  We consider the Item Pricing problem for revenue maximization in the limited
supply setting, where a single seller with $n$ items caters to $m$ buyers with
unknown subadditive valuation functions who arrive in a sequence. The seller
sets the prices on individual items. Each buyer buys a subset of yet unsold
items that maximizes her utility. Our goal is to design pricing strategies that
guarantee an expected revenue that is within a small factor $\alpha$ of the
maximum possible social welfare -- an upper bound on the maximum revenue that
can be generated. Most earlier work has focused on the unlimited supply
setting, where selling items to some buyer does not affect their availability
to the future buyers. Balcan et. al. (EC 2008) studied the limited supply
setting, giving a randomized strategy that assigns a single price to all items
(uniform strategy), and never changes it (static strategy), that gives an
$2^{O(\sqrt{\log n \log \log n})}$-approximation, and moreover, no static
uniform pricing strategy can give better than $2^{\Omega(\log^{1/4} n)}$-
approximation. We improve this lower bound to $2^{\Omega(sqrt{\log n})}$.
  We consider dynamic uniform strategies, which can change the price upon the
arrival of each buyer but the price on all unsold items is the same at all
times, and static non-uniform strategies, which can assign different prices to
different items but can never change it after setting it initially. We design
such pricing strategies that give a poly-logarithmic approximation to maximum
revenue. Thus in the limited supply setting, our results highlight a strong
separation between the power of dynamic and non-uniform pricing versus static
uniform pricing. To our knowledge, this is the first non-trivial analysis of
dynamic and non-uniform pricing schemes for revenue maximization.
