Extreme value theorems for optimal multidimensional pricing

Abstract

We provide near-optimal, polynomial-time algorithms for pricing n items to optimize revenue against a unit-demand buyer whose values are independent from known distributions. For any chosen ϵtextgreater0 and values in [0,1], our algorithm’s revenue is optimal up to an additive ϵ. For values sampled from monotone hazard rate (MHR) or regular distributions, we achieve a $(1−\varepsilon)$-fraction of the optimal revenue in polynomial time and quasi-polynomial time, respectively. Our algorithm for bounded distributions applies probabilistic techniques to understand the statistical properties of revenue distributions, obtaining a reduction in the algorithm’s search space via dynamic programming. Adapting this approach to MHR and regular distributions requires the proof of novel extreme-value theorems for such distributions. As a byproduct, we show that, for all n, a constant or a polylogarithmic (in n) number of distinct prices suffice for near-optimal revenue for MHR and regular distributions, respectively.