Static pricing for multi-unit prophet inequalities

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 \emph{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 pricing scheme achieves a competitive ratio that increases gradually with the supply and approaches to $1$ at the optimal rate. Astonishingly, for $k<20$, it even outperforms the state-of-the-art adaptive pricing for the small-$k$ regime.