Extreme-Value Theorems for Optimal Multidimensional Pricing

We provide a near-optimal, computationally efficient algorithm for the unit-demand pricing problem, where a seller wants to price n items to optimize revenue against a unit-demand buyer whose values for the items are independently drawn from known distributions. For any chosen accuracy > 0 and item values bounded in [0, 1], our algorithm achieves revenue that is optimal up to an additive error of at most , in polynomial time. For values sampled from Monotone Hazard Rate (MHR) distributions, we achieve a (1 − )-fraction of the optimal revenue in poly-nomial time, while for values sampled from regular distributions the same revenue guarantees are achieved in quasi-polynomial time. Our algorithm for bounded distributions applies probabilistic techniques to understand the statistical properties of revenue distributions, obtaining a reduction in the search space of the algorithm through dynamic programming. Adapting this approach to MHR and regular distri-butions requires the proof of novel extreme value theorems for such distributions. As a byproduct, our techniques establish structural properties of approximately-optimal and near-optimal solutions. We show that, when the buyer’s values are independently distributed according to MHR distributions, pricing all items at the same price achieves a constant fraction of the optimal revenue. Moreover, for all > 0, at most g(1/) distinct prices suffice to obtain a (1 − )-fraction of the optimal revenue, where g(1/) is a quadratic function of 1/ that does not depend on the number of items. Similarly, for all > 0 and n> 0, at most g(1/ · log n) distinct prices suffice if the values are independently distributed according to regular distributions, where g(·) is a polynomial function. Finally, if the values are i.i.d. from some MHR distribution, we show that, as long as the number of items is a sufficiently large function of 1/, a single price suffices to achieve a (1 − )-fraction of the optimal revenue