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:我们研究一个价格问题,即卖主的产品的相同副本为K美元,买主按顺序到达,卖主的价款是旨在最大限度地提高社会福利的物品。当美元=1美元时,这就是所谓的“预言不平等”问题,对此,有一个简单的定价计划,其竞争性比率为1/2美元。在另一端,当美元到达无限时,静态和适应性定价的零用性表现是完全可以理解的。我们为小供应制度提供了一个静态定价计划:美元小,但大于1美元。在我们工作之前,这一设定的最佳竞争比率为1/2美元,它源自单一单位先知不平等。我们的定价计划既简单又实用 -- -- 这是匿名的,非适应性的,而且订单模糊。我们选择了单一种价格,它等于所售物品的预期部分,而供应在满足所有客户之前可能没有售出;然后向每个客户提供价格,最后供应者都得到这种价格。这扩大了塞缪尔米洛-卡公司提出的价格比,这个价格方法是单一单位先知不平等的0.2美元。我们的定价计划既容易又实际描述 -- -- -- 以最高价格方式来取得最佳价格。