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 = n 2 . 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 = n 2 / d . We introduce vector quantization and establish that the losses due to ﬁnite menus are signiﬁcantly reduced by oﬀering optimally chosen bundles.
Bergemann, Dirk; Shen, Ji; Xu, Yun; and Yeh, Edmund M., "Nonlinear Pricing with Finite Information" (2015). Cowles Foundation Discussion Papers. 2398.