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Optimal Auction in a Multidimensional World

  • Charles Z. Zheng

A long-standing unsolved problem, often arising from auctions with multidimensional bids, is how to design seller-optimal auctions when bidders' private characteristics ("types") differ in many dimensions. This paper solves the problem, assuming bidder-types stochastically independent across bidders. First, it proves that in any optimal auction, with positive probability, the object is not sold. Thus, a standard auction without a reserve price is not optimal. Second, and more importantly, the paper obtains an explicit formula for optimal auctions in a class of environments. The optimal mechanism is almost equivalent to a Vickrey auction with reserve price, except that the bids are ranked by an optimal scoring rule, which assigns scores to the multidimensional bids. When the hazard rate of a statistic of bidder-types is monotone, this auction is optimal among all "scoring mechanisms," where a winner chooses a multidimensional payment bundle subject to a type-specific rule. Our optimal auction implies that an optimizing seller would not evaluate bid by her own preferences; instead, she would induce downward distortion of nonmonetary provisions from the first-best configuration. Applied to multidimensional nonlinear pricing, our design of optimal auction yields an explicit optimal pricing function.

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Paper provided by Northwestern University, Center for Mathematical Studies in Economics and Management Science in its series Discussion Papers with number 1282.

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Date of creation: Jan 2000
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Handle: RePEc:nwu:cmsems:1282
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  1. 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.
  2. repec:att:wimass:9123 is not listed on IDEAS
  3. 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.
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  5. 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.
  6. Armstrong, Mark, 1996. "Multiproduct Nonlinear Pricing," Econometrica, Econometric Society, vol. 64(1), pages 51-75, January.
  7. McAfee, R. Preston & McMillan, John, 1988. "Multidimensional incentive compatibility and mechanism design," Journal of Economic Theory, Elsevier, vol. 46(2), pages 335-354, December.
  8. Laffont, Jean-Jacques & Tirole, Jean, 1987. "Auctioning Incentive Contracts," Journal of Political Economy, University of Chicago Press, vol. 95(5), pages 921-37, October.
  9. Jehiel, Philippe & Moldovanu, Benny & Stacchetti, Ennio, 1999. "Multidimensional Mechanism Design for Auctions with Externalities," Journal of Economic Theory, Elsevier, vol. 85(2), pages 258-293, April.
  10. 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.
  11. 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.
  12. Yeon-Koo Che, 1993. "Design Competition through Multidimensional Auctions," RAND Journal of Economics, The RAND Corporation, vol. 24(4), pages 668-680, Winter.
  13. Armstrong, Mark, 2000. "Optimal Multi-object Auctions," Review of Economic Studies, Wiley Blackwell, vol. 67(3), pages 455-81, July.
  14. Roger B. Myerson, 1988. "Mechanism Design," Discussion Papers 796, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  15. Roger B. Myerson, 1978. "Optimal Auction Design," Discussion Papers 362, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  16. 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.
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