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.
Bergemann, Dirk; Shen, Ji; Xu, Yun; and Yeh, Edmund M., "Multi-dimensional Mechanism Design with Limited Information" (2012). Cowles Foundation Discussion Papers. 2219.