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Multi-dimensional Mechanism Design with Limited Information

Author

Listed:
  • 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.

Suggested Citation

  • Dirk Bergemann & Ji Shen & Yun Xu & Edmund M. Yeh, 2012. "Multi-dimensional Mechanism Design with Limited Information," Cowles Foundation Discussion Papers 1859, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1859
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    File URL: http://cowles.yale.edu/sites/default/files/files/pub/d18/d1859.pdf
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    References listed on IDEAS

    as
    1. Bergemann, Dirk & Pesendorfer, Martin, 2007. "Information structures in optimal auctions," Journal of Economic Theory, Elsevier, vol. 137(1), pages 580-609, November.
    2. 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.
    3. Armstrong, Mark, 1996. "Multiproduct Nonlinear Pricing," Econometrica, Econometric Society, vol. 64(1), pages 51-75, January.
    4. Rothkopf, Michael H. & Harstad, Ronald M., 1994. "On the role of discrete bid levels in oral auctions," European Journal of Operational Research, Elsevier, vol. 74(3), pages 572-581, May.
    5. Jean-Charles Rochet & Philippe Chone, 1998. "Ironing, Sweeping, and Multidimensional Screening," Econometrica, Econometric Society, vol. 66(4), pages 783-826, July.
    6. Heidrun Hoppe & Benny Moldovanu & Emre Ozdenoren, 2011. "Coarse matching with incomplete information," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 47(1), pages 75-104, May.
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    Cited by:

    1. Wong, Adam Chi Leung, 2014. "The choice of the number of varieties: Justifying simple mechanisms," Journal of Mathematical Economics, Elsevier, vol. 54(C), pages 7-21.

    More about this item

    Keywords

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

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • D43 - Microeconomics - - Market Structure, Pricing, and Design - - - Oligopoly and Other Forms of Market Imperfection
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

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