Multidimensional dynamic pricing forwelfare maximization

We study the problem of a seller dynamically pricing d distinct types of indivisible goods, when faced with the online arrival of unit-demand buyers drawn independently from an unknown distribution. .e goods are not in limited supply, but can only be produced at a limited rate and are costly to produce. .e seller observes only the bundle of goods purchased at each day, but nothing else about the buyer's valuation function. Our main result is a dynamic pricing algorithm for optimizing welfare (including the seller's cost of production) that runs in time and a number of rounds that are polynomial in d and the approximation parameter. We are able to do this despite the fact that (i) the price-response function is not continuous, and even its fractional relaxation is a non-concave function of the prices, and (ii) the welfare is not observable to the seller. We derive this result as an application of a general technique for optimizing welfare over divisible goods, which is of independent interest. When buyers have strongly concave, Holder continuous valuation functions over d divisible goods, we give a general polynomial time dynamic pricing technique. We are able to apply this technique to the se.ing of unit demand buyers despite the fact that in that se.ing the goods are not divisible, and the natural fractional relaxation of a unit demand valuation is not strongly concave. In order to apply our general technique, we introduce a novel price randomization procedure which has the e.ect of implicitly inducing buyers to "regularize" their valuations with a strongly concave function. Finally, we also extend our results to a limited-supply se.ing in which the number of copies of each good cannot be replenished.