Optimal Pricing Schemes for an Impatient Buyer

TL;DR

This paper determines the revenue-maximizing pricing strategy over time for a seller with a impatient buyer, providing an efficient algorithm and comparing the optimality of simple pricing curves versus more complex lottery mechanisms.

Contribution

It introduces an efficient algorithm for computing the optimal pricing curve and compares its revenue with broader classes of mechanisms, establishing the limits of pricing curves.

Findings

Pricing curves are revenue-optimal among simple mechanisms.

Adaptive lottery menus can outperform pricing curves significantly.

The revenue gap can reach the support size of the buyer's value distribution.

Abstract

A patient seller aims to sell a good to an impatient buyer (i.e., one who discounts utility over time). The buyer will remain in the market for a period of time $T$, and her private value is drawn from a publicly known distribution. What is the revenue-optimal pricing-curve (sequence of (price, time) pairs) for the seller? Is randomization of help here? Is the revenue-optimal pricing curve computable in polynomial time? We answer these questions in this paper. We give an efficient algorithm for computing the revenue-optimal pricing curve. We show that pricing curves, that post a price at each point of time and let the buyer pick her utility maximizing time to buy, are revenue-optimal among a much broader class of sequential lottery mechanisms. I.e., mechanisms that allow the seller to post a menu of lotteries at each point of time cannot get any higher revenue than pricing curves. We also show that the even broader class of mechanisms that allow the menu of lotteries to be adaptively set, can earn strictly higher revenue than that of pricing curves, and the revenue gap can be as big as the support size of the buyer's value distribution.