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Optimal Auction in a Multidimensional World

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  • Charles Zheng

    (Northwestern University)

Abstract

A long-standing unsolved problem, often arising from auctions with multidimensional bids, is how to design seller-optimal auctions when bidders' private characteristics differ in many dimensions. This paper partially solves the problem in an auction setting with characteristics stochastically independent across bidders. The solution applies to the multidimensional versions of incentive contracts (Laffont and Tirole (1987) and Che (1993)) and nonlinear pricing (Armstrong (1996)). First, the paper proves that the multidimensionality requires that an optimal auction exclude a positive measure of bidders. Consequently, a standard auction without a reserve price or entrance fee is not optimal. Second, the paper obtains an explicit formula for optimal mechanisms, adopting the assumption of multiplicative separability from Armstrong (1996). Our optimal mechanism is almost equivalent to a Vickrey auction with a reserve price, except that the bids are ranked by an optimal scoring rule, which assigns scores to the multidimensional bids. This ``scoring-rule auction'' is optimal among all mechanisms if incentive compatibility constraints are non-binding (guaranteed by a hazard-rate assumption), and it is optimal among a smaller class of mechanisms if the constraints are binding. Our solution implies that an optimizing seller would induce downward distortion of a bid's nonmonetary provisions from the first-best configuration. Applied to multidimensional nonlinear pricing, our solution yields an explicit optimal pricing function.

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Paper provided by Econometric Society in its series Econometric Society World Congress 2000 Contributed Papers with number 0296.

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Date of creation: 01 Aug 2000
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Handle: RePEc:ecm:wc2000:0296

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  1. Yeon-Koo Che, 1993. "Design Competition through Multidimensional Auctions," RAND Journal of Economics, The RAND Corporation, vol. 24(4), pages 668-680, Winter.
  2. Jean-Jaques Laffont & Jean Tirole, 1985. "Auctioning Incentive Contracts," Working papers, Massachusetts Institute of Technology (MIT), Department of Economics 403, Massachusetts Institute of Technology (MIT), Department of Economics.
  3. Jehiel, Phillipe & Moldovanu, Benny & Stacchetti, E., 1997. "Multidimensional Mechanism Design for Auctions with Externalities," Sonderforschungsbereich 504 Publications, Sonderforschungsbereich 504, Universität Mannheim;Sonderforschungsbereich 504, University of Mannheim 97-04, Sonderforschungsbereich 504, Universität Mannheim;Sonderforschungsbereich 504, University of Mannheim.
  4. Steven A. Matthews, 1995. "A Technical Primer on Auction Theory I: Independent Private Values," Discussion Papers, Northwestern University, Center for Mathematical Studies in Economics and Management Science 1096, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  5. Roger B. Myerson, 1988. "Mechanism Design," Discussion Papers, Northwestern University, Center for Mathematical Studies in Economics and Management Science 796, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  6. Jean-Charles Rochet & Philippe Chone, 1998. "Ironing, Sweeping, and Multidimensional Screening," Econometrica, Econometric Society, Econometric Society, vol. 66(4), pages 783-826, July.
  7. David Zilberman, 1996. "The Economics of a Public Fund for Environmental Amenities: A Study of CRP Contracts," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, Agricultural and Applied Economics Association, vol. 78(4), pages 961-971.
  8. Rochet, J. C., 1985. "The taxation principle and multi-time Hamilton-Jacobi equations," Journal of Mathematical Economics, Elsevier, vol. 14(2), pages 113-128, April.
  9. Chao, Hung-Po & Wilson, Robert, 2002. "Multi-dimensional Procurement Auctions for Power Reserves: Robust Incentive-Compatible Scoring and Settlement Rules," Journal of Regulatory Economics, Springer, Springer, vol. 22(2), pages 161-83, September.
  10. Armstrong, Mark, 1996. "Multiproduct Nonlinear Pricing," Econometrica, Econometric Society, Econometric Society, vol. 64(1), pages 51-75, January.
  11. McAfee, R. Preston & McMillan, John, 1988. "Multidimensional incentive compatibility and mechanism design," Journal of Economic Theory, Elsevier, Elsevier, vol. 46(2), pages 335-354, December.
  12. Roger B. Myerson, 1978. "Optimal Auction Design," Discussion Papers, Northwestern University, Center for Mathematical Studies in Economics and Management Science 362, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  13. Babcock, Bruce A. & Lakshminarayan, P. G. & Wu, J. & Zilberman, David, 1997. "Targeting Tools for the Purchase of Environmental Amenities," Staff General Research Papers 5220, Iowa State University, Department of Economics.
  14. Babcock, Bruce A. & Lakshminarayan, P. G. & Wu, JunJie & Zilberman, David, 1996. "Economics of a Public Fund for Environmental Amenities (The)," Staff General Research Papers 1065, Iowa State University, Department of Economics.
  15. Armstrong, Mark, 2000. "Optimal Multi-object Auctions," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 67(3), pages 455-81, July.
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Cited by:
  1. Lamping, Jennifer, 2007. "The Value of Information in Auctions with Default Risk," MPRA Paper 24375, University Library of Munich, Germany.
  2. Lamping, Jennifer, 2008. "The Value of Commitment in Auctions with Matching," MPRA Paper 24373, University Library of Munich, Germany.
  3. Lamping, Jennifer, 2008. "Ignorance Is Bliss: Matching in Auctions with an Uninformed Seller," MPRA Paper 24374, University Library of Munich, Germany.
  4. Leonardo Rezende, 2009. "Biased procurement auctions," Economic Theory, Springer, Springer, vol. 38(1), pages 169-185, January.

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