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Optimal Auction in a Multidimensional World

  • Charles Zheng

    (Northwestern University)

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. Jehiel, Phillipe & Moldovanu, Benny & Stacchetti, E., 1997. "Multidimensional Mechanism Design for Auctions with Externalities," Sonderforschungsbereich 504 Publications 97-04, Sonderforschungsbereich 504, Universit├Ąt Mannheim;Sonderforschungsbereich 504, University of Mannheim.
  2. McAfee, R. Preston & McMillan, John, 1988. "Multidimensional incentive compatibility and mechanism design," Journal of Economic Theory, Elsevier, vol. 46(2), pages 335-354, December.
  3. Yeon-Koo Che, 1993. "Design Competition through Multidimensional Auctions," RAND Journal of Economics, The RAND Corporation, vol. 24(4), pages 668-680, Winter.
  4. Roger B. Myerson, 1988. "Mechanism Design," Discussion Papers 796, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  5. 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, vol. 78(4), pages 961-971.
  6. 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.
  7. Jean-Charles Rochet & Philippe Chone, 1998. "Ironing, Sweeping, and Multidimensional Screening," Econometrica, Econometric Society, vol. 66(4), pages 783-826, July.
  8. Roger B. Myerson, 1978. "Optimal Auction Design," Discussion Papers 362, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  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, vol. 22(2), pages 161-83, September.
  10. Armstrong, Mark, 1996. "Multiproduct Nonlinear Pricing," Econometrica, Econometric Society, vol. 64(1), pages 51-75, January.
  11. Armstrong, Mark, 2000. "Optimal Multi-object Auctions," Review of Economic Studies, Wiley Blackwell, vol. 67(3), pages 455-81, July.
  12. Laffont, Jean-Jacques & Tirole, Jean, 1987. "Auctioning Incentive Contracts," Journal of Political Economy, University of Chicago Press, vol. 95(5), pages 921-37, October.
  13. 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.
  14. Steven A. Matthews, 1995. "A Technical Primer on Auction Theory I: Independent Private Values," Discussion Papers 1096, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  15. 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.
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