dynamic auction for multiple complements. J. Political Econ. 122, 2 (2014), 422--466.Google ScholarCross Ref
- Rakesh V. Vohra. 2011. Mechanism Design: A Linear Programming Approach. Cambridge University Press.Google Scholar
- H. Zhou, R. A. Berry, M. L. Honig, and R. Vohra. 2009. Complementarities in spectrum markets. In Proceedings of the 47th Annual Allerton Conference on Communication, Control, and Computing. IEEE, 499--506. Google ScholarDigital Library
Index Terms
- Pricing Equilibria and Graphical Valuations
Recommendations
Lottery Pricing Equilibria
EC '16: Proceedings of the 2016 ACM Conference on Economics and ComputationWe extend the notion of Combinatorial Walrasian Equilibrium, as defined by \citet{FGL13}, to settings with budgets. When agents have budgets, the maximum social welfare as traditionally defined is not a suitable benchmark since it is overly optimistic. ...
Optimal pricing for submodular valuations with bounded curvature
AAAI'17: Proceedings of the Thirty-First AAAI Conference on Artificial IntelligenceThe optimal pricing problem is a fundamental problem that arises in combinatorial auctions. Suppose that there is one seller who has indivisible items and multiple buyers who want to purchase a combination of the items. The seller wants to sell his ...
Existence and computation of equilibria of first-price auctions with integral valuations and bids
AAMAS '09: Proceedings of The 8th International Conference on Autonomous Agents and Multiagent Systems - Volume 2We consider existence and computation of symmetric Pure Strategy Nash Equilibrium (PSNE) in single-item, sealed-bid, first-price auctions with integral valuations and bids. For the most general case, we show that existence of PSNE is NP-hard. Then, we ...
Comments