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: namely, 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.

中文翻译：

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不耐烦买家的最佳定价方案

耐心的卖家旨在将商品出售给不耐烦的买家（即，随着时间的推移对效用进行折扣的买家）。买家将在市场上停留 $T$ 的时间，她的私人价值来自公开的分布。卖方的收入最优定价曲线（（价格，时间）对的顺序）是什么？这里是随机化的帮助吗？收入最优定价曲线是否可以在多项式时间内计算？我们在本文中回答了这些问题。我们给出了一种计算收入最优定价曲线的有效算法。我们展示了定价曲线，即在每个时间点发布价格并让买家选择其效用最大化购买时间，在更广泛的顺序彩票机制中是收入最佳的：即，允许卖家在每个时间点发布彩票菜单的机制无法获得比定价曲线更高的收入。我们还表明，允许自适应设置彩票菜单的更广泛类别的机制可以赚取比定价曲线更高的收入，并且收入差距可以与买方价值分配的支持规模一样大。