Nonlinear Pricing with Finite Information

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.