Nonlinear Pricing with Finite Information

This paper considers a seller’s problem in choosing a finite number of combinations of products to offer to buyers, and the loss associated with constraining the seller to offer a finite menu of product combinations rather than allowing buyers to flexibly choose quantities of each available product.

Abstract and Citation

Dirk Bergemann, Edmund Yeh, and Jinkun Zhang. 2021. Games and Economic Behavior: 130, 62-84.

Abstract

We analyze nonlinear pricing with finite information. We consider 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. The limited menu reflects a finite communication capacity between the buyer and seller.
We identify necessary conditions that the optimal finite menu must satisfy, for either the socially efficient or the revenue-maximizing mechanism. These conditions require that information be bundled, or "quantized," optimally. We introduce vector quantization and establish that the losses due to finite menus converge to zero at a rate of 1/n2/d_ In the canonical model with one-dimensional products and preferences, this establishes that the loss resulting from using the n-item menu converges to zero at a rate proportional to 1 /n2 .

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