Dynamic and Non-uniform Pricing Strategies for Revenue Maximization

We study the Item Pricing problem for revenue maximization in the limited supply setting, where a single seller with $n$ distinct 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 multiplicative factor of the optimal social welfare -- an upper bound on the maximum revenue that can be generated by any pricing mechanism. Most earlier work has focused on the unlimited supply setting, where selling an item to a buyer does not affect the availability of the item to the future buyers. Recently, Balcan et. al. (EC 2008) studied the limited supply setting, giving a randomized pricing strategy that achieves a $2^{O(\sqrt{\log n \log\log n})}$-approximation; their strategy assigns a single price to all items (uniform pricing), and never changes it (static pricing). They also showed that no pricing strategy that is both static and uniform can give better than $2^{\Omega(\log^{1/4} n)}$-approximation. Our first result is a strengthening of the lower bound on approximation achievable by static uniform pricing to $2^{\Omega(\sqrt{\log n})}$. We then design {\em dynamic uniform pricing strategies} (all items are identically priced but item prices can change over time), that achieves $O(\log^2 n)$-approximation, and also show a lower bound of$\Omega\left(\left(\log n/\log\log n\right)^2\right)$ for this class of strategies. Our strategies are simple to implement, and in particular, one strategy is to smoothly decrease the price over time. We also design a {\em static non-uniform pricing strategy} (different items can have different prices but prices do not change over time), that give poly-logarithmic approximation in a more restricted setting with few buyers. Thus in the limited supply setting, our results highlight a strong separation between the power of dynamic and non-uniform pricing strategies versus static uniform pricing strategy. To our knowledge, this is the first non-trivial analysis of dynamic and non-uniform pricing schemes for revenue maximization in a setting with multiple distinct items.