Abstract
We consider the _menu size_ of auctions and mechanisms in general as a measure of their complexity, and study how it affects revenue. Our setting has a single revenue-maximizing seller selling two or more heterogeneous goods to a single buyer whose private values for the items are drawn from a (possibly correlated) known distribution, and whose valuation is additive over the goods. We show that the revenue may increase arbitrarily with menu size and that a bounded menu size _cannot_ ensure any positive fraction of the optimal revenue. The menu size turns out also to pin down the revenue properties of deterministic mechanisms: their menu size is at most exponential in the number of goods, and indeed their revenue may be larger than that achievable by the simplest types of mechanisms by a similar factor. Our model is related to a previously studied unit-demand model and our results also answer an open problem in that model.
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