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.
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 Rd+ 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/M2/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.
We analyze the canonical nonlinear pricing model with limited information. A seller offers a menu with a finite number of choices to a continuum of buyers with a continuum of possible valuations. By revealing an underlying connection to quantization theory, we derive the optimal finite menu for the socially efficient and the revenue-maximizing mechanism. In both cases, we provide an estimate of the loss resulting from the usage of a finite n-class menu. We show that the losses converge to zero at a rate proportional to 1/n2 asn becomes large.