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. [4] studied the limited supply setting, giving a randomized pricing strategy that achieves a 2(O(root 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(log1/4 n))-approximation. Our first result is a strengthening of the lower bound on approximation achievable by static uniform pricing to 2(Omega(root log n)). We then design 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(log n/log log n)(2)) 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 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.