COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1859

Multi-dimensional Mechanism Design with Limited Information

Dirk Bergemann, Ji Shen, Yun Xu, and Edmund Yeh

April 2012

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.

JEL Classification: C72, C73, D43, D83

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