  We consider the problem of dynamic pricing with limited supply. A seller has
$k$ identical items for sale and is facing $n$ potential buyers ("agents") that
are arriving sequentially. Each agent is interested in buying one item. Each
agent's value for an item is an IID sample from some fixed distribution with
support $[0,1]$. The seller offers a take-it-or-leave-it price to each arriving
agent (possibly different for different agents), and aims to maximize his
expected revenue.
  We focus on "prior-independent" mechanisms -- ones that do not use any
information about the distribution. They are desirable because knowing the
distribution is unrealistic in many practical scenarios. We study how the
revenue of such mechanisms compares to the revenue of the optimal offline
mechanism that knows the distribution ("offline benchmark").
  We present a prior-independent dynamic pricing mechanism whose revenue is at
most $O((k \log n)^{2/3})$ less than the offline benchmark, for every
distribution that is regular. In fact, this guarantee holds without *any*
assumptions if the benchmark is relaxed to fixed-price mechanisms. Further, we
prove a matching lower bound. The performance guarantee for the same mechanism
can be improved to $O(\sqrt{k} \log n)$, with a distribution-dependent
constant, if $k/n$ is sufficiently small. We show that, in the worst case over
all demand distributions, this is essentially the best rate that can be
obtained with a distribution-specific constant.
  On a technical level, we exploit the connection to multi-armed bandits (MAB).
While dynamic pricing with unlimited supply can easily be seen as an MAB
problem, the intuition behind MAB approaches breaks when applied to the setting
with limited supply. Our high-level conceptual contribution is that even the
limited supply setting can be fruitfully treated as a bandit problem.
