The Limits of Optimal Pricing in the Dark

TL;DR

This paper investigates the strategic manipulation of learning-based pricing by buyers in digital markets, characterizing the limits of seller revenue under optimal buyer imitation strategies and revealing counterintuitive effects of advanced pricing schemes.

Contribution

It provides a full characterization of buyer manipulation strategies and their impact on seller revenue, highlighting fundamental limits and counterintuitive outcomes in strategic pricing.

Findings

Seller with concave costs earns nearly zero revenue at equilibrium.

Seller with convex costs achieves revenue equal to the Bregman divergence of costs.

More powerful pricing schemes can harm seller revenue rather than improve it.

Abstract

A ubiquitous learning problem in today's digital market is, during repeated interactions between a seller and a buyer, how a seller can gradually learn optimal pricing decisions based on the buyer's past purchase responses. A fundamental challenge of learning in such a strategic setup is that the buyer will naturally have incentives to manipulate his responses in order to induce more favorable learning outcomes for him. To understand the limits of the seller's learning when facing such a strategic and possibly manipulative buyer, we study a natural yet powerful buyer manipulation strategy. That is, before the pricing game starts, the buyer simply commits to "imitate" a different value function by pretending to always react optimally according to this imitative value function. We fully characterize the optimal imitative value function that the buyer should imitate as well as the resultant seller revenue and buyer surplus under this optimal buyer manipulation. Our characterizations reveal many useful insights about what happens at equilibrium. For example, a seller with concave production cost will obtain essentially 0 revenue at equilibrium whereas the revenue for a seller with convex production cost is the Bregman divergence of her cost function between no production and certain production. Finally, and importantly, we show that a more powerful class of pricing schemes does not necessarily increase, in fact, may be harmful to, the seller's revenue. Our results not only lead to an effective prescriptive way for buyers to manipulate learning algorithms but also shed lights on the limits of what a seller can really achieve when pricing in the dark.