Dynamic and Non-Uniform Pricing Strategies for Revenue Maximization

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.