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Multi-dimensional Mechanism Design with Limited Information

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  • Dirk Bergemann

    ()
    (Cowles Foundation, Yale University)

  • Ji Shen

    (Dept. of Finance, London School of Economics)

  • Yun Xu

    (Dept. of Electrical Engineering, Yale University)

  • Edmund M. Yeh

    (Dept. of Computer Science and Electrical Engineering, Northeastern University)

Abstract

We analyze a nonlinear pricing model with limited information. Each buyer can purchase a large variety, d, of goods. His preference for each good is represented by a scalar and his preference over d goods is represented by a d-dimensional vector. The type space of each buyer is given by a compact subset of R_d^+ with a continuum of possible types. By contrast, the seller is limited to offer a finite number M of d-dimensional choices. We provide necessary conditions that the optimal finite menu of the social welfare maximizing problem has to satisfy. We establish an underlying connection to the theory of quantization and provide an estimate of the welfare loss resulting from the usage of the d-dimensional M-class menu. We show that the welfare loss converges to zero at a rate proportional to d/M^{2/d}. We show that in higher dimensions, a significant reduction in the welfare loss arises from an optimal partition of the d-dimensional type space that takes advantage of the correlation among the d parameters.

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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1859.

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Length: 22 pages
Date of creation: Apr 2012
Date of revision:
Publication status: Published in 13th ACM Conference on Electronic Commerce (EC '12)
Handle: RePEc:cwl:cwldpp:1859

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Postal: Yale University, Box 208281, New Haven, CT 06520-8281 USA
Phone: (203) 432-3702
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Web page: http://cowles.econ.yale.edu/
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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

Related research

Keywords: Mechanism design; Multi-dimensional private information; Limited information; Nonlinear pricing; Quantization; Information theory;

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References

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  1. Jean-Charles Rochet & Philippe Chone, 1998. "Ironing, Sweeping, and Multidimensional Screening," Econometrica, Econometric Society, vol. 66(4), pages 783-826, July.
  2. Bergemann, Dirk & Pesendorfer, Martin, 2007. "Information structures in optimal auctions," Journal of Economic Theory, Elsevier, vol. 137(1), pages 580-609, November.
  3. Dirk Bergemann & Ji Shen & Yun Xu & Edmund M. Yeh, 2010. "Mechanism Design with Limited Information: The Case of Nonlinear Pricing," Cowles Foundation Discussion Papers 1775, Cowles Foundation for Research in Economics, Yale University.
  4. Heidrun Hoppe & Benny Moldovanu & Emre Ozdenoren, 2011. "Coarse matching with incomplete information," Economic Theory, Springer, vol. 47(1), pages 75-104, May.
  5. Armstrong, Mark, 1996. "Multiproduct Nonlinear Pricing," Econometrica, Econometric Society, vol. 64(1), pages 51-75, January.
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