Multi-Dimensional Mechanism Design with Limited Information

23 Pages Posted: 19 Apr 2012  

Dirk Bergemann

Yale University - Cowles Foundation - Department of Economics; Yale University - Cowles Foundation

Ji Shen

London School of Economics & Political Science (LSE)

Yun Xu

affiliation not provided to SSRN

Edmund M. Yeh

Northeastern University

Date Written: April 18, 2012

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 Rd 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.

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

JEL Classification: C72, C73, D43, D83

Suggested Citation

Bergemann, Dirk and Shen, Ji and Xu, Yun and Yeh, Edmund M., Multi-Dimensional Mechanism Design with Limited Information (April 18, 2012). Cowles Foundation Discussion Paper No. 1859. Available at SSRN: https://ssrn.com/abstract=2041968 or http://dx.doi.org/10.2139/ssrn.2041968

Dirk Bergemann (Contact Author)

Yale University - Cowles Foundation - Department of Economics ( email )

28 Hillhouse Ave
New Haven, CT 06520-8268
United States
203-432-3592 (Phone)
203-432-2128 (Fax)

HOME PAGE: http://www.econ.yale.edu/~dirk/

Yale University - Cowles Foundation

Box 208281
New Haven, CT 06520-8281
United States

Ji Shen

London School of Economics & Political Science (LSE) ( email )

Houghton Street
London, WC2A 2AE
United Kingdom

Yun Xu

affiliation not provided to SSRN ( email )

Edmund M. Yeh

Northeastern University ( email )

Boston, MA 02115
United States

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