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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
    as

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    File URL: https://cowles.yale.edu/sites/default/files/files/pub/d18/d1859.pdf
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    References listed on IDEAS

    as
    1. Mussa, Michael & Rosen, Sherwin, 1978. "Monopoly and product quality," Journal of Economic Theory, Elsevier, vol. 18(2), pages 301-317, August.
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    Cited by:

    1. Alex Smolin, 2023. "Disclosure and pricing of attributes," RAND Journal of Economics, RAND Corporation, vol. 54(4), pages 570-597, December.
    2. Bergemann, Dirk & Yeh, Edmund & Zhang, Jinkun, 2021. "Nonlinear pricing with finite information," Games and Economic Behavior, Elsevier, vol. 130(C), pages 62-84.
    3. Ran Eilat & Kfir Eliaz Eliaz & Xiaosheng Mu, 2021. "Bayesian Privacy," Working Papers 2021-65, Princeton University. Economics Department..
    4. 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.

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    More about this item

    Keywords

    Mechanism design; Multi-dimensional private information; Limited information; Nonlinear pricing; Quantization; Information theory;
    All these keywords.

    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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