Sequential Item Pricing for Unlimited Supply

We study prior-free revenue maximization for a seller with unlimited supply of n item types facing m myopic buyers present for k < log n days. We show that a certain randomized schedule of posted prices has an approximation factor of O(log m+log n/k). This algorithm relies on buyer valuations having hereditary maximizers, a novel natural property satisfied for example by gross substitutes valuations. We obtain a matching lower bound with multi-unit valuations. In light of existing results [2], k days can thus improve the approximation by a Θ(k) factor. We also provide a posted price schedule with the same factor for positive affine allocative externalities, despite an increase in the optimal revenue.