Publication Date: January 2015
We analyze nonlinear pricing with ﬁnite information. A seller oﬀers a menu to a continuum of buyers with a continuum of possible valuations. The menu is limited to oﬀering a ﬁnite number of choices representing a ﬁnite communication capacity between buyer and seller.
We identify necessary conditions that the optimal ﬁnite menu must satisfy, either for the socially eﬀicient or for the revenue-maximizing mechanism. These conditions require that information be bundled, or “quantized” optimally. We show that the loss resulting from using the n-item menu converges to zero at a rate proportional to 1 = n2.
We extend our model to a multi-product environment where each buyer has preferences over a d dimensional variety of goods. The seller is limited to oﬀering a ﬁnite number n of d-dimensional choices. By using repeated scalar quantization, we show that the losses resulting from using the d-dimensional n-class menu converge to zero at a rate proportional to d = n2/d. We introduce vector quantization and establish that the losses due to ﬁnite menus are signiﬁcantly reduced by oﬀering optimally chosen bundles.
Mechanism design, Nonlinear pricing, Multi-Dimension, Multi-product, Private information, Limited information, Quantization, Information theory