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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.