Approximate Revenue Maximization with Multiple Items

Maximizing the revenue from selling _more than one_ good (or item) to a single buyer is a notoriously difficult problem, in stark contrast to the one-good case. For two goods, we show that simple "one-dimensional" mechanisms, such as selling the goods separately, _guarantee_ at least 73% of the optimal revenue when the valuations of the two goods are independent and identically distributed, and at least $50\%$ when they are independent. For the case of $k>2$ independent goods, we show that selling them separately guarantees at least a $c/\log^2 k$ fraction of the optimal revenue; and, for independent and identically distributed goods, we show that selling them as one bundle guarantees at least a $c/\log k$ fraction of the optimal revenue. Additional results compare the revenues from the two simple mechanisms of selling the goods separately and bundled, identify situations where bundling is optimal, and extend the analysis to multiple buyers.Comment: Presented in ACM EC conference, 201

Sampling and Representation Complexity of Revenue Maximization

We consider (approximate) revenue maximization in auctions where the distribution on input valuations is given via "black box" access to samples from the distribution. We observe that the number of samples required -- the sample complexity -- is tightly related to the representation complexity of an approximately revenue-maximizing auction. Our main results are upper bounds and an exponential lower bound on these complexities

Bounding the Optimal Revenue of Selling Multiple Goods

Using duality theory techniques we derive simple, closed-form formulas for bounding the optimal revenue of a monopolist selling many heterogeneous goods, in the case where the buyer's valuations for the items come i.i.d. from a uniform distribution and in the case where they follow independent (but not necessarily identical) exponential distributions. We apply this in order to get in both these settings specific performance guarantees, as functions of the number of items $m$, for the simple deterministic selling mechanisms studied by Hart and Nisan [EC 2012], namely the one that sells the items separately and the one that offers them all in a single bundle. We also propose and study the performance of a natural randomized mechanism for exponential valuations, called Proportional. As an interesting corollary, for the special case where the exponential distributions are also identical, we can derive that offering the goods in a single full bundle is the optimal selling mechanism for any number of items. To our knowledge, this is the first result of its kind: finding a revenue-maximizing auction in an additive setting with arbitrarily many goods