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.

翻译：一位耐心的卖家意图将商品出售给一位不耐烦的买家（即其效用随时间折现）。该买家将在市场中停留一段时间 $T$，其私人价值从公开已知的分布中抽取。对于卖家而言，最优收益的定价曲线（即一系列（价格, 时间）对）是什么？随机化是否有助于此？最优收益的定价曲线能否在多项式时间内计算得出？我们在本文中回答这些问题。我们给出了一种高效算法，用于计算最优收益定价曲线。我们证明，在每时刻设定价格并让买家选择其效用最大化的购买时机的定价曲线，在更广泛的顺序抽奖机制中是最优收益的。也就是说，允许卖家在每时刻发布抽奖菜单的机制无法获得比定价曲线更高的收益。我们还证明，允许抽奖菜单自适应设定的更广泛机制，可以获得严格高于定价曲线的收益，且收益差距可与买家价值分布的支持集大小相当。