We study a pricing problem where a seller has $k$ identical copies of a product, buyers arrive sequentially, and the seller prices the items aiming to maximize social welfare. When $k=1$, this is the so called "prophet inequality" problem for which there is a simple pricing scheme achieving a competitive ratio of $1/2$. On the other end of the spectrum, as $k$ goes to infinity, the asymptotic performance of both static and adaptive pricing is well understood. We provide a static pricing scheme for the small-supply regime: where $k$ is small but larger than $1$. Prior to our work, the best competitive ratio known for this setting was the $1/2$ that follows from the single-unit prophet inequality. Our pricing scheme is easy to describe as well as practical -- it is anonymous, non-adaptive, and order-oblivious. We pick a single price that equalizes the expected fraction of items sold and the probability that the supply does not sell out before all customers are served; this price is then offered to each customer while supply lasts. This extends an approach introduced by Samuel-Cahn for the case of $k=1$. This pricing scheme achieves a competitive ratio that increases gradually with the supply. Subsequent work by Jiang, Ma, and Zhang shows that our pricing scheme is the optimal static pricing for every value of $k$.

翻译：我们研究一个定价问题：卖家持有某产品的$k$份相同副本，买家按顺序到达，卖家通过定价来最大化社会福利。当$k=1$时，这就是所谓的“先知不等式”问题，存在一个简单定价方案，其竞争比可达$1/2$。在另一个极端，当$k$趋于无穷大时，静态定价与自适应定价的渐近性能已得到充分理解。我们为小供给情形（即$k$较小但大于$1$）提出了一种静态定价方案。在此工作之前，该情形下已知的最佳竞争比是源于单单位先知不等式的$1/2$。我们的定价方案易于描述且实用——它是匿名的、非自适应的且与订单顺序无关。我们选择一个单一价格，使得预期售出的物品比例与供应在服务完所有顾客前不会售罄的概率相等；随后，在供应持续期间，每位顾客都会收到这一价格。这扩展了Samuel-Cahn针对$k=1$情形提出的方法。该定价方案实现的竞争比随着供给量增加而逐步提高。后续Jiang、Ma和Zhang的研究表明，对于每个$k$值，我们的定价方案均是最优的静态定价。