The menu-size complexity of revenue approximation

We consider a monopolist that is selling n items to a single additive buyer, where the buyer's values for the items are drawn according to independent distributions F₁, F₂,⋯, F_n that possibly have unbounded support. It is well known that - unlike in the single item case - the revenue-optimal auction (a pricing scheme) may be complex, sometimes requiring a continuum of menu entries. It is also known that simple auctions with a finite bounded number of menu entries can extract a constant fraction of the optimal revenue. Nonetheless, the question of the possibility of extracting an arbitrarily high fraction of the optimal revenue via a finite menu size remained open. In this paper, we give an affirmative answer to this open question, showing that for every n and for every ϵ > 0, there exists a complexity bound C = C(n, ϵ) such that auctions of menu size at most C suffice for obtaining a (1 - ϵ) fraction of the optimal revenue from any F₁,⋯, F_n. We prove upper and lower bounds on the revenue approximation complexity C(n, ϵ), as well as on the deterministic communication complexity required to run an auction that achieves such an approximation.